US 2016.0328253A1 (19) United States (12) Patent Application Publication (10) Pub. No.: US 2016/0328253 A1 MAJUMDAR (43) Pub. Date: Nov. 10, 2016
(54) QUANTON REPRESENTATION FOR (52) U.S. Cl. EMULATING QUANTUM-LIKE CPC ...... G06F 9/455 (2013.01); G06F 17/18 COMPUTATION ON CLASSICAL (2013.01); G06N 99/002 (2013.01) PROCESSORS (57) ABSTRACT (71) Applicant: KYNDI, INC., Redwood City, CA (US) The Quanton virtual machine approximates Solutions to (72) Inventor: Arun MAJUMDAR, Alexandria, VA NP-Hard problems in factorial spaces in polynomial time. (US) The data representation and methods emulate quantum com puting on classical hardware but also implement quantum (73) Assignee: KYNDI, INC., Redwood City, CA (US) computing if run on quantum hardware. The Quanton uses permutations indexed by Lehmer codes and permutation (21) Appl. No.: 15/147,751 operators to represent quantum gates and operations. A (22) Filed: May 5, 2016 generating function embeds the indexes into a geometric object for efficient compressed representation. A nonlinear Related U.S. Application Data directional probability distribution is embedded to the mani fold and at the tangent space to each index point is also a (60) Provisional application No. 62/156,955, filed on May linear probability distribution. Simple vector operations on 5, 2015. the distributions correspond to quantum gate operations. The Publication Classification Quanton provides features of quantum computing: Superpo sitioning, quantization and entanglement Surrogates. Popu (51) Int. Cl. lations of Quantons are evolved as local evolving gate G06F 9/455 (2006.01) operations solving problems or as solution candidates in an G06N 99/00 (2006.01) Estimation of Distribution algorithm. The Quanton repre G06F 7/8 (2006.01) sentation and methods are fully parallel on any hardware.
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(3) (3) (3)(E) Patent Application Publication Nov. 10, 2016 Sheet 47 of 47 US 2016/0328253 A1 i US 2016/0328253 A1 Nov. 10, 2016 QUANTON REPRESENTATION FOR data volume or size of the description of the data, its EMULATING QUANTUM-LIKE high-dimensionality, dynamics or intertwined relationships COMPUTATION ON CLASSICAL with other data. Altogether, these issues make it very com PROCESSORS plicated to separate data into distinct classes and analyze. 0008 Training data samples are assumed to exist when CROSS-REFERENCE TO RELATED many times there are no training data for synthesizing new APPLICATIONS Solutions to creative problems (such as synthesis of new 0001. This application is based upon and claims the drugs or materials design). The same underlying distribution benefit of priority to provisional U.S. Application No. in classical approaches draws training samples from the same sets as the problem solution data: however, due to the 62/156.955, filed May 5, 2015, the entire contents of which sparsity and noise in data collection and sometimes-large are incorporated herein by reference. variations of the inputs, the assumptions about the congru BACKGROUND ence between training and solution data cannot be made or when a new model that did not exist before has to be created. 0002 1. Field of the Disclosure 0009. In settings where decisions, based on stratified 0003. The present disclosure relates to a probabilistic data, need to be made (Such as data in ontologies, and polynomial Turing Machine computing model that emulates taxonomies or, other strata) we can expect an effect on the quantum-like computing and performs several practical data response due to the stratum to which data belongs. In a processing functions, and, more particularly, to a system of model framework, a shift due to stratum membership can representation and computing method in both classical significantly influence the estimate of distribution of some probabilistic and quantum computing or quantum emulation outcome given a particular set of covariates and data. on classical computers by reformulating the computation of Membership within a particular stratum can impact the value functions by permutations and embeds these into a prob of the distribution of interest. However, in many cases one ability space, and incorporating methods of topological does not wish to explicitly estimate these stratum-level quantum computing. effects: rather, one seeks to estimate other parameters of 0004 2. Description of the Related Art interest—such as linear coefficients associated with other 0005. The universal model of computing can be referred features that are observed across strata and account for any to as a virtual machine that is traditionally called the Turing non-linearity in the estimates. For a concrete example, Machine, and in accordance with the Church-Turing thesis, consider the case of a classical conditional likelihood model the virtual machine is built on the foundation of computation approach, wherein one conditions on the histogram of the as function evaluation whose archetype is the lambda observations in the stratum. This conditional likelihood is calculus. There are many other ways in which to design a invariant to any stratum-level linear effects, thus removing Turing Machine. Such as, for example, using pi-calculus as them as a contributing factor (in the likelihood) in the model a foundation, or using probability theoretic models to build described herein. One may then proceed to use the model a probabilistic Turing machine or quantum theory to build a with maximum likelihood estimation to recover the remain quantum Turing machine. ing parameters of interest. For example, conditioning on the 0006. Many classes of artificial intelligence techniques histogram of the responses in a case-control study, such as often draw inspiration from philosophy, mathematical, in analyses of clinical psychological trials, a stratum physical, biological and economic sciences: however, it is in amounts to considering all permutations of the response the novelty of combining parts of these different disciplines vector. This results in a Summation over a combinatorially, that while may have been common knowledge to those and thus factorially, growing number of terms. Computation skilled in these arts, are non-obvious when integrated becomes infeasible in this classical approach. together. Therefore, the study of how to make a good choice 0010 Patterns that are hard to find in data, however, must when confronted with conflicting requirements, and which have some related features else one would be dealing with selections of which disciplines (e.g. biologically inspired or pure noise. These patterns are hard to identify because they economically inspired models etc.) is the fundamental prob are often confounded with other patterns; they can appear lem in all data analysis: this is where our techniques apply. distorted by noise but once recognized could be restored and 0007 Choosing good data underlies the problem of how classified despite of the noise. Patterns sometimes need to be to make good decisions: in addition, there are the issues of learned where no pattern models had existed before by handling conflicting data, weak-signals or very high com speculatively hypothesizing a pattern model structure and binatorial complexity. Conflicting data simply means that then testing for the existence of patterns in data against this the data indicate inconsistent logical inference results. Weak Structure. signals are those that are hard to distinguish from noise or 0011. In the late nineteen-nineties, the “Estimation of deceptively strong signals that mask the weak signals. Distribution Algorithms' (EDA) was introduced and goes by Classical approaches, such as frequentist statistical analyses, several other terms in the literature, such as “Probabilistic work with strong signals or simple complexity classes so Model-Building Genetic Algorithms’, or "Iterated Density that when a result is found, then it is guaranteed to be a best Estimation Algorithms. Due to its novel functionality, it has Solution. In the case of weak-signals, or high complexity become a major tool in evolutionary algorithms based on classes, however, there is usually a balance of tradeoffs that probabilistic model learning by evolution, biologically must be achieved because some kind of approximation will inspired computing in spirit similar to genetic algorithms. be used in place of an exact answer. In these cases, the data 0012. EDAs estimate a joint probability distribution asso characteristic is not that it is weak or strong but that it is very ciated with the set of training examples expressing the complex. Complexity arises due to fundamental algorithmic interrelations between the different variables of the problem structure that defines a computational complexity class, or via their probability distributions. Therefore, EDAs are US 2016/0328253 A1 Nov. 10, 2016 ideally suited to iterative probabilistic machine learning Such computed prototypes can be used to reason about, techniques. Sampling the probabilistic model learned in a classify or index very large-size structural data so that previous generation in an EDA estimation cycle breeds a queries can be efficiently answered by only considering new population of Solutions. The algorithm stops iterating properties of those prototypes. The other important applica and returns the best solution found across the generations tion is that a prototype can be used in reconstructing objects when a certain stopping criterion is met, Such as a maximum from only a few partial observations as a form of com number of generations/evaluations, homogeneous popula pressed knowledge or sensing. Prototypes can be used to tion, or lack of improvement in the Solutions. identify general, though hidden, patterns in a set of disparate 0013 Most of the approaches to NP-Hard problems such data items, thus relating the data items in non-obvious ways. as inference in factorially large data spaces, ranking objects A Software data structure for prototype learning and repre in complex data sets, and data registration rely on convex sentation can be used for structure matching, for example, a optimization, integer programming, relaxation and related set of points whose pair wise distances remain invariant classical approximation techniques to reach a solution that is under rigid transformation. inexact but closely representative of the decision solution. In 0017 Many of the prototype learning algorithms use an addition, the standard approach to Solving Such a problem embedding of data into a low dimensional manifold, that assumes a linear objective function. However, often this is produces a low granularity representation, that is often, but just an approximation of the real problem, which may be not always, locally linear, and that, hopefully, captures the non-linear and allowing for non-linear functions results in a salient properties in a way that is likely to be computation much broader expressive power. However, the methods for ally useful to the problem at hand, without generating false combining both linear and non-linear structure in the same negatives and positives. Missing data or inferring that data model have largely been ad-hoc. is missing is a major problem with many of the manifold 0014 Probabilistic inference becomes unwieldy and embedding techniques because the structural composition of complex as correlations between data need to be taken into data is lost during the embedding process—hence, the account as opposed to the assumption of data independence manifold has no way to take missing structure or put back that is usually used, for example, in Bayesian reasoning. missing structure, as part of its operations. Methods to reason within complex associated networks 0.018. The “background description provided herein is often rely on simulated annealing and various approxima for the purpose of generally presenting the context of tions while enforcing the need to process data sequentially the disclosure. Work of the presently named inventors, and under the null hypothesis. Methods such as Markov to the extent it is described in this background section, Random Fields, Markov Logic, Bayesian Belief Networks as well as aspects of the description which may not and other similar structures fall into the models of process otherwise qualify as prior art at the time of filing, are ing just presented. neither expressly or impliedly admitted as prior art 00.15 Machine learning applications often involve learn against the present disclosure. ing deep patterns from data that are inherently directional in SUMMARY nature or that the data are correlated, or stratified or latently seriated and, most often in real world cases, the data is both 0019. The Quanton virtual machine approximates solu seriated, stratified, directional and correlated without any tions to NP-Hard problems in factorial spaces in polynomial a-priori knowledge of the cluster size. Spectral clustering time. The data representation and methods emulate quantum techniques have been used traditionally to generate embed computing on classical hardware but also implement quan dings that constitute a prototype for directional data analy tum computing if run on quantum hardware. The Quanton sis, but can result in different shapes on a hypersphere uses permutations indexed by Lehmer codes and permuta (depending on the original structure) leading to interpreta tion-operators to represent quantum gates and operations. A tion difficulties. Examples of such directional data include generating function embeds the indexes into a geometric text, medical informatics, insurance claims analysis, and object for efficient compressed representation. A nonlinear Some domains of most sciences that include directional directional probability distribution is embedded to the mani vector fields (winds, weather, physical phenomena). Various fold and at the tangent space to each index point is also a probability densities for directional data exist with advan linear probability distribution. Simple vector operations on tages and disadvantages based on either Expectation Maxi the distributions correspond to quantum gate operations. The mization (EM) strategies or Maximum a Posteriori (MAP) Quanton provides features of quantum computing: Superpo inference, for example, as in graph models. The main sitioning, quantization and entanglement Surrogates. Popu difficulty is learning the posterior, which is usually not lations of Quantons are evolved as local evolving gate directly accessible due to incomplete knowledge (or missing operations solving problems or as solution candidates in an data) or the complexity of the problem and hence, approxi Estimation of Distribution algorithm. The Quanton repre mations have to be used. The output of learning is some sort sentation and methods are fully parallel on any hardware. of prototype. 0020. The foregoing paragraphs have been provided by 0016 Learning prototypes from a set of given or way of general introduction, and are not intended to limit the observed objects is a core problem in machine learning with scope of the following claims. The described embodiments, a large number of applications in image understanding, together with further advantages, will be best understood by cognitive vision, pattern recognition, data mining, and bio reference to the following detailed description taken in informatics. The usual approach is to have some data, often conjunction with the accompanying drawings. sparse input from relatively few well understood real world examples, and learn a pattern, which is called the prototype. BRIEF DESCRIPTION OF THE DRAWINGS The prototype minimizes the total difference (when differ 0021. A more complete appreciation of the disclosure and ences are present) between input objects that are recognized. many of the attendant advantages thereof will be readily US 2016/0328253 A1 Nov. 10, 2016 obtained as the same becomes better understood by refer 0048 FIG. 22 illustrates an example of a Bitonic sorting ence to the following detailed description when considered network operation and design; in connection with the accompanying drawings, wherein: 0049 FIG. 23 illustrates a Bitonic sorting network opera 0022 FIG. 1 illustrates an exemplary flowchart of the tion as a polytope design; overview of the system; 0050 FIG. 24 illustrates an example of Braidings acting 0023 FIG. 2 illustrates an exemplary representation of a as a sorting network; permutation sequence at vertices of polytope (Permutrix); 0051 FIG. 25 illustrates an example of a permutation 0024 FIG. 3 illustrates an exemplary Fibonacci lattice matrix and permutation sign inversion matrix: used to generate vertices on the polytope; 0.052 FIG. 26 illustrates an example of a permutations 0025 FIG. 4 illustrates an exemplary geometry of a matrix and permutation pattern matrix; specific probability density distributions on a surface of a 0053 FIG. 27 illustrates a relationship between a point Hypersphere: and a tangent at a point to angular parameter; 0026 FIG. 5 illustrates an exemplary mixture of distri 0054 FIG. 28 illustrates geometries of specific probabil butions on tangent spaces to the Hypersphere; ity density distributions on the sphere (or hypersphere); 0027 FIG. 6 illustrates a specific tangent point and 0055 FIG. 29 illustrates example of embedding permu tangent space on the spherical Fibonacci lattice of the tations onto an Orbitope with a probability density; Quanton; 0056 FIG. 30 illustrates a flowchart for a Quanton data 0028 FIG. 7 illustrates a schema of iterative feedback structure construction for a hypersphere; data processing for estimation of distribution algorithms for 0057 FIG. 31 illustrates a re-Interpretation of vertices of the Quanton; a permutation matrix Zonotope as combinatorial patterns; 0029 FIG. 8 illustrates probability path densities with a 0.058 FIG. 32 illustrates an exemplary enumeration for lookup table for mapping Qubits to classical bits for the the 3-Permutation Orbitope patterns; Quanton; 0059 FIG. 33 illustrates encoding the 3-permutation 0030 FIG. 9 illustrates an example of the topological Orbitope patterns in rank ordered indices using the Lehmer structure of the Quanton and embedding a permutation state index; space as an Orbitope on a sphere; 0060 FIG. 34 illustrates digitizing a signal by pattern 0031 FIG. 10 illustrates an operational model for esti based sampling and showing the numerical encoding: mation of distribution algorithms for the Quanton; 0061 FIG. 35 illustrates patterns that are mapped to the 0032 FIG. 11 illustrates an exemplary calibration opera sphere; tion and resolution for Quanton construction; 0062 FIG. 36 illustrates that a pattern at the center of the 0033 FIG. 12 illustrates exemplary details of Quanton sphere is a mixed state while the Surface is pure; calibration construction processing: 0063 FIG. 37 illustrates an example of patterns mapped 0034 FIG. 13A illustrates an exemplary topological to the Quanton; structure of the space of the 4-Permutation Orbitope; 0064 FIG. 38 illustrates a hierarchical structure of the 0035 FIG. 13B illustrates an exemplary topological Quanton with nested Quantons; structure of the space of the 5-Permutation Orbitope; 0065 FIG. 39 illustrates an exemplary flowchart sche 0036 FIG. 14 illustrates a flowchart describing recursive mata for Quantons in the estimation of distribution algo evolution and estimation; rithm; 0037 FIG. 15 illustrates a polyhedral structure of the 0.066 FIG. 40 illustrates an exemplary hardware design space of a few Braid group and permutation Orbitopes for Quantons in a System on a Chip (SOC); (Zonotopes); 0067 FIG. 41 illustrates an overall master flow chart for 0038 FIG. 16 illustrates a polyhedral structure illustrat Quanton PMBGA consensus output; and ing a Quantized polyhedral probability density distribution; 0068 FIG. 42 illustrates an exemplary illustration of a 0039 FIG. 17 illustrates a polyhedral structure illustrat computer according to one embodiment. ing a projection of a single Quantized polyhedral probability density; DETAILED DESCRIPTION OF THE 0040 FIG. 18A illustrates an example of Quantum Gates EMBODIMENTS (Quantum Circuits); 0069 Performing an approximation to quantum comput 0041 FIG. 18B illustrates an equivalence between per ing by treating permutations as representative of model mutation representations and Quantum circuits; states provides the interpretation that all states are simulta 0042 FIG. 19 illustrates an example of a classical irre neously computed by iteration. Treating distributions as versible full adder circuit and reversible (Quantum) full approximating density functionals, estimating distributions, adder circuit; coupling these distributions to State spaces represented by 0043 FIG. 20A illustrates a flowchart for a permutational permutations, computing based on these distributions, rea model of computing: soning with these distributions over the symmetric group 0044 FIG. 20B illustrates another flowchart for the per and structure learning using the present Quanton model are mutational model of computing: the central ideas as described by embodiments of the present 004.5 FIG.20C illustrates another flowchart for the per disclosure. mutational model of computing: 0070 The search space of solutions in permutation prob 0046 FIG. 20D illustrates another flowchart for the per lems of n items is n factorial. The search space is usually mutational model of computing: denoted as Sn, in reference to the symmetric group of size 0047 FIG. 21 illustrates an example of Bitonic sorting n. In general, permutation problems are known as very hard network polynomial constraints and corresponding imple problems when n goes above a relatively small number and mentation; their computational complexity demonstrated that many of US 2016/0328253 A1 Nov. 10, 2016 the typical permutation problems is NP-hard. In view of for creating the permutations is detailed in the section of the their complexity, computing optimal Solutions is intractable present disclosure corresponding to FIG. 20A. in general. For this reason, invented the Quanton in order to (0077 STEP-3: There is a deep and fundamental concept put in place a data structure designed to work, at worst at both the local and global levels which that of “navigation approximately, and at best in certain special cases, exactly, of the informational geometry space. In the local case of an at the factorial sizes of the search space. individual Quanton this amounts to choosing how to embed 0071. Furthermore, noting that Quantum computing also a regular grid or lattice into a shaped geometry. Details of has a very large space in terms of Solution possibilities, the choices for the lattice and the geometry are given in sections Quanton data structure and methods, using the unique, of the present disclosure in TABLE 1 of the present disclo efficient and computable model described in the present Sure. In the global case, the informational geometry is the disclosure built on the idea of permutation as computation sampling procedure which explores the search space of the (aka permutational quantum computing), the Quanton is population or, put another way, the sampling function walks devised herein to emulate Quantum computing as a virtual in the geometry of the Quanton population to select Solution machine. candidates as the space is explored. 0072 Now, referring to FIG. 1, which provides a Quan (0078 STEP-4: In the local model of the Quanton, a ton Overview, there are two parts to the overall procedure: non-linear directional probability density function according first, there is the local procedure for creating the Quanton for to TABLE 5 of the present disclosure and is assigned to the use in emulating localized (to the Quanton) computational Quanton which results in each permutation, embedded at operations and then there is the global procedure for evolv each lattice point have a transition probability to the next ing the Quanton or a population of Quantons to learn about permutation or, at the user's discretion, the probability can incoming data problems. This is done in order to produce also represent the likelihood of permutation observation. optimal solutions based on the procedure of Estimation of This enables the use of the L2 norm in operations of the Distribution (EDA) algorithms, also known as Probabilistic Quanton. For example, the methods of general Bayesian Model Building Genetic Algorithm (PMBGA). reasoning and probability density redistribution on mani 0073. The Quanton uses embeds permutations in special folds, such as the hypersphere, are known as Bayesian way that allows the permutations to each have a unique filtering and can use distributions such as the Kent or von index (by using a lattice) into a continuous probability space. Mises Fisher distributions and its simpler versions, Kalman The produces a unique encoding for operations that enable filtering to update the distributions until some convergence it to mimic quantum gates. Hence quantum gates are embed or fixed point is reached. ded in a continuous probabilistic vector space in which fast (0079 STEP-5: A linear probability density is also asso vector computations perform the equivalent of complex ciated to the Quanton by associating the tangent space at quantum gate operations, transforming inputs to outputs, or, each lattice point of the manifold, which allows the use of equivalently, computing quantum transitions from state to the L1 norm in operations of the Quanton. Hence the state. Given that all permutations are simultaneously avail Quanton combines both a linear and non-linear component able as indexed on the Quanton, every continuous vector associated to each permutation. The update of the Quanton space operation, therefore, updates all permutations simul proceeds by using the classical and flexible mixture models taneously since it is the probability density distribution that based on the Dirichlet process mixture model of Gaussian is performing the update. In this sense the Quanton repre distributions in distinct tangent spaces to handle an unknown sents a Superposition of all potential Solutions. The Quanton number of components and that can extend readily to represents quantized discrete structure because of its lattice high-dimensional data in order to perform the Quanton and entanglements are represented by correlations between probability updates in its linear space. variables that emerge as a result of an evolutionary process 0080 STEP-6: The most unique part of the present dis that Surfaces the entangled States as Solution sets to the input closure is that, having set up the Quanton using the Landau query state (i.e. the data to be learned or solved). numberings to generate permutations, a permutation gate 0074 STEP-1: The main procedure for computation with operator, directly equivalent to Quantum Gates or Quantum the Quanton is initialization and setup, whether an indi Circuits is associated to the permutations so that permutation vidual Quanton is being instantiated, or a population of relationships on the Quanton correspond to the operation of Quantons is instantiated, item 1. Either the problem size is a Quantum Operation. Further details of this mapping are known or unknown, item 1. If the problem size is known, or provided in the disclosure corresponding to at least FIG. 19. estimated by an external source, then this number is used to I0081) STEP-7: The Quanton can be directly used as a set the size of the Quanton. The problem size often correlates Quantum Emulator (i.e. a Quantum Virtual Machine) or, it in some with problem complexity and if this is know, then can be then used in the Quanton Population where the the population size as the population of Quantons can be set. Quantum Gate Operators replace the conventional notion of If this is unknown, then a random population size is set. crossover and mutation operations on the usual bit-strings of 0075. The section of the present disclosure corresponding classical computation: in this sense, the Quanton provides an to at least FIG. 2 and TABLE 4 presents further detail on the evolutionary path to Quantum Circuits that produce problem design of the Quanton for instantiation. solving quantum circuits using PMBGA as further detailed 0076 STEP-2: As part of the design of the local and in the section corresponding to FIG. 39 of the disclosure. global structures, the next procedure allocates local structure This is significant because the operations of the present and global distributional population structure. The local disclosure all occur with very high efficiency at low poly Quanton structure uses the Landau number to generate the nomial cost and hence the system can be seen as an size of the largest permutation group that fits the problem ultra-fast, Scalable quantum emulator for problem solving size while global structure of the Quanton populations is set using the new paradigm of quantum computing. If the by choosing the distribution function. The Landau process Quanton is used in Solving hard problems, it works as an US 2016/0328253 A1 Nov. 10, 2016 approximate Turing machine model to approximate solu of iterations, or that the Quantons have achieved a desired tions to NP-Hard problems by probabilistic polynomial level of fitness, then they are returned as solutions to be computation steps in an iterative population estimation of utilized. distribution algorithm to identify candidate solutions. I0087. The probability model is built according to the 0082) STEP-8: The evolutionary step is the main process distribution of the best solutions in the current population of in which new individuals are created within a population. As Quantons. Therefore, Sampling solutions from a Quanton explained in the further details in the section of this disclo population model should fall in promising areas with high sure corresponding to at least FIG. 18A, the evolution probability or be close to the global optimum. proceeds by application of Quantum Gate operators that will I0088 Generally, a Quanton virtual machine, which act on the gate states of the Quanton represented by permu applies the Quanton computational model, has all the prop tations as described by embodiments of the present disclo erties required of a model of computation based on the Sure. Because this disclosure has a system and method for original Turing machine. Additionally, the Quanton compu very fast operations to build and execute quantum gates, tational model represents various computational problems using Quantum Gate operators in permutational form, it is using probability distribution functions at lattice points on a efficient to combine this quantum application and methods high-dimensional Surface (e.g., a hyper-sphere oran n-torus) within classical frameworks like PMBGA in order to evolve that are represented by permutations. The unique combina quantum computing circuits within the population the serve tions of features provided by the Quanton computational as problem solvers. model overcome many of the above-identified challenges I0083 STEP-9: Once the Quantum Gate operators have with more conventional computational and machine learn been selected, as in Step-8, the quantum circuits of the ing models. Quanton are updated and the estimate of distribution is I0089 For example, as identified above, conventional updated. models of machine learning can falter when assumption 0084 STEP-10: A permutation distance function is used assumptions about the congruence between training and to measure the Quantum Gate Solution. If the distance is Solution data cannot be made or when a new model that did Small, then a solution is near. If the distance is far, then not exist before has to be created. In these cases, the solution is still to be found. A critical piece of the Quanton Quanton computational model can advantageously mitigate is, therefore, the choice of a permutational distance function. these limitations of conventional methods because, without There are several choices for the distance functions. Such as ignoring the nice linear properties of the traditional methods, the Hamming Distance between the output bit-strings of gate local features and intrinsic geometric structures in the input operations or Levenstein Distance as an edit distance data space take on more discriminating power for classifi between bit strings of the gate operations. However, the cation in the present invention without being overly fitted disclosure uses a permutational distance that is more closely into the assumption of congruency because non-linearity is aligned with the probabilistic nature of quantum systems: also accounted for. the distance measurement on permutations is based on the 0090. Additionally, while the EDA model has many ben generalized Mallows model following the teachings of J eficial properties, as described above, the Quanton compu Ceberio, E Irurozki, A Mendiburu, J A Lozano, "A review of tational model can improve upon these. The unique differ distances for the Mallows and Generalized Mallows estima ence with the Quanton model relative to the standard EDA tion of distribution algorithms. Computational Optimiza model is that the Quanton model combines the directional tion and Applications 62 (2), 545-564 and is incorporated (non-commutative, geometric, and usually complex) prob herein in its entirety. ability density functions in a representation of structure 0085. The distance measure is an analog to the Leven based on a lattice of permutations (state spaces) with a Stein Edit distance measure between Strings except that in probability density on the locally linear tangent space. The this case, The Mallows model is use which is a distance permutations can index or represent directly, any other based exponential model that uses the Kendall tau distance model or pattern structure as will be shown in the present in analogy with the Levenstein measure: given two permu disclosure. The directed probability density represents the tations O1 and O2, the measure counts the total number of non-linear components of data and the tangent space repre pairwise disagreements between O1 and O2 which is equiva sents the linear components. Therefore, the Quanton model lent to the minimum number of adjacent Swaps to convert O1 distinguishes between directional or complex probability into O2. As noted in section of the present disclosure densities and structure while the conventional EDA model corresponding to FIG. 39, this is actually equivalent to a uses only isotropic probabilities and without and kind of Quantum Gate operator in the Quantum Permutational com state-space structure or lattice. putation regime presented in this disclosure. Hence, the 0091. As described in detail later, the permutations are evolution, using this distance measure, seeks optimal quan generated and embedded on a lattice that tessellates the tum circuits performing the problem solving. manifold: an example of Such as lattice is the Fibonacci I0086) STEP-11: Those Quantons that produce the solu lattice and an example of a corresponding assignment of tions are evaluated for fitness either by computing an output permutations to points of the lattice is the Permutohedron or, Solution state as a bit-string or that are computing the preferentially, its more optimal representation as the Birk Solution State as population are injected back into the hoff Polytope of permutation matrices. Further details of the population based on a distance function that measures the structure are provided in several embodiments of the present fitness. If the fitness is sub-optimal with respect to a thresh disclosure; however, it is important to note that, in the old then the population is injected back and parents are Quanton, every discrete structure has a unique non-linear as deleted, leaving in place more fit child Quantons. If he well as linear probability density function space that is system has exceed as user defined threshold for the number associated to it. US 2016/0328253 A1 Nov. 10, 2016 0092. The nearest related concept to the Quanton is that document, the sentence in which the word occurs, a fixed of a probability simplex, which has been used in natural window of words, or a specific syntactic context. However, language processing, for example, for topic analysis. Points while distributional representations can simulate human in the probability simplex represent topic distributions. The performance (for example, LSA models) in many cognitive differences between two probability distributions results in tasks, they do not represent the object-relation-object triplets topic similarity. Distance metrics are not appropriate in the (or propositions) that are considered the atomic units of probability simplex since probability distribution differences thought in cognitive theories of comprehension: in the are being compared and the divergence based measurements Quanton model, data are treated as permutation vectors. based on information-theoretic similarity, such as Kullback Therefore, in the case of linguistics, words are simply Leibler and Jensen-Shannon divergence and Hellinger dis permutations over phrases, which are permutations of sen tance, which are used do not conform to the triangle inequal tences, which are themselves permutations of paragraphs ity. However, the probability distributions over K items, and texts in general. In the case of image data, the permu Such as topics, are simply vectors lying in the probability tations are based on pixels to produce texels, and permuta simplex to which a single probability distribution is tions of texels produce the image. assigned. Therefore, large datasets represent documents by points (i.e. vectors) in the simplex: these cannot be 0096. As mentioned above, conventional computational addressed with the usual methods of nearest neighbors or models can Suffer because they use classical approximation latent semantic indexing based approaches. The inability to techniques to reach a solution that is inexact but closely perform fast document similarity computations when docu representative of the decision solution. A challenge with this ments are represented in the simplex has limited the explo type of approximation is that the real problem may be ration and potential of these topological representations at non-linear and allowing for non-linear functions results in a very large scales. much broader expressive power. However, conventional methods for combining both linear and non-linear structure 0093. To further illustrate and emphasize the difference in the same model are largely ad-hoc. The Quanton model between the Quanton model and the EDA model, a plain provides a homogeneous method for combining both rep Euclidean Sphere is utilized as an example, emphasizing that this is simply a special case of the more general resentations and therefore simple procedures for probabilis hypersurfaces as defined in the present disclosure. Firstly, tic learning or inference are used. Further, as mentioned there is a directional probability density that can be assigned above, learning methods such as Expectation Maximization onto the surface of the sphere which is itself a non-linear (EM) strategies or Maximum a Posteriori (MAP) inference space; and, secondly, at any point on the Surface of the can each have their own sets of challenges. sphere, a tangent space at the point of tangency can be 0097. The Quanton model also uses an embedding defined that is a linear Subspace, which can also contain its approach, while leveraging the benefits of higher granularity own probability density function. The Quanton thus com in being able to handle higher-dimensionality, reducing false bines both linear (in the tangent space) and non-linear positives or negatives and dealing with missing data. Given components (in the spherical Surface space) probability an appropriate choice of sampling points, noisy partial data density functions with respect to the structure of data. The can be reconstructed in O(dNK) time, where d is the structure of the data is itself indexed or represented by dimension of the space in which the filter operates, and K is permutations. a value independent of N and d. 0094 Furthermore, the Quanton model is a hybridization 0098. For example, recent work on accelerating high of the ideas of the probability simplex and the EDA dimensional Gaussian filters has focused on explicitly rep approach with a new fast encoding so that computations can resenting the high dimensional space with point samples, be achieved in low polynomial time. The encoding relies on using a regular grid of samples. When the space is explicitly using geometric algebras to represent the Quanton and to represented in this way, filtering is implemented by resam simplify computations, when further needed, by re-repre pling the input data onto the high-dimensional samples, senting the Quanton in a conformal space: in effect nearest performing a high dimensional Gaussian blur on the neighbors and comparative search becomes represented by a samples, and then resampling back into the input space. This distance sensitive hash function, which encodes related process is usually defined in the literature as the three stages documents or topics into the linear Subspace of the Quan of Splatting, blurring, and slicing. Furthermore, unlike other ton's non-linear base space. systems, the present system can be used in machine learning 0095. A geometric algebra (GA) is a coordinate free to induce filter responses based on training data inputs. algebra based on symmetries in geometry. In GA, the Instead of using only a regular grid of samples, one can geometric objects and the operators over these objects are augment the approach and use the permutohedral lattice, treated in a single algebra. A special characteristic of GA is which tessellates high-dimensional space with uniform sim its geometric intuition. Spheres and circles are both alge plices. Further, one can apply a probability density measure braic objects with a geometric meaning. Distributional to this tesselation in a machine-learning phase to configure approaches are based on a simple hypothesis: the meaning of the lattice with priors for inference. The tessellating simpli an object can be inferred from its usage. The application of ces of the permutohedron are high-dimensional tetrahedra, that idea to the vector space model makes possible the and therefore the enclosing simplex of any given point can construction of a context space in which objects are repre be found by a simple rounding algorithm (based on the sented by mathematical points in a geometric Sub-space. quantization resolution). Using the permutohedral lattice for Similar objects are represented close in this space and the high-dimensional of data with n values in d dimensions has definition of “usage' depends on the definition of the context a time complexity of O(d2n), and a space complexity of used to build the space. For example, in the case of words, O(dn), but with the encoding of the present invention as a with words as the objects, the context space can be the whole sorting orbitope, this complexity reduces to kilog(d). US 2016/0328253 A1 Nov. 10, 2016 0099. Once the machine learning phase is completed, between an ideal analytic result, and its approximation can and, the application commences the use of the learned be made negligible. The data structure and algorithms col patterns, it is important that these learned patterns are robust lectively define a virtual machine that is referred to herein as and resilient to noise as well as capable of running at high a Quanton or the Quanton Virtual Machine (QVM). rates, high data throughput in the goal of large-scale online 0103) In what follows is provided a description of the real-time knowledge processing. A key learning problem is QVM as a quantum-inspired data representation, computa that of structure identification, which is finding the best tion model and algorithm, for performing indexing, cluster structure that represents a set of observed data. While the ing and inference in very high dimensional and combinato problem is in general NP-Hard, the data representation rially complex domains. The Quanton encodes data in a techniques provided in the present disclosure provide the manner that algorithms as specified herein perform efficient means for fast approximation as a solution. quantum-like computing on arbitrary data: it must be appre 0100. The Quanton can be applied to anticipatory com ciated that the term quantum-like is used herein, because the puting, and in particular, abducing plausible causal states of action of the Quanton is to act on all possible solution paths scenarios, or forecasting potential likely surprise outcomes. within a single virtual clock cycle of the Quanton's virtual The Quanton can be used to compute sets of reachable states machine operation as defined by aspects of the present as defined by learned trajectories on its manifold: if this set disclosure. does not intersect with a specified set of hypothetical (i.e. 0104. The Quanton emulates quantum-like computing by possible counterfactual) states, the Quanton will not predict encoding high-dimensional parallelism into a single com a likely surprise outcome—however, if the set does intersect, putation step. The Quanton provides a new approach for then the outcomes are plausible and could constitute a efficiently performing high-quality, high-dimensional, infer Surprise, given by the trajectory. ence, missing-data interpolation, synthesis and filtering 0101. In general, quantum computing, machine learning, while using sparse training samples. The Quanton avoids the and artificial intelligence as well as classical probabilistic shortcomings found in mainstream techniques. Quantons computing can benefit from reversible circuits and high can be fabricated on current software as well as hardware speed, high efficiency, compact computational representa chip technologies (e.g., Field Programmable Gate Arrays tions and algorithms that can scale to Super exponential (i.e. (FPGA), Very Large Scale Integrated (VLSI) Circuits) and beyond exascale) inference and data space sizes. Revers the present disclosure for the Quanton includes both the ibility leads to substantially improved lower power con preferred embodiment of software as well its hardware Sumption compared to current technologies both for soft specification. ware and hardware design. The representation and synthesis 0105. The Quanton model provides efficient approxima of reversible logic circuits is the key step for quantum tions to any degree of user defined accuracy in the following computing: it is also of interest in new types of encryption example applications as listed below. and computation. The major challenge in synthesizing gen 0106 1. learning and ranking preferences: eral types of probabilistic reversible circuits that carry out 0107 2. cognitive image understanding: general computation, machine learning, and inference, is 0108. 3. computer-music (e.g. automated accompani that the State space becomes exponentially large as the ment or musical phrase synthesis); large complex air number of inputs increases. Embodiments of the present traffic control; disclosure describe a system of representation and comput 0109 4. automatic target recognition and target track ing method in both classical probabilistic and quantum ing: computing or quantum emulation on classical computers 0.110) 5. high-dimensional factorially Big Data infer that is fast, efficient, and uses a highly compact representa ence, radar and other signal track identity assignment; tion. Any computable function can be embedded into a 0111 6. computer graphics and video analysis; reversible function: this means that it can be represented as 0112 7. densely coded point-set registration (pixel, a permutation. Aspects of the present disclosure reformulate Voxel or quantum lattice structures); the computation of functions by permutations and embed 0113 8. robotics, biometrics, machine learning, such these into a probability space for realization of a probabi as clustering, experimental design, sensor placement, listic polynomial Turing Machine computing model that graphical model structure learning, Subset selection, emulates quantum-like computing and performs several data compression, resource allocation, object or target practical data processing functions. tracking and other problems with factorially large 0102 The present disclosure provides techniques of (state) spaces, alignment problems for rigid objects and reversibly mapping discrete or symbolic data representa detecting similarities between multi-dimensional point tions onto a permutation representation embedded in a SetS. special probabilistically configured space Such that compu 0114. 9. Semantic or textual applications such as the tation steps simulating various decision or logic gate pro discovery of paraphrased paragraphs across different cesses can be calculated by shortcutting arbitrarily complex documents that share similar topics and; operation sequences into one. The space is quantized and 0115 10. Cryptography (i.e. pattern discovery in hid each quantization provides an index that enables the revers den codes). ible mapping to be efficiently computed. The quantization is 0116. The Quanton model can process both linear and performed by a construction that builds a lattice. The quan non-linear correlations for estimating the distribution of tized space and the reversible discrete to spatial embedding Some outcome (i.e. probability measure) from data that are operations provide a data structure for approximate machine naturally observed as groupings or strata given a set of learning and approximate computation. The computations correlated variables under conditional likelihoods. For can be exact or approximate. In the case of approximation, example, in case-control clinical studies the conditioning is the quantization can be made fine enough that the difference performed on the histogram of the observations in the US 2016/0328253 A1 Nov. 10, 2016 stratum, which is equivalent to considering all permutations embedded to the geometric object, Such as the n-sphere, of the unique elements of the case-control response vector: which can be complex, but its real output is a directional this produces the need to computationally perform a sum probability distribution on the manifold of the object rep mation over a factorially growing number of terms, which is resenting a pattern of possible states as a function of the infeasible on classical computers. The Quanton solves these permutations. problems by geometric approximation: it can perform an I0121 Permutations represent a state model. Of specific embedding of this stratified conditional likelihood into a importance is the type of permutation operation (i.e. how a permutation orbitope, onto a hypersphere or hypertorus, permutation of elements is performed algorithmically), as which, ultimately, replaces the messy and lengthy combi the choice of permutation operation defines the class of natorial Sums with a single simple vector term that produces models to which the Quanton is optimally applicable. A the solution in (n)log(n) steps (instead of factorial n). specific position value on the sphere defines a state, which 0117 The Quanton is not a quantum-computer in the pure represents a Surrogate for a quantum particle state. The technical sense of Quantum Computing as usually defined, ensemble of all values on the countable set of paths on which is a "computer that makes use of the quantum states manifold of the geometric object (for example the n-sphere), of subatomic particles to store information” (Oxford English therefore, is a Surrogate for a Hilbert-space quantum wave Dictionary) but is inspired by the properties and the concepts function. of quantum systems. The present disclosure defines the I0122. By one embodiment, the Quanton is a software Quanton in terms of a probabilistic (geometric) Turing element that may represent any data structure using a machine based on a discrete state representation encoded permutation representation coupled with an embedding into using permutations with a continuum representation using a probability density space (manifold), with algorithms as either real or complex Estimation of Distribution Algorithms specified in aspects of the present disclosure. The Quanton (EDA). The Quanton model represents an approximation forms a Surrogate model for approximating quantum-like model of quantum-like computing to emulate the kinds of computing on classical processors, and in future quantum results that could be achievable on a topological quantum processors the Quanton is an exact model for complex computer (TQC). In the TQC model, any computation can inference in high dimensional spaces. be solely defined by permutations (i.e. that permutations I0123. The Quanton on a classical processor produces solely define computations on a Quanton based virtual very fast results with minimal iteration, in which the itera machine) as described by Stephen P. Jordan. in “Permuta tion (i.e. generations of population evolution and Kalman tional quantum computing'. Quantum Info. Comput. 10, 5 like iteration steps in the probability manifold) in a machine (May 2010), 470-497, which is incorporated by reference learning of data cycle, can in many cases, be performed in herein in its entirety. a single pass. The Quanton data representation enables space 0118. The Quanton virtual machine enables classical and time efficient representation of extremely highly com computers to approximate solutions to problems that might plex data: it also enables algorithms presented herein for be infeasible or certainly highly non-obvious to achieve by reducing the computation space and time of quantum simu any other approach except by true quantum computing (i.e. lations based on genetic algorithms, most notably, the esti using the states of Subatomic particles). The Quanton is best mation of distribution algorithm (EDA), also known as described as an approximating Turing machine in which any probabilistic model building genetic algorithm (PMBGA). state representation is mapped to a state represented by 0.124. By one embodiment, the Quanton uses the follow permutations: each computational step is represented by a ing unique embeddings: a permutation or association orbi permutation transition model where the transition is a tope, a nonlinear manifold with a probability density func directed probability from one permutation to another as tion, a lattice for indexing points of the manifold equivalent to the transition between one state representation corresponding to points of the orbitope, and a tangent-space (or state space) and another. at the indexed point for representing the linear Subspaces. As 0119 The continuous model of the Quanton is created by one example of a specific instance of a Quanton, consider embedding the permutations in a probability density distri the Quanton built using a spherical Fibonacci lattice. The bution or a mixture model of probability densities via Fibonacci lattice defines the indexed points, corresponding building quasicrystals (quasi-lattices) using numbers such as to the points of a permutohedron. These points are inscribed the Fibonacci numbers or sampling from other sequences, in the continuous manifold of a hypersphere. The manifold Such as from the Champernowne number, or utilizing the of the hypersphere is associated with a geometric directional techniques as described by T. N. Palmer in “A granular probability density function, such as the von-Mises Fisher permutation-based representation of complex numbers and function. A tangent space at the lattice point is also associ quaternions: elements of a possible realistic quantum theory, ated with its linear sub-space and probability distribution, Proc. R. Soc. Lond. A 2004 460 1039-1055; DOI: 10.1098/ Such as a Gaussian. rspa. 2003.1189, Published 8 Apr. 2004, which is incorpo 0.125 Quantons can be assembled hierarchically by rated herein by reference in its entirety. embedding Quantons into Quantons or forming networks of 0120. The general idea is to associate numbers (from Quantons: in the case of networks, the Quantons provide permutations) derived from models of numberings, such as path-anticipation-evolution (PAE) in which connectivity is sequences (such as Fibonacci or Champernowne) to create a anticipated as data enters the network because the probabil quasicrystal lattice that embeds a quantized (or granular) ity density and permutation functions of a “learned Quan version of a geometric object, Such as the Riemann Sphere, ton provide an structural anticipatory capability. An example and onto which probability density can be assigned. More of path anticipation, as used in real time tracking and specifically, the lattice points are associated to permutations collision avoidance is provided by J. Park, S. Kang, N. that represent states or data structures of Some type as will Ahmad and G. Kang in "Detecting collisions in an unstruc be explained herein. Further, a probability density can be tured environment through path anticipation, 2006 Inter US 2016/0328253 A1 Nov. 10, 2016 national Conference on Hybrid Information Technology, object in order to illustrate, disclose, and explain the core Cheju Island, 2006, pp. 115-119, which is incorporated by working algorithms and methods of the present disclosure. reference herein in its entirety. However, it must be appreciated that other geometric objects 0126 Quantons are developed in order to handle infer (e.g. hypertorii) are equally applicable. ence in extremely massive high-dimensional complicated 0.133 Selecting, ranking and inference using Such large domains in multimedia or multisensory data as well as in sets of data objects lead to rule-structures in which the difficult combinatorial optimization problems where data permutations of data objects, and therefore, the number of spaces can be factorial in the size of inputs. Because of the rules or possible chains of inferences becomes factorial in huge number of these data and their high dimension, con the number of entities (n!). ventional classical indexing or search methods as well as I0134. The Quanton data representation and algorithm traditional inference procedures are far too inefficient to be provides an embedding of all in permutations of any object useful even with modern Supercomputing power. For these set, for a given number of objects (entities) in a Surface of kinds of problems which include, but are not limited to, a combinatorial hypersphere is defined in R(n-1) of a space optimization problems, Quantum computing has been pro in R(n+1). The Quanton provides n(log(n)) time algorithms, posed as well as various other heuristic approximation at worst case, knclog(n)), between the combinatorial hyper models, such as EDA models. sphere representation, quantum computing Surrogate data 0127. Two versions of EDA models are Particle-Swarms representations, and the n-element combinatorial time and and Ant-Colonies for optimization problems. However, none space complexity reduced to a simple polynomial, that is of these various disparate approaches has integrated a required in inference. Aspects of the present disclosure robust, flexible software data representation beyond bit provide a way to use continuous Morton Codes that can be vectors as the fundamental data structure on which the associated to directional probability densities and other probabilistic, mutation, crossover and other evolutionary methods for establishing probability densities over arbitrary operators function upon. states or structures that can be represented as permutations 0128. The Quanton is made up of a continuous direc using the algorithms described herein. tional probability distribution whose discrete substructure is 0.135 The method can handle a large number of objects, made up of the vertices of an orbitope. This geometric shape emulate traditional probabilistic data association algorithm could be inscribed within a hypersphere (a manifold), and its results, differentiate weak signals in conditions of high associated locally linear tangent space, which is a Surrogate noise, handle chirality and can scale to data sizes that are not for a non-directional probability space on the directed prob by the state of the art mainstream inference methods known ability space. The States form a Surrogate to outcomes of to those skilled in the art. The Quanton is ideally suited to corresponding quantum states. Examples of Such outcomes implementation on both Quantum and Classical computers. include results, inferences or configurations in data cluster 0.136 Referring now to the drawings, wherein like ref ing, analysis and reasoning applications as mixtures of erence numerals designate identical or corresponding parts probability densities. throughout the several views. In describing the embodi 0129. The data representation and method of the present ments in detail some conventional notation is used that is system is also ideally Suited as a bridge from implementa defined in the foregoing. tions on classical computer architectures (i.e. Von Neumann 0.137 n is the length of a permutation. models) to analog-computer, hybrid analog-digital and fully (0138 (1,2,3,..., n)' is the Identity Permutation. quantum computer models. 0.139 P” is the set of all permutation vectors of length n. 0130. One of the key ideas is to connect the way that 0140 teP" is any permutation. orbitopes (i.e., combinatorial polytopes). Such as the per mutohedron and associated Zonoids, can be compiled by 0141 p, is the i-th indexed element of the permutation embedding them into the Surface of a hypersphere or hyper vector, p. torus or other geometric object with probability distributions (0.142 it is the vector of length n whose entries are all that are oriented in ways that represent data behaviors, while set to 1. at the same time providing an efficient indexing method (as I0143 P is the set of nxn permutation matrices contain provided by the lattice of the present disclosure). ing ones and Zeroes with a single one in each row and 0131 By one embodiment, the Quanton of the present column. disclosure combines five key ideas: (i) combinatorial poly 0144. The convex hull of the set of nxin permutation topes, (ii) manifold embedding, and (iii) directional prob matrices is the Birkhoff polytope denoted E'. The set of all ability density (iv) lattices; and, (v) tangent spaces. The doubly-stochastic nxn matrices is: Quanton provides a quantum Surrogate data representation (for example, by combing the permutation structure here with the teachings of Palmer as referred earlier in this (0145 The permutahedron, PHC " is the convex hull of disclosure, on which the novel and non-obvious result is that permutions, P', has 2'-2 facets and is defined as follows: a variety of (non-parametric) inference processes can be applied as Surrogate quantum computing approximators. f IS These approximators provide a degree of approximation PH = {x eR', X Wi = 12.) Wi sX (n + 1 - i)WS Cn producing results that are close to but not perfectly matching i=1 ieS i=1 the fidelity of a true quantum computer, but easily amenable to a quantum computer for artificial intelligence (also known in the art as Quantum Artificial Intelligence). 0146 The permutohedron is the projection of the Birk 0132) For the purposes of simplifying the disclosure, we hoff polytope from IR* to IR” by will use the n-Sphere (i.e., hypersphere) as our geometric US 2016/0328253 A1 Nov. 10, 2016 0147 Let V, V, V. . . . . V.) and u, F(u, u, us . . . , u,) S is a d-dimensional hypersphere in IR' with dimension: be two vectors pointing to vertices in the permutohedron and d=(n-1)-1 let the distance between u and v be d(u,v). 0.161 All permutation matrices, M, on n objects are set 0148. Further, define the e-neighborhood of u as: to be on the surface of a hypersphere S of radius V(n-1) in R(n-1) using the previous rules. Therefore, for example, 3 permutations are in a 4-dimensional space. In the conformal for example: model, 3 permutations are in a 5-dimensional space since the for 0 0171 Definition 1: The information-carrying degrees of software data structure composed of at least, the PDF, a freedom are referred to herein as observable, denoted as o discrete lattice and certain algorithms as presented herein for and the other degrees of freedom in the system that may computation or learning deep patterns in complex data. carry virtual information as hidden, denoted as h. The 0186 The Quanton may be an object that is composed of description of the State that results from a quantum state a polytope whose kissing vertices inscribe a hypersphere tomography procedure is given by the density matrix, TD. Such that the kissing vertices correspond to points on the 0172. The key approach is to separate the state explicitly Fibonacci spherical lattice as well as to permutations. The into observable and hidden parts and to calculate the observ surface of the Quanton will usually have a probability able parts and their constraints using the quantum statistical distribution associated with it such as the von Mises Fisher, needs of the whole state. Watson, Gaussian or a mixture, as well as complex versions 0173 Definition 2: The real observable permutation that of these probability density functions, that the present dis includes a sequence of symbols in a particular order we call closure defines as the Software data structure and associated the permutant and another, hidden sequence of symbols quantum-inspired algorithms for learning deep patterns in (possibly acting as fixed constraints) that we shall call the complex data. One example is a tangent plane to one of the permuton. An operation of applying the permuton to the vertices, a normal vector at the point and a probability permutant will result in a permutrix. distribution, not limited to, for example, a Gaussian distri 0.174 Definition 3: The probability measures are observ bution, about the point on the tangent space. This structure able at the vertices of the lattice (e.g. Fibonacci Lattice) and enables a mix of probability distributions on either or both the points within the subspace bounded by the lattice points the Surface and the tangent spaces to the points on the are undefined: in other words, we interpret the space in surface for a variety of mixtures to be represented. analogy to a quantum Psi-Epistemic theory in which the 0187. The Fibonacci spiral lattice permits fully analytic lattice points provide observables while the intervening Solutions to the distribution of points as coordinates on the regions are transition spaces, remain undefined. Surface of a hypersphere and these points number in the (0175 Definition 4: The Quanton same quantity as the permutation sequence. The relationship 0176 A Quanton comprises the following data elements: between a permutation sequence the Fibonacci lattice, as 0177 (1) A geometry that is symmetric and in the form explained in the present disclosure, is such that the vertices of a continuous manifold; of the permutation polytope intersect the Fibonacci lattice so 0.178 (2) A lattice (and therefore a topology) that that all point on hyper Surface to permutation and Vice-verse partitions the Symmetry group of the manifold and an are indices of the Fibonacci lattice, in analogy with space optional tangent plane at the point of the lattice that filling curve indexing, such as the Hilbert Curve. The provides a linear subspace embedded in the manifold; arrangements of the vertices is also important so the vertex (0179 (3) An index to the points of the lattice that to vertex relationships are not random: for example, given a associate a permutation of the integers to each; set of four integers, (1,2,3,4) there are 4–24 ways of 0180 (4) A rule for transition from one permutation to arranging them, however, the geometries by which these 24 another between lattice points; and, numbers may be written down, which is to say, their 0181 (5) A probability density function that associates ordering, is itself 24!. Therefore, the Quanton requires a a transition probability between points of the lattice. certain kind of regular ordering using an ordering rule. 0182 Given the elements of the Quanton, one can iden These rules will be stipulated further in the embodiments tify the specific operational rules by which the QVM oper under FIGS. 13A and 13B for exemplary topologies of the ates to perform its functions. For the purpose of simplicity orbitope characterizing the Quanton. and clarity of the exposition of the method of the present 0188 Table 1 shows the types of geometric objects disclosure, a spherical geometry in the form of the n-Sphere usable for design of the Quanton: will be used throughout to show the exact algorithms and processes, noting that these shall be obvious to apply to TABLE 1. other geometries. 0183) Definition 5: The Quanton Language is defined Lookup Table for Quanton Construction herein as a reversible language of operations that correspond Manifold to movements between permutations and these movements Type Lattice Constructor Probability Distribution are reversible between lattice points. Calabi-Yau Bravais Lattice Lattice Green Function, Lp 0184 An n-Sphere is an exemplary geometry and topol (3Sle ogy for the Quanton data structure. Other geometries such as n-Sphere Fibonacci Univariate Directional, L2 (3Sle the n-Torus or other manifold structures, such as the Calabi n-Torus Fibonacci, Fermat Bivariate Directional, L2 Yau can be used also. The Quanton is an object that is (3Sle composed of a polytope whose kissing vertices inscribe a n-Cube Binary Uniform, L1 measure hypersphere such that the kissing vertices correspond to n-Simplex Simplicial polytopic number Any points on the Fibonacci spherical lattice as well as to 2-polygon Figurate Numbers Any permutations. Zonotope Any Any 0185. The surface of the Quanton will usually have a probability distribution or probability density function 0189 FIG. 2 shows the unique methodology and repre (PDF) associated with it such as the von Mises Fisher, sentation of a permutation sequence 16 at the vertices 12 Watson, Gaussian or a mixture, as well as complex versions called a Permutrix 15. The permutrix 15 is composed of a of these probability density functions such as the complex permutation pattern, the permutant 13, which is the visible Bingham or complex Watson distribution on the complex part of a permutation pattern, and other entries, called the n-sphere. The Quanton is, therefore, by one embodiment, a permutons 14, whose purpose is to serve as the invisible US 2016/0328253 A1 Nov. 10, 2016 parts of a permutation pattern. The Permuton 14 can serve as second is de{0,1}, and the n-th is de0... n. For instance, a constraint system by making its positions invariant to fr(42)-> 0, 0, 0, 3, 1->0*0!--0*1!+0*21+3*3!+1*4!. permutations of the other elements, the Permutant 13. One 0.195 The left-to-right, decreasing order variant fl is analogy is to think of the permutant as the allowed moves obtained by reversing the digits of fr. For instance, fr and the permuton as the blocked (disallowed) moves in (42)=0, 0, 0, 3, 1 and, therefore, fl(42)=1, 3, 0, 0, 0). The Some game: in 2-dimensions, the game can be Chess, as one inverse of the procedure fr that transforms a factoradic into example of many. a list of numbers is denoted as rf that transforms a list of 0190. The permutrix like the preceding description for numbers back into a factoradic. Similarly, the inverse off is the Quanton, also can generate, for n-permutants and d-per 1f. mutons (n+d), and the number of orderings of the arrange ment of (n+d)) geometric orders. This point bears some Example importance as the ordering determines the starting sequence (0196) fr: 42->0,0,0,3,1: for any index, and the order itself determines the relation 0.197 rif: (0,0,0,3,1->42: ships that govern the way the index behaves. Given that the 0198 fl: 42-> 13,0,0,0; and d-permutons could (as one possible rule of many) act as (0199 lif: 1.3,0,0,0)->42 “dummy” variables and if deleted, results in some sequences 0200. The rank and un-ranking of permutations is per that will be repeating. The role of the permutons will become formed using Lehmer codes and their Factoradics. The clear in the foregoing but the permutons can also play the function perm2nth generates the n-th permutation of a given role of virtual-particles that enable other operations and size and, therefore, naturally represents a rank for any relationships to control the permutants and the factors permutation of size greater than Zero. It starts by first depending on permutant relations. As one concrete example, computing its Lehmer code Lh with perm2lehmer. Then it permutants can be indexed by the natural integers (1.2.3 . . associates a unique natural number n to Lh, by converting it . N), while permutons can be indexed by imaginary valued with the function lf from factoradics to decimals. Of special natural integers (i1, 12, i3 ... iN) so that that other operations note for the present disclosure is that the Lehmer code Lh is can result in output indices by combining the permutants and used as the list of digits in the factoradic representation. permutons in certain ways (e.g. complex conjugation). Of 0201 The function nth2perm is the inverse of perm2nth importance is that the consecutive permutations in which the and provides a matching un-ranking that associates a per numbers of inversions that differ by one, forms a Gray code mutation to a given size (greater than Zero) and a natural based on the Lehmer code (factoradics) from the in Stein number N. A permutation can therefore be reconstructed haus-Johnson-Trotter algorithm. Therefore, using the Leh from its Lehmer code which in turn can be computed from mer code, the n'-inversion is deterministically computable. the representation of the permutation as a factoradic. 0191) Any data structure of any kind can be represented Examples of the bijective mapping follow: by a permutation and the permutation can represent (0202) nth2perm (5,42)-> 1.4,0.2,3] sequences, sets and multisets: therefore, one can use per 0203 perm2nth 14.0.2.3->(5, 42) mutations as the most fundamental building block for both 0204 nth2perm (8, 2008)->0,3,6,5,4,7,1,2 data representation and computation by the method and 0205 perm2nth 0,3,6,5,4,7,1,2->(8, 2008) algorithm expressed below. 0206 Given the preceding process and application of 0.192 The factoradic numeral system replaces digits mul methods of encoding permutations and numbers in with tiplied by a power of a base n with digits that multiply respect to Lehmer codes and factoradics, the present disclo successive values of the factorial of n. The Lehmer code, Sure uses these facts to build Succinct integer representations which is a sequence of factoradics, represents a permutation of data structures such as lists of natural numbers, arbitrary uniquely, and also can be represented using a natural num lists, sets and multisets. ber. The natural number is computing by first computing a 0207 Procedure-1: Encoding Lists of Natural Numbers rank for the permutation of size greater than Zero by first 0208 (1) For any input multiset, first sort the multiset computing its Lehmer code, and then associating a unique it as and then compute the differences between con natural number to it by converting it from its factoradic secutive elements i.e. Xo . . . XX, . . . Xo . . . representation into decimal. In this way, it easy to recon X-X, . . . . As an example: given multiset 4.4.1.3. struct any permutation from its Lehmer code, which in turn 3.3 sort into 1.3.3.3.4.4 and compute pair wise is computed from the permutation's factoradic representa difference list as 1,2,0,0,1,0). tion. 0209 (2) Given the result of step (1), re-encode the 0193 Given the preceding, therefore, the procedure for difference list by elements in the sequence of incre converting between arbitrary lists, sequences, sets and mul ments replaced by their predecessors: therefore, prede tisets into permutations is represented as index numbers (i.e. cessors of the incremental Sums of the Successors of a natural number from which the permutation and Subse numbers in the sequence, return the original set in quently the source data structure such as multiset is recov sorted form. For example, the first element of 1.2.0, erable without loss). The procedures for encoding and 0,1,0) is the number 1 and is followed by its successors decoding between data structures, permutations and Succinct as the list 2,0,0,1,0). Prefix sums of the numbers in the integer representations covers all the other cases (sequences, sequence return the original set in Sorted form. For sets and lists) and is given by the following set of transfor another more obvious example, given 1.2.3.4.5, the mation rules: resultant difference list is 1,1,1,1,1). 0194 The factoradic numeral system as described by 0210 (3) Given any set of natural numbers, such as Knuth replaces digits multiplied by a power of a base n with {7, 14.3}, the application of the first two operations digits that multiply successive values of the factorial of n. In preceding produce the sort (1.3.4.7) and then comput the increasing order variant fr the first digit do is 0, the ing the differences between consecutive elements gives US 2016/0328253 A1 Nov. 10, 2016 1.2.1.3), in which, with the first element 1 followed by refer to a structure, Such as a graph or fragment or have any the increments 2,1.3 we convert it into a bijection, other mapping, like a hash to other objects, then these can be including 0 as a possible member of a sequence, by encoded, decoded, and represented using the methods of taking the elements in the sequence of increments and Table 3. This feature is important in mapping structures in replacing by their predecessors to produce 1.1.0.2 the Quanton. Such that this can be encoded using the Lehmer or factoradic into a natural number. TABLE 3 0211 (4) Taking the output of (3), the predecessors of the incremental sums of the Successors of numbers in Mapping Permutations to Data Structures Such a sequence, return the original set in sorted form. Rule Procedure Example The (big-endian) binary representation of any natural num List to encodes a sequence of Input List = 1, 1, 2, O ber can be written as a concatenation of binary digits of the Permutation natural numbers as Output Perm = 4, 1, 3, 0, 2 form a permutation of n=bb'... b. *... ben (1) natural numbers Permutation Decodes a permutation Input Perm = 4, 1, 3, 0, 2 with be {0,1}, b,zb, and the highest digit b, 1. o List to a list Output List = 1, 1, 2, O Therefore, by one embodiment, a process is defined List to Set nput List = 1, 1, 2, O Output Set = 1, 3, 6, 7 wherein: Set to List input Set = 1, 3, 6, 7 an even number of the form Ojcorresponds to the operation Output List = 1, 1, 2, O 2 and an odd number of the form 1 corresponds to the List to Multiset nput List = 1, 1, 2, O operation 2'(i+1)-1. Thus, the following equation holds true: Output MultiSet = 1, 2, 4, 4) Multiset to List input MultiSet = 1, 2, 4, 4) f(i)=2(j+1)-1 (2) Output List = 1, 1, 2, O List to Natural compose nput List = 1, 1, 2, O Therefore, each block 1 in n, shown as 1 in representation Number Output Number = 140 (1), corresponds to the iterated application of f. i times, Natural uncompose nput Number 140 Number Output List = 1, 1, 2, O n=f(). o List 0212. It follows from this fact that the highest digit (and List to nput List = 1, 1, 2, O therefore the last block in big-endian representation) is 1 and Permutation Output Perm = 4, 1, 3, 0, 2 the parity of the blocks alternate that a number n is even if Permutation input Perm = 4, 1, 3, 0, 2 and only if it contains an even number of blocks of the form o List Output List = 1, 1, 2, O b. in equation (1). A number n is odd if and only if it contains an odd number of blocks of the form b, in equation 0219 Procedure-2: Quanton Combinatorial Design (1). 0220. The combinatorial design of the Quanton builds on 0213 Define the procedure compose(i, j, N) that outputs the relationship as established between data structures and a single natural number, N, given two separate input natural permutations to provide a transition model between permu numbers as follows: tations to produce resultant permutations as outputs. There 10214) compose(i,j)-2'iiffjis odd and 2'' (i+1)-1 if are several elements in the combinatorial design: j is even 0221 (i) Design of the permutations of the Quanton in Therefore, in general, define procedure: terms of a choice of integer sequence that relates compose(ii,N)=is0.jeO.D-mod(j+1,2).N=(2' (i+ permutations to Successors and predecessors; D))-D 0222 (ii) Design of the permutation transition operator 0215 Specifically, defined herein is that the exponents between permutations in the design produced by Step are i+1 instead of i as counting is defined herein to start 0. (i) preceding; and, Note also that compose (i,j) will be even when j is odd and 0223 (iii) Associating the permutation transition odd when j is even. The operation compose (i,j) is reversible operator from step (ii) to an operational semantics. and can be used to iteratively convert a list of numbers to a 0224 First, the design of the structure, ordering and size single number or can reverse from a single number back to of the Quanton is based on a user defined requirement in the the original list of numbers. size of computations, type of operations and data represen 0216 Define the procedure uncompose(N)-i,j] that tations needed. For example, and in analogy with traditional accepts a single natural number as input, and outputs two CPU, a 4-element Quanton can represent 2 bits, and a numbers i and j that composed it. The definition of the 10-element Quanton can represent 2 bits (e.g. 10!). The uncompose is the inverse of compose and utilizes a helper issue is to have a method to choose the right size and design subroutine, divine(N.i,j) to identify the largest exponent of 2 of the permutation in terms of the appropriate design of the dividing N. It is computed as follows noting that the symbol symmetric group of Sn that provides optimality. The answer "/7” means integer division. is given by determining the Landau Numbers in terms of permutations: these guarantee, with the design of the per mutation operator design, that the operational Semantics of 0217. Therefore, in general, define procedure to uncom transforming permutations is bijective and reversible, as will pose a natural number reversibly into two other natural be shown in the later sections of this disclosure. numbers: 0225. The choice of the integer sequence is not obvious uncompose(N,i,j)->is0.js0.B=mod(N2).X=N+B.di and follows from the twelvefold way as originally defined vine(Xp,q).imax(0,p-1).j=q-B by Gian-Carlo Rote and explained in the reference: Richard 0218. The preceding procedures are used to define rules P. Stanley (1997). Enumerative Combinatorics, Volume I. and are summarized in Table 3. Provided that the numbers Cambridge University Press. ISBN 0-521-66351-2. p. 41, US 2016/0328253 A1 Nov. 10, 2016 which is incorporated by reference herein in its entirety. 0229. In the case of generating the vertices of a permu Preferred sequences for the combinatorial design of the tation, the nth-permutation is given by the Lehmer code and Quanton are illustrated in Table 4. However, it must be the nth-Fibonacci number is given by the Binet Formula. appreciated that, any permutation that is decomposable into Hence there is an analytic correspondence between permu regular patterns of some series, or set of operations, can tations and coordinates on the Fibonacci Lattice. Exemplary serve to build the permutation lattice. For instance, the options as to how these can be further distributed to points pattern operator may be Catalan automorphism, signature are further expounded upon for FIGS. 13A and 13B. For the permutations and the like. 2-sphere, the spherical Fibonacci point sets are directly TABLE 4 Preferred Sequences for the Combinatorial Design of a Quanton Pattern Operator Description Example Blockwise pattern operator The simplest blockwise Odd-And-Even Swaps = {2, 1, permutations are obtained by 4, 3, 6, 5, 8, 7, 10, 9, ... } acting on blocks of fixed size, p(1) = 2. p(2) = 3. p(3) = 1. p(n + 3) = always with the same 3 + p(n), for all positive permutation operation, which integers n = {2, 3, 1, 5, 6, 4, 8, can include permutation 9, 7, ... } operations as part of the operator Alternating pattern, ZigZag Stanley, Richard P. (2010). A pattern, or updown pattern Survey of alternating operator permutations. Contemporary Mathematics 531: 165-196. Retrieved 11 February 2016 Landau Number based largest order of permutation of n Permutation Sequence elements. Equivalently, largest LCM of partitions of n" (LCM means Lowest Common Multiple) Baxter Permutation They satisfy a pattern avoidance all permutations of length 4 are property. These permutations Baxter except {2, 4, 1, 3} and are defined by prohibited Subsequences in the permutation 0226. The Landau Sequence, based on the Landau Num defined using the angles (and these can be generalized to ber, provides the largest cycle into which the Permutation n-spheres as stated earlier for the hyperspherical coordinates can be naturally structured in a natural ordering in a cyclic by simple substitution): Subgroup of Sn: in this case, for example, the usual arith metic addition operation becomes just the application of permutations, as will be shown. This is important because 6; = arccosi 1 - 2i) the Quanton relies on the design of the permutations in order - F ) Osi < F; to be able to represent and compute by using transitions (i.e. = 2it . Fm-1 F is the m” Fibonacci number movements) between the lattice points by which the permu e = 2 (j-i} tations are indexed. 0227. The main tool used in the distribution of the 0230. For each base angle, G, the points are distributed vertices of the polytope onto the surface of the hypersphere on the vertical (Z-axis): is illustrated in FIG. 3: Fibonacci Lattice. The Fibonacci Lattice and the Fermat Lattice (also a spiral) are analytic and 0231 And the angle of rotation, (p. is given by the enable fast generation of points with variable geometries. Fibonacci ratio: Other Lattices could also be used. For example, FIG. 3 item 17 illustrates a geometry generated by the Fibonacci lattice that is evenly distributed while FIG. 3 item 18 shows how the lattice generator has been modulated to increase the density of the points generated on the equator. 0228. The representation of the Fibonacci lattice for 0232. The limit of the Fibonacci inverse ratio is the arbitrary dimensions is given in the form of angular param Golden Ratio as m increases. By adding 1/N to Z coordinates eters, with one angular direction per point on the n-dimen we achieve uniformity at the poles and hence: sional Surface. The use of angular parameters is preferred because of the symmetry of rotation operations using angles in periods of tradians. Given this representation, in hyper spherical coordinates, then any movement from one permu tation to another is simply and analytically given by move N is the number of spherical points ment either along the Fibonacci hyperspiral or between windings of the Fibonacci hyperspiral. US 2016/0328253 A1 Nov. 10, 2016 0233. It must be appreciated from the preceding descrip permutation of “0123. This can be repeated until the tion that a circle can be evenly divided into points that permutation cycles back to "0123. correspond to the numbers of points representing a single 0237. It must be noted that for each permutation gener permutation, or that a two dimensional Fibonacci spiral, or ated in order, that it can be associated to the indices (0.1.2.3 Euler Spiral can also be used. For example, with three . . . N) of the lattice generating function (i.e. the Fibonacci permutations, there will be six points on the circle, corre sequence). In this way, permutations can be associated to sponding to a hexagon. For a permutation of 3 objects, any point on the lattice. indexed using the set {1,2,3}, and treating the integers as 0238. In FIG. 4, items 19 and 20, show respectively a von coordinates in a vector space, then, the six vectors are Mises Fisher circular directional statistic and a mixture (1.2.3), (2.1.3), ... (3.2.1) which are coordinates in 3-space. model of the multivariate Watson distribution from a sample However, from this it must be noted that various ways of directional data set, though other distributions such as mapping from high-dimensional coordinate systems to Gaussian or Generalized Mallows distributions could also be lower dimensional coordinate systems can be used without depicted. loss of information, such as the Z-Order (Morton Code) to convert from high dimensional to single dimension num 0239 FIG. 4 depicts the sphere without the vertices of the berings bijectively. Therefore, for example, coordinates of inscribed correlation polytope (e.g. Permutohedra, Associa the permutations for a permutation of 4-objects, {1,2,3,4} hedra), but serves to emphasize that the basic underlying would result in a 4-dimensional space, but these can be implementation and design of the Software Quanton is a projected as indices using the Morton Code onto a Fibonacci centered on the use of probability density functions (PDFs) spiral and distributed onto the 3-sphere. The teachings of on the n-sphere, or, on an n-torus. Formally, the depiction is Benjamin Keinert, Matthias Innmann, Michael Sänger, and a mixture model, which represents a distribution of data, Marc Stamminger. in “Spherical fibonacci mapping, ACM seen as the circles on item 19, and the trajectory on item 20, Trans. Graph. 34, 6, Article 193 (October 2015), and incor which are modeled as a weighted sum of individual prob porated herein by reference in its entirety describe the ability distributions of the generating data. mapping of points to spheres as the means to associate 0240. In other words, the observed data is a mixture of a permutations as points to various desired manifolds includ number, M, of probability distributions, and defined by the ing spheres using the Fibonacci series. formula: 0234. If the choice is that the permutation represents the coordinates in a space, then, these discrete coordinates represent the Zonotope: a polytope or hypercube. In the case i of 3 points, these would form the vertices of a hexagon that P(v, M) =X w.f. (v, p.) are the kissing vertices of the surface of a sphere in three dimensions. In general, therefore, any Subset of permuta tions can be represented in a dimension lower than that where v is a d-dimensional vector (datum) and M is the required if the permutation elements were each taken as a number of mixture components with w as the evidence dimension. (weight) and p as the probability density of the k-th com 0235. In generating the permutations themselves, and ponents respectively. especially, in providing a generator that results in permuta 0241 The Quanton can also be built on a complex space, tions in a sequence that can be indexed starting at 0 (no therefore, for example, the Quanton on a Riemannian mani permutation) or starting at 1 (the first permutation) to N (the fold coupled with a discrete structure as permutations of N" permutation, regardless of whether 0 or 1 was chosen), objects. The Probability Density Function (PDF) of a poten the simplest generator is the additive permutation generator tially large dataset that lies on a known Riemannian mani using the integers as follows from the teachings of John R. fold is approximated by the methods of the present disclo Howel, “Generation Of Permutations By Addition’-Math Sure. The Quanton data structured thus provides a ematics of Computation Vol. 16—Issue 78–1962 p. completely data-driven representation and enables data 243 and incorporated herein by reference in its entirety: K. driven algorithms that yields a PDF defined exclusively on permutations can be generated by addition of a constant, C the manifold. to a permutation generation operator. The integers, {0, 1, 2, 0242. Now, turning to FIG. 5, we describe the relation . . . . (K-1)} or {1,2,3,... K} can be concatenated to form ship between the tangent space, the nonlinear manifold the “digits of a base K integer. Using the base K integers, space 21, probability and, later, its relationship to permuta the repeated addition of 1 will generate integers whose tions, and finally, the use of permutations to define the 'digits represent permutations of length K. Quanton in the computation of State space 22 to state space 0236. This process will also generated numbers which are 23 transitions in the Turing machine sense. not permutations. The correct number, C, greater than 1 to 0243 FIG. 5 illustrates specifically the tangent spaces at add to this integer is a multiple of C=(K-1) radix K. The the vertices of the inscribed Fibonacci spherical lattice arithmetic difference radix K between an integer composed whose points represent the vertices of the inscribed orbitope. of mutually unlike digits and another integer composed of a The figure is important because it illustrates a key point permutation of the same digits will be a multiple of (K-1). about the Quanton, which is that the Quanton represents a This algorithm will generate all K. permutations in lexico non-linear manifold PDF as well as a linear PDF that resides graphic order or that any permutations “between two given in the tangent space at a point on the manifold. Alternative permutations can also be directly computed. For example, representations use a single tangent space located at the for the 4 permutations for K-4, using the set of digits, geodesic mean, which can lead them to have significant D={0,1,2,3}, then C-3 radix 4. First concatenate D to get the accuracy error when input data is spread out widely on the digit “0123. Second add C get “0132 which is the first manifold, or when the data sizes are very large, the associ US 2016/0328253 A1 Nov. 10, 2016 ated algorithmic complexity, for example of using Gaussian processes, makes them unfeasible for large datasets. 0244. By one embodiment of the present disclosure, a f(u, x) = - -e" logarithmic map is utilized to project multiple input data (2)? Is (K) points using an unsupervised algorithm that automatically 2 computes the number of model components that minimize the message length cost in the linear Euclidean tangent 0250. The Minimum Message Length (MML) requires a space. Data inputs are represented by distributions on the prior distribution or at least an approximation if the prior is tangent spaces at corresponding mean points, quantized to unavailable easily. The easiest prior that we can use as an the nearest vertex point (this is produces an approximation assumption is uniform and independent of k, written simply with a small error). The accuracy loss produced using a aS single tangent space versus multiple tangent spaces over comes the cost of the quantization error at Vertices on the manifold. However, the quantization error is addressed sin(a) herein as a separate term by itself, which is a non-linear hog (a, f3) = 47t correcting component, also a PDF of the surface of the manifold. This PDF is itself approximated since it plays the 0251. The MML estimate are values of (C.B.K) minimiz role of the non-linear part of the posterior of the total model ing a Message Length Expression (MLE) for 3-dimensional of the Quanton. von Mises Fisher distributions and were D represents the 0245 Tangent space statistics are computed and pro data and h represents the hypothesis: jected back to the manifold using the exponential map. This process iterated until convergence of the statistics. The Quanton can handle an arbitrarily large number of Samples in contrast to existing non-parametric approaches whose hog (a, f3, K), f(D|a, f3. K) -- complexity grows with the training set size. The Quanton manifold mixture model handles both non-linear data mod eling as well as regression to infer Subsets of missing where C are constants. components without any specific training (i.e. a missing data The expression det(F(C.B.K)) is the determinant of the gap identification solution). In other words, the Quanton can expected Fisher Information Matrix, whereas the always produce hypotheses on demand. expression he (C.B.K) is the prior in 3-dimensions: 0246 The analytical estimates of the parameters of a multivariate Gaussian distribution with full covariance matrix, using the Minimum Message Length (MML) prin ciple are preferred in approximating Gaussian PDFs using hog, (a, f3, K) = hog (a, f3)X hk = 21-22 the tangent space. The Riemannian center of mass of N points, x, sometimes also called the “Karcher Mean' on a manifold is: 0252 Finally, the expression f(DC.B.K) is given in 3-dimensions following the VMF distribution from: f(a, f3, K) = 4tsinho (sindisina-cos(8:-b)+costicosa) pu(t + 1) = Riemann Expo Therefore, for n data elements, D, we have: This is iterated until: f(Da, f3, K) = || 47tsinh(K) k(sindisina-cos(9-É)+costicosa) 0247 The value e is a threshold and the value Ö is the quantization (resolution or step size) of the Fibonacci lattice 0253) A generalized search heuristic for minimum mes (or other quasi-crystalline structure) used to construct the sage length to infer the optimal number of mixture compo manifold. Essentially, the directional PDF defines a vector nents and their corresponding parameters that explain the field on R' and this can be interpreted as averaging the observed data is used and based on the search used in vectors from an arbitrary point XeR" to the mass points p. various versions of the Snob classification program as the 0248 Thus, while a single Zero-mean Gaussian in the preferred embodiment for the Quanton. tangent space around a point, p, provides an effective model 0254 The situation illustrated in FIG. 5: Mixtures of for the within-cluster deviations from p provided it is the Distributions on Tangent Spaces to the Hypersphere, pro Riemannian center of mass of this cluster. vides the Quanton mixture model that ability to represent 0249. The formula for the expression to infer the con use both linear tangent spaces to the sphere providing full centration parameter k and mean vector, u, of any d-dimen covariance matrices as well as isotropic covariances asso sional VMF distribution on a d-dimensional sphere is given ciated with the spherical von-Mises-Fisher (VMF) distribu aS tions of FIG. 4. US 2016/0328253 A1 Nov. 10, 2016 0255. The end result is that the Quanton model, in the present disclosure to reduce the computational burdens and inference processes of the present disclosure, which involve bring speed as well as precision to estimators given some Quantons as particles and as models in the context of data set. estimators of distributions that is parallelizable and handles 0262 The overall model of use of the preceding diagrams both isotropic as well as anisotropic distributions with is summarized by the overall schema depicted in FIG. 7: adaptive model complexity to observations. As will be Quanton Schema of Iterative Feedback Data Processing for shown, each point of tangency is the vertex of a correlation Estimation of Distribution Algorithms. In item 27 of FIG. 7, polytope built on permutations. This space is denoted we have a hypothesis that models some observational data, and represented as POX) where represents the hypotheses T and is the linear Subspace that best approximates a and X represents the observable data. The data X, in item 28, region of the sphere in a neighborhood of the point p and can is mapped to the Quanton. This can be done for several, or be thought of as a directional derivative. Vectors in this a mixture of various POulx). space are called tangent vectors at p. 0263. A few key points are in order. First the basic 0256 Every tangent point on the sphere, at the point of approach consists of maintaining information relative to the tangency is defined to map to its linear tangent space using first order marginals, which express the probability of some the Riemann transforms as follows: item e being at Some position n in the permutation O of the 0257 Let TS be a point on the sphere and T be the data X. In the case of the Quantons, item 29, shown as S2 and tangent point on the sphere, and p be any tangent point, S2 connected at a tangent point, we provide the concept of where the sphere is in dimension D-1. Hence: TS eS''': permutants and permutons, that together form the permutrix, and therefore, the Quanton extension consists of higher tangent points are in T; ; and peT order marginals on the permutrix, II, item 30 in FIG. 7, which corresponds to the probability of a specific set of items in the permutrix (e. e. . . . . e.) being at Specific Wve T; |y| = Wg (V, V) ag Riemannian Norm positions (O), O2, . . . . O). d 0264. Using the permutrix, the Quanton can therefore Riemannian inner product tg ge (u, v) = (u, v) maintain the probability of an element being at a position following another element without specifying an exact posi 0258. The mapping from point on sphere to tangent space tion, thus relative information without absolute information as the Riemannian logarithmic map: is captured. Thus, the permutrix represents the X data items that are in the Quanton. The output of the Quanton, item 31 of FIG. 7, after the computation model is executed, is the 8 model of the permutrix, II, as well as the hypothesis and data TSs - T = (TS pCos(0)sino, that go together, P(III) .x). In other words, raw data and its underlying model and results are probabilistically induced as 6 as Sin(6) for limo Sin(0) = 1:8 = d. (p,(p. T)T outputs. 0265. In order to represent arbitrary distributions over large X, given that X is factorial in size, the permutations of where d(p, TS) is the geodesic distance between any point X are referenced, not stored, by an analytic relationship and a point of interest. given by the index on the Fibonacci spherical lattice and the 0259. The inverse transform is defined herein as: Lehmer code representing the n-th index permutation at that point of the lattice. In this way, a uniform distribution over the X data items, for any size of X can be represented, where s T a is the permutation and U(O) is the uniform distribution T - TS = pCos(IIT II.) -- |T|| Sin(T.I.) OVer O. p 12 1 where the 12 norm is the Riemannian norm IIT, SII for U(r) = vote SP-1 T SE-1 which is the geodesic distance between TS and p (a point of tangency). In this representation, each probabilistic mixture component is in its own unique tangent space. 0266 The structure of the Quanton encodes permutations 0260. The geodesic distance transformation is: d(p,q) and their relations to each other with respect to certain ->arc cos(p,q). primitive operations such as, but not limited to, Swap 0261 FIG. 6 depicts a Specific Tangent Point and Tan operations or addition operations. Each permutation is gent Space on the Quanton's Spherical Fibonacci Lattice— indexed by a lattice that tessellates an underlying manifold this figure shows a neighborhood indicated by item 24, of to which is associated a probability distribution, that implic the tangent point indicated by item 25, as well as its linear itly is a distribution, therefore, over the permutations. Subspace indicated by item 26. The points on the linear space 0267 Distributions can be complex directional statistics are indicated on item 26 as O'', o', o', and O' with their and can serve as a proxy for the density functional, which is corresponding points on the manifold of the sphere as O'', itself used, in this disclosure, as an approximating Surrogate O’, O., and O'. These sigma points form an ellipse for a quantum wavefunction, in which case each permuta that is the covariance, which means that statistics on the tion represents a state space configuration. tangent space are linear while being embedded on the 0268. This situation is illustrated in FIG. 8: Quanton n-sphere which is non-linear. This bridge between linearity Probability Path-Densities, with Lookup Table for Mapping and non-linearity is specifically taken advantage of in the Qubits to Classical Bits. In FIG. 8, item 32, we illustrate the US 2016/0328253 A1 Nov. 10, 2016 Poincare-Bloch sphere and use a special representation in Algorithms, in which we have a set of parameters, 40, that terms of line-segments with an open and closed endpoint for determine the structure of the Quanton, specifically the size each of the Superposed compositions of the basis of C. and of the permutrix as a permutation size, which may or may B: the line segment. In item 33 of FIG. 8, we depict the usual not originate from the training data, 41, the number of Qubit model in terms of the angular parameter. Note that the dimensions, the Fibonacci spherical grid and the choices of permutrix, which is a tangent point on the sphere, specifies control parameters that are required by the choice of prob the linear tangent hyperplane from which samples can be ability density model, 42: the choices of probability distri selected as well as approximating the nearby-random axis butions not limited to but preferentially include the von aligned hyperplanes of its neighborhood. Mises Fisher, Watson, Mallows and Gaussian distributions, 0269. Item 34 of FIG. 8 illustrates the tangent point, though others may be chosen as needed and any mixture T and the oriented segment, 1, that represents the geodesic thereof. distance with respect to 0 to another point and showing that (0274 The choice of Probability Model, 43, includes the interval of the point is somewhere between 0 and 1 with Bayesian models, Kalman Models and Quantum Models the orientation given by the filled and unfilled circles at the with a preferential use of Time Dependent Density Func endpoints. Finally, item 35 shows the rules defined by which tionals (TDDF) or Direction Dependent Density Functionals to determine if the line segment, based on 0 is deemed to (DDDF) using pseudo-potentials in lieu of formal quantum map to a classical bit, 0 or 1. The line segment is drawn to wavefunctions. While TDDF may be known to those skilled represent the Subspace of the Superposition and to emphasize in the art, DDDF is not known as it pertains to the use of the geometric specification of the Quanton and its relation directional statistics provide an vector direction for transi ship to the Qubit. tions, while, in TDDF, time provides the directionality. 0270. In other words, just as a Qubit represents a prob 0275. The Quanton data mapping, 44, will depend on the ability density, so too does the Quanton. The geometry of the chosen probabilistic model, 43 and the Table 5 shows the Quanton is a hypersphere though for illustrative purposes a preferred mapping models. sphere and circle have been drawn to depict lower dimen sional cases. However, the Quanton, unlike the Qubit, but TABLE 5 similar to the Qudit, represents a set of Subspaces in which each permutrix is a set of independent, mutually orthogonal, Lookup Table for Quanton Probability Density k-dimensional Subspaces labeled by an integer aligned with Quasicrystal Type Probability Measure the coordinate axes for efficiency: in the simplest case, the n-Sphere Von Mises, Kent, Watson, Gaussian k-dimensional Subspace is 1, but in more complex situations, n-Torus Bivariate Von Mises the Quanton represents Subspaces greater than 1. In the case n-Cube Uniform, Viterbi of the subspace in which the permutrix represents the binary n-Simplex Bayes, Markov-Chain, Viterbi 2-polygon Markov space 0.1, then the space is proportional to binary exponent Zonotope Any of the number of dimensions as is the usual interpretation. 0271 To make this clear, and to show that the Quanton is a complex data object of unique design, “FIG. 9: Topologi 0276 One model developed for quantum computing cal Structure of Quanton and Embedding Permutation State defines quantum circuitry in terms of topological models: Space as Orbitope on Sphere' illustrates the subspaces and these topological quantum computing models, in one the mappings. In FIG. 9, 36, a simple sequence of 4 Qubits instance, use permutations in an operation called “braiding is shown with the circle diagram of FIG. 8, 33 representing in order to perform computations. A braid operation can be the states. The structure of a quantum permutation register is represented by a matrix, called a generator, that acts on the defined and constructed by taking sets of Qubits, and an qubit space to perform the logical operations of a quantum example of a register with pairs of 2-qubits at time in a gate. A braid can represent a product of the generators that quaternary Qudit register of 4 pairs of Qubits is shown, with encode the individual braid operations. their respective vector subspaces, in 37 of FIG. 9. Finally, 0277. Therefore, using the permutation representation of for the set of four pairs (i.e. 4 Qudits) there are 4-factorial the Quanton, the braiding operation is a path through the possible sequences (4=24), with 24-factorial possible order permutation space driven to a solution path by learning ings of the permutations. using the evolution of the estimation of distributions of 0272. The permutations are represented by integers probabilities. The details are presented in the foregoing. which represent the Subspaces and the integers are mapped 0278 FIG. 10, 46 receives the noisy partial data, which to coordinates which define the Permutohedron, a permuta can include image data, directional data, probabilistically tion Zonotope, as shown in 38. However, as described by associated data, partial graph data, time series and state embodiments of the present disclosure, there is an extended representation data. Computation on the Quanton data struc compact representation of the Permutation polytope as a ture, constructed as 45, and explained further in FIG. 11, sorting-orbitope, which is derived from the Birkhoff-poly produces a solution to a tomography of the data relation tope, an equivalent representation of the Permutohedron, but ships, 47, in the form of an estimated distribution and an the Birkhoff-polytope is further extended to the sorting output from the tomograph of the sought results, item 48. orbitope which provides n(log(n)) computation complexity Therefore, the principle components are a density model, in contrast to the n complexity of the Birkhoff-polytope. with a set of parameters, and a database of valid observa The final orbitope for the Quanton is schematically illus tions, though if the database is not available, the system will trated as the output in 39 of FIG.9, wherein the details of its be shown to converge to an approximate solution. construction will be explained later in this disclosure. (0279. In the preceding FIG. 10, we have provided the (0273. The operational model is fully shown in FIG. 10: high-level model description, with reference to the Quanton Quanton Operational Model for Estimation of Distribution at FIG. 10, 45, and now in “FIG. 11: Flowchart for Quanton US 2016/0328253 A1 Nov. 10, 2016 Data Structure Construction', a description is provided with 0286 Given the transformation, and for sake of explana fine granularity, of the operational steps and flow chart to tion, n 3 is used to simply for notational convenience with construct the Quanton. the direct generalization to all hyperspheres of dimension of 0280. Now referring to FIG. 11, training data, item 49, is l. represented as permutations by the method of Table 3 of the 0287 Also, the radius is assumed to be unity for the unit present disclosure and these resultant permutation patterns hypersphere in order to simplify the foregoing without any are resolved to their index locations on the Quanton as, for loss of generality to the higher dimensional cases. example, the k-th index of Fibonacci series which is the k-th 0288. Now, the arc-segment (for n=3 dimensions) is Lehmer Code representing the permutation (pattern), item given by: 50. The Orbitope, item 51, is constructed as per the methods of Table 1 of the present disclosure and optimally, as the sorting polytope, as shown later in the description for FIG. 23. The probability density, item 52, is chosen using the method of Table 5 and a step of compilation comprising did writing the Binnet Formula and the list of its indices in a ds = 1 + Sin2 (e)(2) de basic data structure, item 53, such as set of relational predicates or standard relational database system using the index as the primary key as well as an estimate (for example using a Bayesian recursive filtering process), item 61, of the Therefore: maximum a-posterior (MAP) probability distributions of the 0289. A slope of a spiral curve on the sphere is defined to training set. Once this has reached a fixed point, as per the be a constant given as: methods of FIG. 5, then the calibrated Quanton is returned. Now, with resolution which is a generalization of the Quan ton using noisy partial data to produce results, the noisy data, did item 56, is also represented as permutations by the method of Table 3 of the present disclosure: in this case, however, a filtering process may be applied to reject data using a variety of user chosen methods, to reduce the data presented for the 0290. Therefore, (p=k0 Quanton, item 57. An estimation function, item 58, such as Permutation embedding into surface of Hypersphere: IR't' a recursive Bayesian filtering process to produce an output Dimension d: d=(n-1)-1=n-2n+1-1=n-2n Space: IR n that can be reduced, using a Gaussian for example, as a Permutation Embedding into hypersphere of dimension: Kalman Filter, to allocate the distributions of the noisy data (n-1) to the Quanton by projecting onto the sphere, if a spherical 0291 Radius of hypersphere: V(n-1) geometry had been chosen as the embedding geometry, item 0292 All permutation matrices on n objects belong to the 60. The estimate may be updated and then the nearest surface of a radius: V(n-1) on hypersphere S rt-2n in Rn quantized points on the lattice returned as candidate Solu tions, with a conversion back to the data from their permu 0293 Permutation space: IP tational representation by the methods of Table 3. 0294 Center of mass of all permutation vectors: This general process is used to locally update the Quanton probability distribution with respect to a set of training data and is a typical well understood idiom to those skilled in the art of machine learning with probabilistic methods and MAP computation. 0281. The construction of the Hypersphere can be described as follows: 0295) Let the index be qr so if the position q has a value, 0282) S is embedded in IR'; dimensions=d-n-2n+ it will be r. As an example, IP = IP means that the first 1=(n-1); S''-xe IR"Ix"x=1}. position in the permutation vector is set to value 1. For n (0283. The hypersphere $" is represented by a set of vectors, there are, therefore, (n-1) vectors with (1,1) per hyperspherical polar coordinates in n-dimensions as follow: mutations. 0284 Let IR"={x1, x2, x . . . x} be the Cartesian 0296. Therefore coordinates in n-dimensions Therefore: 1 1 1 0285 C IP = ii), P- (n-1)!= (n+1) X = r. sin(6-1)sin(62) ... sin62 sin6 The radius of the hypersphere is: X2 = r. sin(6-1)sin(6–2) ... sin62cosé X3 = r. sin(6-1)sin(6–2) ... cosé2 X3 = r. sin(6-1)sin(62) ... cosé2 x = r. cos(6-1) US 2016/0328253 A1 Nov. 10, 2016 20 0297. The nth-Fibonacci Number is given by the Binet 0309 4. Apply the von Mises Fisher or other (Mal Formula: lows, Watson) probability model to the hypersurface on which the hypercurve is embedded. 0310 5. Therefore, each index on the hypersphere, as -- given by the Fibonacci number at that point, represents V5 2 2 the respective permutation and, therefore, also a locale with the indexed directional distribution indexed by the area patch around the point. 0298. The motivation is drawn from the design of a 0311) 6. Use unit spherical coordinates in IR''. probability density estimator for data of arbitrary dimension, 0312 7. Associate the indices to the index terms of the more generally from Source coding. The following proce Fibonacci spherical map. Algorithm From Hypersphere dures define a PDF estimation model that maps (n-1)- dimensional samples onto S". The PDF estimation is back to Permutation Polytope: S-> IP performed in the new domain using a kernel density derived 0313) Given an arbitrary point, in R (on or near the estimation technique. A Smoothing kernel function repre surface of the hypersphere), we find the closest point that is sents each sample at its place and sums their contributions. a permutation. As per the teachings of Plis et al., an arbitrary point on the hypersphere can be matched to the nearest 0299. This is equivalent to a convolution between the permutation point by using a minimum weighted bipartite kernel and the data; therefore, the derived convolution result represents the estimation. Note that convolution is used only matching algorithm in which a cost matrix between the input to compute the estimate, not to handle the dependency. In the point and its nearest neighbors is minimized to produce the VMF transition model, the marginalization can be performed best nearest neighbor choice. using the fact that a VMF can be approximated by an angular 0314. However, in the present disclosure, the situation is Gaussian and performing analytical convolution of angular simpler because the Quanton generator is analytic and the Gaussian with Subsequent projection back to VMF space Fibonacci lattice tessellates (i.e. quantizes) the space in a (Mardia & Jupp). regular index so that any arbitrary point at any quasi-cell is 0300. In related work, the teachings of S. M. Plis, T. Lane easy to identify and compute a distance in order to choose and V. D. Calhoun, “Permutations as Angular Data: Efficient the best lattice (i.e. permutation) point. In other words, the Inference in Factorial Spaces,’ 2010 IEEE International index points of the Fibonacci lattice polytope quantizes the Conference on Data Mining, Sydney, N S W. 2010, pp. space and, therefore, the nearest point on the Fibonacci 403-410 are incorporated here by reference in its entirety. curve, to an arbitrary point, is found using the minimum However, while the Plis et al. teachings show a way to distance along the curve to the arbitrary point. If the arbi embed the discrete permutation structure into an n-Sphere, trary point is a perfect midpoint bisection (for example, they do not show how to make the process efficient and between two lattice points), then randomly select the nearest general by using an indexed quasicrystalline lattice. Such as neighbor lattice point. The procedure for solving for the the way of the present invention. The present disclosure distance to the curve is as follows: provides for one way of several ways, but generally all using 0315 (1) The Fibonacci curve spirals down the sphere a number-series, such as the Fibonacci Series, to build an from North Pole to South Pole. It remains a constant indexed embedding space on the manifold that can gener distance from neighboring windings for Z (-1 at South alize to other kinds of geometries, such as the n-Torus, and Pole to +1 at North Pole): not just the n-Sphere. 0316 in a constant defining a given spiral 0301 Algorithm From Permutation Polytope to Hyper 0317 k=Vnt sphere: P --> S 0318 r V, 0302) The general algorithm to map from the vertices of the polytope to the surface of a sphere is given by the 0319 0=k Arcsin(z) following three simple generic steps: 0320 x=r-Cos(0) 0303 1. Given polytope, IP transform integer coordi 0321) y=r-Sin(0) nates to origin. 0322. It makes k/2 revolutions around the sphere, with 0304 2. Change basis by transforming the vectors to each winding V4t/n from adjacent windings, while the slope spherical coordinates in IR''. 0323 (2) Set k such that the inter-winding distance 0305 3. Rescale by the radius to get unit vectors. covers the largest tile on the sphere. Now, the preferred embodiment of this general procedure 0324 (3) For every point in the main set, calculate the presented below avoids re-scaling and many other compu theta of the nearest point on the curve, and index the list of tations as otherwise shown by Plis et al: points by those numbers. For a given test point, calculate its 0306 1. Transform the ppermutation polytopepolvtope II”,IP into theta of the nearest point on the curve, which is the nearest a Lehmer Code index, so any code from 0 . . . n neighbor distance limit, and find that point in the index. provides the permutation, P-th index: 0325 (4) Search outward (in both directions) from there 0307 2. Associate each Lehmer code to a unique index (i.e. from the index), to theta values that are as far away as from 1 to N. your current nearest neighbor using linear Scanning. After 0308) 3. Let this p-th index, be the point on the reaching that limit, if the distance to that neighbor is less Fibonnaci hypercurve on the hypersurface of the hyper than the distance from the test point to the next adjacent sphere. Use the Binet formula to compute this P-th winding, one has obtained the nearest neighbor. If not, jump integer. the theta value by 2pi and search that winding the same way. US 2016/0328253 A1 Nov. 10, 2016 0326 (5) Then, the arc length, s, on an n-sphere is 0334 FIG. 14 depicts a Recursive Evolution and Esti computed as follows: mation Flowchart. 0335 The model has two main parts: 0336 prob(h,Ih); which describes the stochastic evo lution of the hidden permutation s = Rc-acosts. ( ) 0337 prob(o,x); where o, is the noisy observation of the hidden permutation 0327 (6) Transform from point on surface of hypersphere to nearest lattice point back to nearest permutation using the marginalization for this equation can be computed with by relation that the index of the nearest lattice point produces approximating the VMFs with angular Gaussians, convolv the Lehmer code for the permutation at that point; however, ing analytically, and projecting back to VMF space. 0328 (7) In the case that the point on the surface is 0338 All the inference steps operate only on S repre equidistant from multiple lattice points, choose the nearest sentations of permutations, avoiding unnecessary transfor point at random and return the permutation at that point. mation overhead. 0329 Quanton Just-In-Time Calibration Algorithm. 0339. The transition model is defined by a possibly mixed Now, referring to FIG. 12: Detail of Quanton Calibration conditional probability distribution over the permutations Construction Processing, 64, starts with the structure model h(o'o''), and might be that elements belonging to two that defines how the permutation if generated. The permu different data are swapped with some probability by a tation structure is defined in terms of the type of permutation mixing event. The observation model is defined by the and how it represents a given problem or how it represents distribution P(h'o'), which might, for example, capture a Some data as well as how one permutation leads to the next distribution over a specific data feature that is observable for permutation. Element 65, the state transition model, speci each data element. fies the type of probabilistic transition between permutations 0340 Given the distribution h(O(t)|Z(1), . . . . Z(t)), we and may be learned from data or provided a-priori: the recursively compute the posterior at time t+1, h(O(t+1)|Z(1), model, in effect, describes a transition state vector. The . . Z(t+1)), in two steps: a prediction/rollup step and a Temporal Model, 66 specifies a function that determines conditioning step. Taken together, these two steps form the how the permutation or their allocation to the manifold Forward Algorithm. The prediction (induction)/rollup step changes over time. The Measurement Model, 67, provides a multiplies the distribution by the transition model and mar weighting or Scaling factor and a function to measure ginalizes out the previous time step: distances between permutations: in effect, it provides a 0341 Closely related permutations have more informa measurement vector of lower dimensionality than observa tion in common and hence more compressible than the tions at the State vector. For example, the Hamming Distance permutations that are poorly related. This information theo could be used. retic framework can be adapted to individual contexts by 0330 Algorithm Parameters, 68, contain any constants or accommodating prior knowledge about rankings in those heuristic or concentration or other parameters needed by the settings. The specific types of measurements that can be probabilistic model, as given by the choice of the probability made are: density function, 69. 0342 (1) the measurement of the extent of non-overlap 0331. The probabilistic model is composed of the obser in the two permutrixes: vation, transition and posterior 70 probabilities in accord 0343 (2) the measurement of disarray of its overlap with the standard definitions to those skilled in the artin, for ping elements in permutrixes: example, Bayesian inference. 0344 (3) the displacement of the positions (ranks) of 0332. Now, the items referred in FIG. 12 as 71, 72,73, 74 these elements in permutants; and, and 75 collectively constitute a process of calibration, also 0345 (4) A surrogate for entanglement based on the known as machine learning. The Quanton is calibrated using size of the permutons and the alignment between their some known reference data as the observation: the calibra Subsequence avoiding patterns or their subsequence tion is based on iterative estimation of the probability isomorphisms. density distribution between the permutations on the mani 0346 Continuous parameters (which are projections to fold until they correspond to an acceptance threshold in the n-sphere) can necessarily only be stated only to finite which the observed training data is satisfied. The marginal precision (due to the Fibonacci lattice structure and permu ization step, item 71 produces a new distribution and this is tations). MML incorporates this in the framework by deter used, as in the traditional case, in an update estimation mining the region of uncertainty in which the parameter is process, item 73, that is recursive (like recursive Bayesian located: The value: estimation) until the data are learned, item 72 and calibration is completed, 74, 75. 0333. Now, referring to FIG. 13A: Topological Structure the space of the 4-Permutation Orbitope. We note that the structure of the permutation as movements of the numbers defining a permutation, forms a certain configuration. In one example shown herein, a four permutation may start at 76 gives a measure of the Volume of the region of uncertainty and results at 77. in the parameter 0 is centered. FIG. 13B illustrates the topological structure of the space of (0347. When 0 is multiplied by the probability density, the 5-Permutation Orbitope. In one example, a five permu h(0) gives the probability of a specific 0 and is proportional tation may start at 78 and end at 79. tO: US 2016/0328253 A1 Nov. 10, 2016 22 be exemplified herein using the von Mises-Fisher distribu tion. Further, the initialized parameters can include the relative percentage of hidden permutations. The Quanton can be represented as a Birkhoff polytope and/or as a hypersphere. 0348. This probability is used to compute the message 0356. In step 81 of method 1400, observation and tran length associated with encoding the continuous valued sition modes are defined. The transition model is defined by parameters (to a finite precision). a possibly mixed conditional probability distribution over 0349 The vector of parameters 0 of any distribution the permutations h(olo'''), and might be that elements given data, D. A prior on the hypothesis, h(0), is chosen (e.g. belonging to two different data are Swapped with some Gaussian). The determinant of the Fisher information matrix probability by a mixing event. The observation model is is also necessary to evaluate: F (0) of the expected defined by the distribution P(h')lo'), which might, for second-order partial derivatives of the log-likelihood func example, capture a distribution over a specific data feature tion: - C (D10). that is observable for each data element. 0357 Given the distribution h(O(t)|Z(1), . . . , Z(t)), we recursively compute the posterior at time t+1, h(O(t+1)|Z(1), I(0, D) = Sloga.) - lo h(0) |- f(DO) + f . . Z(t+1)), in two steps: a prediction/rollup step and a VIf (O) conditioning step. Taken together, these two steps form the Forward Algorithm. The prediction (induction)/rollup step 0350 Note that p is the number of free parameters in the multiplies the distribution by the transition model and mar model. ginalizes out the previous time step: 0351 Convex Relaxations for Permutation problems 0358. In step 82 of method 1400, the posterior of the state Given: pairwise similarity information A, on n variables. is generated. 0352 Let It be a seriated permutation ordering (poset): 0359. In step 83 of method 1400, the marginalization for then, the stochastic evolution of the hidden permutations is cal A decreases with li-ji (aka R-Matrix). culated. For the von Mises-Fisher distribution, this can be Define the Laplacian of A as L(A)=diag(A1)-A calculated using the approximation that the convolution The Fiedler vector of A is: f(A)=argmin x'L(A)x; 1" x=0; between two von Mises-Fisher distributions represents the |x|=1 marginalization. xe IR" is in PH " if and only if the point Z is obtained by 0360. In step 84 of method 1400, the observations are sorting the coordinate-wise values of X in descending order generated. The observations can be generated by first con satisfies the equations: verting the permutation to its matrix operator representation. Next, the matrix operator representation can be used to project the matrix operator representation onto the Surface of k k n(n + 1) X 3, s (n + 1 - i)W ken i a hypersphere in the Subspace spanned by orthogonal basis i=l i=l vectors that represent the subspace of the Birkhoff polytope to generate a mean vector of the distribution. prob(h,h)-VMF(hh...K.) (see element 81) 0361. In certain implementations of step 84, the mean prob(o,h)=VMF(oh, K) (see element 81) vector of the distribution is used to generate random draws prob(ho)-VMF(hu.K.); posterior model (see element 82) from von Mises-Fisher distribution to generate a statistical Use the new observation to update the estimate through the sample to generate a modified permutation distribution on observation model (see element 86): the Birkhoff polytope. 0362. In certain other implementations of step 84, the mean vector of the distribution is used to calculate, for We use probability distributions for the transition, observa various object locations, flip probability matrices. The flip tion, and posterior models. probability matrices are then used as a probability model to 0353 When a partial observation of o objects becomes determine the changes to the permutations represented on available, the dimension of the unknown part of the the hypersphere to generate modified permutations. The observed permutation matrix O is reduced from n to (n-o). modified permutations can then be converted to modified Therefore, the algorithm provides for the abduction (infer permutation distribution on the Birkhoff polytope. ence) of a hidden permutation from its noisy partial observ 0363 The modified permutation distribution on the Birk ables. hoff polytope can then be mapped onto the nearest vertices 0354 FIG. 14 shows a flow chart of one implementation of the hypersphere using the inverses of the orthogonal basis of a method 1400 Quanton model simulation. vectors that represent the subspace of the Birkhoff polytope 0355. In step 80 of method 1400, various input param in order to solve for the nearest vertex of the permutations eters are initialized. For example, the initialized parameters matrix, wherein nearest is determined using a Euclidean can include the maximum the stopping criteria (e.g., the distance. maximum number of iterations), the dimension the structure 0364. In step 85 of method 1400, the partial observations model, the state transition model, the directional probability are performed to generate a new observation. density function (e.g., the mean and the variance for a Gaussian distribution, and for a von Mises-Fisher distribu 0365. In step 86 of method 1400, the new observation is tion the mean and the value of K). While many directional used to update the estimate through the observation model: probability density functions can be used, method 1400 will US 2016/0328253 A1 Nov. 10, 2016 0366. In step 87 of method 1400, an inquiry is performed operation of a complicated set of quantum gates) can be whether the stopping criteria have been reached. If the directly emulated by paths in the Quanton. Specifically, we stopping criteria have not been reached, method 1400 pro address the fact the both non-computable numbers and ceeds from step 87 to step 83. Otherwise, method 1400 is computable numbers result in abductive hypotheses and complete. 2 deductive computation while machine learning that adjusts 0367. An arbitrary point in R' corresponds to a point on the probability transitions results in inductive learning. the hypersphere and therefore to a permutation. Let TS be 0375 Included herein by reference in its entirety is the any point in in R "therefore this point must correspond to teachings of Michael A. Nielsen and Isaac L. Chuang. 2011. a point on S which is to say, on S''' Quantum Computation and Quantum Information: 10th 0368. In terms of a PDF on permutations the VMF Anniversary Edition (10th ed.). Cambridge University Press, establishes a distance-based model, where distances are New York, N.Y., USA. geodesic on Sd. However, while the VMF works for isotro 0376. It must be appreciated that a group (register) of n pic distributions the lack of a closed-form conjugate prior bits can contain only one of 2" different numbers at a given for the concentration parameter in the VMF distribution time and that also for any constant, k>1, and any natural complicates posterior inference and the concentration is number n, that the following holds: k"sn'sn'. Therefore, for usually parameter is arbitrarily fixed and not inferred from example, consider a 4 Qubit state which contains 2-16 the data. possibilities, as contained in the permutation, 4–24 possi 0369. The addition of a tangent space model allows for bility state. Therefore, it is clear to see that the state space anisotropic distributions and can act as a correction to other of permutations scales astoundingly quickly compared to distributions on the sphere while adapting the model com traditional qubits and lastly qubits scale fast than the State plexity to the data, as driven by the data. The inference space of simple bits (or even probabilistic bits, such aspbits algorithms are parallelizable while leveraging the geometry known to those skill in the art). of the sphere and the Fibonacci lattice. 0377. Just as real numbers can be seen as a subset of the 0370. In the present disclosure, in the tangent space complex numbers, it is possible to rewrite Qubits in terms of model, each mixture component exists in its own tangent real numbered Rebits as the subset of quantum states of space, and each tangent space is unique. Qubits that have density matrices with real entries in the 0371. Now, referring to FIG. 15: Polyhedral Structure the computational basis. Based on the teachings of Rudolph, T space of a few Braid Group and Permutation Orbitopes and Grover, L., “A 2 rebit gate universal for quantum (Zonotopes), the different permutation layouts are shown computing, we can state that n+1 Rebits (real bits), with the which identifies the paths or relations in transitioning from additional coding for the real and imaginary parts and an one permutation to another. The purpose of FIG. 15 is to initial state of a n-qubit state can represent any state: draw attention to the structure of the geometry with respect to the ordering of the permutations. This is important because permutation ordering in the Quanton model pro vides the transition ordering between State spaces. For (b) = X re', y) example, in 88, label a, sits at a “square' arrangement of veZ; permutations and is adjacent to a hexagonal arrangement of permutations. However, in 89, label a, sits at a “square' 0378. The corresponding encoded state is where IR) = 0 arrangement of permutations and is adjacent to an octagonal ) and II) =1) is just notation that makes explicit the real arrangement of permutations at label b. In 90, several and imaginary parts in the encoding: different polygonal locales are shown which tile the surface of the Quanton. 0372. Now, referring to FIG. 16: Polyhedral Structure Illustrating Quantized Polyhedral Probability Density Dis tribution, we can see that the lighter shaded patches of equal probability density 91 are interpreted to mean that all the permutations covered associates the external semantics as 0379 Changing a Qubit state by a global phase is observables to the State spaces as permutations. As shown in reflected by the existence of infinitely many Rebit states this disclosure, this can be learned by the Quanton using encoding the same Qubit state. In simple terms, the resulting Bayesian like approaches. formula is similar to the complex case, except that an overall 0373 FIG. 17: Polyhedral Structure Illustrating Projec absolute value sign is missing. Hence, the two-Qubit state tion of a Single Quantized Polyhedral Probability Density. In may be rewritten as a three-Rebit state in the Rudolph and the case of a learned State represents a trajectory, or a Shor model. This is important because the Quanton is an sequence of causal states on the Quanton 92, these states can approximation model of quantum systems in the real-am be unrolled 93 out in a sequence and the set of the states and plitude variant of quantum theory. their transitions constitutes a model of causality as learned 0380 Any given Quantum model state may be trans by standard probability based learning methods. formed into a unique Quanton which serves as the normal 0374 FIGS. 13.14.15 and 16 illustrate the general idea form of the quantum state. The normal form depends on that permutations are embedded in patterns according to a algebraically independent Stochastic Local Operations and permutational pattern sequence and that this is associated to Classical Communication (SLOCC) which are the equiva a geometry and topology given by the quasi-lattice gener lent groups under probability on the Quanton. These invari ating procedure. At this point, we now show how any ants, therefore, constitutes a family of orbits parametrized by Quantum Gate and circuit can be represented by the per these probabilities. By one embodiment, this idea is inter mutation pattern and that any Quantum Gate process (i.e. the preted to set the stage for entanglement representation by a US 2016/0328253 A1 Nov. 10, 2016 24 model of equivalent group sharing Such that for example, 0388 Recall that a transposition is a cycle of length 2, given that at least one group (i.e orbit) must be shared, also hence, any permutation can be written as a product of known as a pattern avoiding sequence that that is share or as transpositions and that therefore any permutation can be a shared pattern conforming sequence, that a N-Qubit written as a product of disjoint cycles, it is sufficient to write entanglement state is defined as (N-1) entangled families. each cycle as a product of transpositions: So for N=4, there are 3=6 families and for N=3, there are 2 families (i.e. two non-isomorphic patterns). (n1, m2, ..., ni)–(n1,n)(n1,n-1)... (n1, m2) 0381 Procedure for Mapping Quantum Logic Circuits to 0389. Note that there are several ways to write a given Permutation Representation: permutation as a product of transpositions. For instance, let 0382 Quantum circuits, following Nielsen and Chuang P=(3, 5, 4)(1, 2, 3, 4). Then applying formula O0183 above as earlier cited and incorporated herein, can be seen as n-bit to each of the cycles (3, 5, 4) and (1, 2, 3, 4), we get P=(3. reversible logic gates. These logic gates can form compli 4)(3, 5) (1, 41, 3)(1, 2), so P is a product of 5 transpositions. cated functional circuits, however, they are all equivalent to On other hand, we can first write P as a product of disjoint permutations acting on the 2'' possible states of bits that pass cycles (in this example there will be just one cycle) and then through them. Therefore, specifying a reversible logic gate use (T). This gives P=(3, 5, 4)(1, 2, 3, 4)=(1, 2, 5, 4)=(1, is defined as specifying the mapping that it performs in terms 4)(1, 5)(1, 2), so now P is a product of 3 transpositions. of permutations. In quantum computing, these computa tional states are represented by the quantum mechanical 0390 The C-NOT gate 94 is a universal quantum gate as States. explained in Nielsen and Chuang. FIG. 18A illustrates the 0383) Define a finite set S having n-elements which are mapping of C-NOT circuits to permutations written in well defined quantum mechanical states: therefore, any permutation cycle notation. The product of cycles of a reversible logic is expressed by introducing a permutation permutation is equivalent to reversible logic. The n-bit operator OeS' where O is a one-to-one mapping of S onto CNOT gate (i.e., controlled-NOT or C-NOT) operates on itself (i.e. a permutation). any one bit of n while the control bits are the other (n-1) bits 0384 Let the set of bits be represented by the lower case or (n-2) and so on. For n=3, there 9 possible C-NOT gates letters as follows: a,be(10); and let the set of sets be 100 as shown in FIG. 18B, where each permutation opera represented by i (such that i=0, 1, 2, 3) which are interpreted tion between inputs and outputs is actualized by a unique as the four computational states expressed by two input bits product of transpositions. of the measurement gate. Therefore, we can write the 0391) For quantum computation, we need universal specification of the logic as follows: reversible gates. One of such gates is the Toffoli Gate 95 is a 3-bit gate with 3 inputs (a, b, c) and corresponding outputs TABLE 6 (a", b', c'). 0392 The relationship between the input and output is Quantum Gate State Transition Mapping (Truth Table the following a control b target li input > a control b target li output > 0393 a'—a O O O O O O O 1 1 O 1 1 1 O 2 1 1 3 0395 c'=ceD(a-b) 1 1 3 1 O 2 0396 Here 6 denotes the Exclusive OR (i.e. addition modulo 2) and denotes the AND function. The variables a 0385. This representation is interpreted in this disclosure and b are control bits where c is called the target bit. As seen as an element of the permutation group S. Now, the from the logic, the control bit has no change while passing important idea is that any universal gate, such as the C-NOT through the gate. However, the target bit flips if both control gate can be represented as a one-to-one mapping in S. So that bits are logic 1. The permutation corresponding to a Toffoli the above table can now be represented solely by permuta tions. S contains 4 permutations that implies 24 different Gate gate is, logic gates for a 4-state system. As noted earlier in the present disclosure, any N-level quantum system can be 000 OO1 010 011 100 101 110 111 permutated in N computationally distinct logic sets: these ''' 000 001 010 011 100 101 111 110 permutations of degree N form a group of order N under the composite operation of permutation. 0386 Therefore, in what follows is illustrated how the The transposition representation in binary is (110, 111) present disclosure represents a virtual machine for Quantum computation by (probabilistic) Permutation (aka Computa which in decimal is (6.7). tion by Permutation). 0397) The Peres gate 96 is a combination of the behaviors 0387 Any permutation group, S" is the collection of all of a Toffoli and C-NOT gate as illustrated in FIG. 18A in 97 bijective maps of the set of permutations (p: added to 98 to result in 99: therefore, it can realize more complex logic Such as addition, Subtraction etc. The Peres gate is a 3-bit gate which maintains relationship between the See S input and output as follows: of interval S=(1,2,3 ... N), with composition maps (p as the group operation over O. US 2016/0328253 A1 Nov. 10, 2016 25 0401 The permutation corresponding to this gate is, processing unit” referenced in U.S. Pat. No. 6,256,656 B1 shows how modular integer representations can be produc tively used for enhancing computing basic irreversible com | 000 OO1 010 011 100 101 110 111 puter operations. The difference between our use in the CPERES F | 000 001 010 011 110 111 101 100 present disclosure of a similar concept of modular integer representations is that we use the Landau function to create a numbering sequence for permutation operations, and that 0402. In decimal, the corresponding cycle has a length of it is with these permutation operations that enhancing com 4 and can be decomposed in 3 transpositions as: O, is puting basic fully reversible computer operations that are (4.6.5,7)=(4, 7)(4, 5)(4, 6). also fully parallelizable as seen in 102 of FIG. 19. Further 0403. Any reversible logic can be rewritten in terms of a more these reversible computation by permutation opera permutation group: therefore, we use the concept of the tions enable high speed emulation of quantum computing permutation cycle works the building block of reversible circuits, and, when combined with the probabilistic model, gates. For every n-bit reversible logic there exist a unique enable very fast approximate quantum-like computation as a permutation of degree 2". Therefore, all properties and the general approximating probabilistic Turing machine model. operations on permutation groups apply or implement reversible logic operations. Therefore, all gates can be 0408 FIG. 20A shows a flow chart of a method to choose defined by permutations, such as other useful gates such as a size and design of the permutations of the symmetric group the Fredkin gate which is also known as a controlled of S, that provides optimality. All steps are fully paralleliz permutation gate. Further, all gates are general in that they able. The method uses the Landau numbers that, when can represent n-bit operations, such as the n-bit Toffoligate combined together with a corresponding design of the also known as the Multiple Control Toffoli (MCT) gate permutation operator design, provide that the operational which is a reversible gate that has n inputs and in outputs. semantics of transforming permutations to ensure that the 04.04 Since permutation mimics reversible logic, there transforming permutations are bijective and reversible. For fore, the compositions of permutations are used to define example, by using the Landau numbering to design the operations performed by reversible gates or quantum circuits permutations, an addition operator can be written as an composed ot the reversible gates given that any reversible operation between permutations where the numbers must fit gate is expressible by corresponding permutation, therefore within the Landau numbering. The result is that the Quanton any quantum circuit is simply a product of disjoint cycles can perform all the traditional instructions, such as addition and necessarily as a product of transpositions. We will show, and subtraction, in terms of the movements and operations in the present disclosure, that for the primitive operation of in the design. Using this design, the Quanton provides a a full-adder Quantum circuit, that certain know-how is model of computation for the instructions, which can rep needed in order to build quantum circuits by permutation resent the Smallest Turing computing model. operators corresponding to circuits. The naive approach of 04.09. In step 103 of method, partitions are determined for simply composing products of transpositions is the ground the dimension n of the Symmetric Group S. The partitions work on which we build general efficient and simple com are determined as a sum of positive natural numbers. This putation operators on the Quanton. step and the remaining steps of method are exemplified 04.05 Any irreversible function is mapped to a reversible below using functions written in Haskell code. The method function just by adding constant inputs and garbage outputs. as illustrated in FIG. 20A makes possible for performing However, the constant inputs must have certain values to arithmetic with a subset of the permutations in the Symmet realize the functionality in the outputs and the garbage ric Group S. To achieve that end, the Landau number can outputs must have no impact on the end computation. The be used to obtain an ideal candidate for the order of the minimum number of garbage outputs required to convert an largest cyclic Subgroup of Sn. Relatedly and as is described irreversible function to a reversible function is log(Q), by method illustrated in FIG. 20B, cyclic groups are iso where Q is the number of times an output bit pattern is morphic with Zn which can be decomposed as a product of repeated in a truth table representing the irreversible func subgroups. These products of subgroups each have orderp., tion. In the case of a full adder 101, as shown in FIG. 19, it wherein p a prime factor of n, and the Subgroups can be used is clear to one skilled in the art that the output bit patterns to emulate operations with these permutations using modu 01 and 10 are repeated three times and have the highest lar arithmetic in order to parallelize and rapidly perform number of occurrences. Therefore, for Q–3; at least log(3) these calculations. =2 garbage outputs and one constant input are required to 0410. To initialize method of FIG. 20A, a maximum make the full adder function reversible as shown in the FIG. integer size can be set (e.g., the maximum integer can be set 19. to 100). 0406 Procedure for Computing in Permutation Repre sentations: 0411. In step 103, the Landau numbers are calculated 0407. The Quanton design and procedure for computing using the determined partitions. Landau number can be in the permutation representation is presented that includes calculated using partitions of n as a sum of positive natural and extends the teachings of Deleglise, Marc, Nicolas, numbers. The Landau number can be determined using the Jean-Louis, and Zimmermann, Paul. “Landau's function for lowest common multiple (LCM) of the elements of a par one million billions.” Journal de Théorie des Nombres de tition. Here, the least common multiple of two integers a and Bordeaux 20.3 (2008): 625-671. Prior art in the teachings of b, which can be denoted by LCM(a, b), is the smallest Carroll Philip Gossett, Nancy Cam Winget in their patent positive integer that is divisible by both a and b. "Apparatus and method for extending computational preci 0412. The partitions and the Landau number in steps 103 sion of a computer system having a modular arithmetic and 104 can be defined in Haskall code, in one example. Any US 2016/0328253 A1 Nov. 10, 2016 26 other programing language could also be used. Also shown 0442 1.2,0.6.7.3,4,514,15,8,9,10,11,12,13 below are examples of function calls and the output gener 0443 2.0.1, 7,3,4,5,6, 15,8,9,10,11,12,13,14 ated by the function calls. 0444 Finally, in step 109, computations are performed in the cyclic Subgroup of the Symmetric Group S using classic Example computations and modular arithmetic or using permutation 0413 landau 19 operations on a quantum or reversible computing frame 0414 (420, 3,4,5,7) work. 0415 landau 42 0445 FIG. 20B shows a flow chart of a method to choose 0416 (32760, 5,7,8,9,13) a size and design of the permutations using an emulation of 0417. In step 103 of method in FIG. 20A, a generator and the symmetric group of S. The method as illustrated in FIG. a template are defined for a cyclic group of the order given 20B performs a similar function as method of FIG. 20A, by the Landau number using the first partition that has a except method (FIG. 20B) is faster and can be parallelized Landau number greater than n. The generator uses the first because the modular arithmetic happens independently in partition for which Landau number is above n. As shown below, the generator and the template in step 1230 can be each Subgroup of which Z, is a product. defined in code. Also shown below are examples of function 0446. In step 110 of method as depicted in FIG. 20B, the calls and the output generated by the function calls. partitions for the dimension n of the Symmetric Group S. are emulated using factors of pirmorials. This emulation is Example based on an isomorphism of the cyclic subgroup of S, with Z. As stated above, cyclic groups are isomorphic with Z. 0418 “landau function 42 which can be decomposed as a product of Subgroups. These 0419 (60, [3,4,5) products of subgroups each have order p?, wherein p a 0420 “landau function 100 prime factor of n, and the Subgroups can be used to emulate 0421 (105,3,5,7) operations with these permutations using modular arithmetic 0422, 1Generator 100 in order to parallelize and speedup performance of these 0423 (0,1,2,3,4,5,6,7,8,9,10,11,12,13,14 0424. In step 106, the successor and predecessor opera calculations. tors are determined for a permutation in cyclic form. The 0447 The partitions and the primorials in steps 110 can Successor operators for a permutation in cyclic form can be be defined in code. Also shown below are examples of represented as a right rotation. Similarly, the predecessor function calls and the output generated by the function calls. operators for a permutation in cyclic form can be repre sented as a left rotation. Example 0425. In step 107, the transformations between natural numbers and permutations are determined in cyclic form. 0448 “partition function 42 0426. As shown below, the successor and predecessor 0449 (60, [3,4,5) operators in step 106 as well as the transformations between 0450 “partition function' 100 natural numbers and permutations in step 107 can be defined 0451 (120, [3,5,8) in code for example. Also shown below are examples of The function “partition function returns the first partition function calls and the output generated by the function calls. with such that a primorial times 2 is above n. Example 0452 Further, in step 111, a generator and a template are defined for the emulation of the cyclic group. The generator 0427 generator and a template are for a cyclic group of the order given by 0428 (0,1,2,3,4,5,6,7,8,9,10,11,12,13,14, 15 a product that is greater than n, the product being a primorial 0429 "right operator generator times 2". As shown below, the generator and the template in 0430) 1.2.0. 4,5,6,73, 9,10,11,12,13,14, 15.8 step 111 can be defined in code. Also shown below are 0431 “left operator it examples of function calls and the output generated by the 0432 (0,1,2,3,4,5,6,7,8,9,10,11,12,13,14, 15 function calls. 0433 “Ouatnton Add” (“number to permutation cycle' 10) (“number to permutation cycle' 20) Example 0434 0,1,2,3,4,5,6,714, 15,8,9,10,11,12,13 0435 “permutation cycle to number” it 0453 “primorial Generator” 100 0436 30 0454 (0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15 Also included in the Haskell code above is the definition of 0455. In step 112, the successor (Z) and predecessor (Z) Successor arithmetic. operators are determined for an emulation of a permutation 0437. In step 108, the orbit for permutations in cyclic in cyclic form. For example, each rotation of the cycles of form are calculated using the Successor operator. As shown a permutation can be mapped to a corresponding modular below, the orbits in step 108 can be defined in code. Also addition by 1 in these subgroups. In the Haskell code below, shown below are examples of function calls and the output the function Z implements successor and the function Z' generated by the function calls. implements predecessor. 0456. In step 113, the transformations between natural Example numbers and emulations of the permutations are determined 0438 mapM print “Landau Orbit function' in cyclic form. 0439 (0,1,2,3,4,5,6,7,8,9,10,11,12,13,14, 15 0457. As shown below, the successor and predecessor 0440 (1.2.0. 4,5,6,73, 9,10,11,12,13,14, 15.8 operators in step 112 and the transformations between 0441 natural numbers and emulations of the permutations in step US 2016/0328253 A1 Nov. 10, 2016 27 113 can be defined in code. Also shown below are examples Example of function calls and the output generated by the function 0487 mapM print (map “number to primorial based calls. permutation 0 . . . 7) Example 0488 (0,0,0) 0489 (1,1,1) 0458 Z 0,0,0) 0490 2.2.2 0459 (1,1,1] 0491 0,3,3] 0460 Z it 0492 (14.4 0461) 2.2.2 0493 (2,05 0462 Z it 0494 (0,1,6 0463 (0.3.3 0495 12.7 0464 Z it 0496 map (“primorial based permutation to number. 0465 2.2.2 “number to primorial based permutation') 0 . . . 7 0497 (0,1,2,3,4,5,6,7 0466 Z it 0498. Additionally, an extended gcd-based algorithm for 0467 (1,1,1] Chinese Remainder Theorem can be expressed in code. 0468 Z it 0499. In step 116B, a bijection is defined between lists of 0469 (0,0,0) residues and permutations by mapping the residues to rotate 0470 mapM print (map n2Zs 0 . . . 7) each cycle. The bijective mappings of the above functions 0471 (0,0,0) can be extended to the actual permutations in cyclic form. To 0472 (1,1,1] achieve this, a bijection is defined between lists of residues 0473 2.2.2 and permutations by mapping the residues to rotate each 0474 0,3,3] cycle. As shown below, this can be defined in code, wherein 0475 (14.4 the function ZS2cp generates the cyclic permutation corre 0476 2.05 sponding to a list of residues, and the inverse function ZS2cp 0477 (0,1,6 generates the list of residues corresponding to a cyclic 0478 12.7 permutation. Note that an assumption is made that the 0479 map (Zs2nn2Zs) (0. .. 7 permutations are in a canonical form with cycles, each 0480 (0,1,2,3,4,5,6,7 represented as a rotation of a slice from . . . to . . . . Also 0481. In step 114, the orbit for permutations in cyclic shown below are examples of function calls and the output form are calculated using the Successor operator. As shown generated by the function calls. below, the orbits in step 108 (FIG. 20A) can be defined in code. Also shown below are examples of function calls and Example the output generated by the function calls. 0500 map (“primorial based permutation to number. “number to primorial based permutation') 0 . . . 7 Example 0501 (0,1,2,3,4,5,6,7 0482 “Primorial Orbit” 0502 Zs2cp (0,0,0) 0503 (0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15 0483 (0,0,0,1,1,1]. . . . .1.3.6.2.47 0504 s it 0484 Finally, in step 115, computations are performed in 0505 (1.2.0, 4,5,6,73, 9,10,11,12,13,14, 15.8 the cyclic subgroup of the Symmetric Group S using the 0506 cp2Zs it primorial-based approximation as modular arithmetic. 0507 (1,1,1] 0485 The above methods of determining the permuta 0508 Z it tions each perform functions of determining the operational 0509 (2.2.2 semantics of transforming permutations to ensure that the 0510) Zs2cp it transforming permutations are bijective and reversible. 0511 2.0,1,5,6,7,3,4, 10,11,12,13,14,15,8,9] However, an open question exists as to whether a faster 0512. In step 116C, a bijection is defined. Having made conversion mechanism can be realized, rather than the above the above definitions, one can now express the bijection described methods that rely on Successor and predecessor between natural numbers and permutations in cyclic form. functions. The answer is yes, and the faster conversion As shown below, the bijection between natural numbers and mechanism cam be realized using the Chinese Reminder permutations in cycle form can be defined in Haskall code. Theorem, which allows us to recover a number form the residues efficiently, as shown in method of FIG. 20O. Thus, Also shown below are examples of function calls and the fast conversion to a list of modular remainders can be output generated by the function calls. achieved by recovering a natural number from the residues. Example 0486 Referring to FIG. 20O, in step 116A, natural num bers are recovered from the residues using the Chinese 0513 “number to permutation in cycle form” 0 Reminder Theorem. As shown below, the fast conversion 0514 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15 functions can be defined in code. Also shown below are 0515 “permutation in cycle form to number it examples of function calls and the output generated by the 0516 0 function calls. Note that, the inverse n2Zs' finds a list of 0517 “number to permutation in cycle form” 42 residues simply by applying the mod function to each of the 0518 (0,1,2,5,6,7,3,4,10,11,12,13,14,15,8,9] elements of the primorial factors (or the Landau-number 0519 “permutation in cycle form to number” it equivalent). 0520 42 US 2016/0328253 A1 Nov. 10, 2016 28 0521. In step 116D, the faster bijection described above Base List is defined. This function can be referred to as from is applied to defining various arithmetic operations. Using Base List, and the function from Base List reverses the the faster bijection between natural numbers and permuta above described process by converting the list of digits tions in cycle form, various arithmetic operations can be “number list all assumed bounded by the bases in bs into a defined and performed efficiently on permutations in cycle single natural number n. form. These arithmetic operations can include, for example, 0543 For a given base list, the two functions of step 116F the addition and multiplication operations. As shown below, define a bijection to tuples of the same length as the list and these arithmetic operations can be defined in Haskall code. with elements between 0 and b,-1 for each element b, of the Also shown below are examples of function calls and the base list. Further, these functions enumerate the element output generated by the function calls. First, a higher order tuples in lexicographic order (with rightmost digit most function ZSOp can be defined. The higher order function can significant) and that this if different from the order induced then be specialized for each arithmetic operation. For by applying the Successor repeatedly, which, on the other example, addition and multiplication operations can be hand, matches the fast conversion provided by the Chinese defined in Haskell code. Reminder Theorem. 0544. As shown below, the functions of step 116F can be Example defined in code. Also shown below are examples of function 0522) “number to permutation based primorials' 42 calls and the output generated by the function calls. 0523 0.2.2 0524 “number to permutation based primorials' 8 Example 0525 2.30 0545 mapM print (map (to Base List (snd "cyclal 0526 ZSAdd 0.2.2 2.30 permutation Template')) 0 . . . 7) 0527 2.0.2 0546 (0,0,0) 0528 “permutation based primorials to number it (0547 (1,0,0) 0529) 50 0548 2,0,0) 0530. Then these operations can be applied from the 0549 (0,1,0) isomorphic residue list, to work on permutations in cycle 0550 1,1,0) form as shown in Haskell code. 0551 (2,1,0) 0552 (0.2.0) Example 0553 (1.2.0) Procedure for Embedding Permutations into Quantons as 0531 “number to permutation in cycle form 11 Minimal Orbitopes 0532 2.0.1, 4,5,6,7,3], [11,12,13,14, 15,8,9,10 0554. The permutation orbitope of transition relation 0533 n2.cp 7 ships between permutation as per TABLE 6 (see earlier) 0534 (1.2,0.5,6,7,3,415,8,9,10,11,12,13,14 requires, for data of size n a number of data points factorial 0535 cpMul 2,0,1), 4,5,6,7,3], [11,12,13,14,15,8,9, in the size of n. In order to improve the efficiency of memory 10 requirements, the Birkhoff polytope is can be used which 0536 1.2,0.5,6,7,3,4,15,8,9,10,11,12,13,14 reduces the number of data points to n variables and 0537 2,0,1,5,6,7,3,4, 13,14,15,8,9,10,11,12 constraints, but this is still significantly more than the n 0538 “permutation in cycle form to number it variables one could use to represent a permutation as a 0539 77 vector or in the cycle forms shown in the present disclosure. 0540. As an alternative to the above method for bijective 0555 However, by combing the recent teachings of mapping between natural numbers and permutations in cycle Michel X. Goemans, “Smallest compact formulation for the form, another method can be used that generates lexico permutahedron'. Mathematical Programming, October graphically ordered representation. For example, a converter 2015, Volume 153, Issue 1, pp 5-11, Springer Publishing, we to a lexicographically ordered representation can be defined, combine our use of the Fibonacci lattice and the teachings Such that the converter enumerates residues in reverse based on Goemans in C. H. Lim and S. Wright, Beyond the lexicographic order, as shown in method of FIG. 20D. Birkhoff polytope: “Convex relaxations for vector permu 0541. In step 116E, to achieve the result of method, a tation problems, in Advances in Neural Information Pro function is defined that extracts one bijective base-b digit cessing Systems, Vol. 27, MIT Press, Cambridge, Mass., form a number n. Thus, the “get base number digit' function 2014, pp. 2168-2176, in order to produce a reduced repre achieves the effective result of reducing the information sentation. The representation of the present disclosure is stored in n by the information consumed for making a choice further reduced and compressed by utilizing the mapping of a digit between 0 and b-1. Relatedly, step 116E also between the Fibonacci lattice, and its indices to the Lehmer defines a function “put base number digit' that achieves the code representing permutations so that that where only result of adding to m the information stored in a digit b from in log(n) permutation vectors represent the permutahedron in 0 to b-1. As shown below, the functions of step 1410 can be the work of Goemans and Limetal, with the embedding, on defined in code. in points are required. Accordingly, this forms the minimal 0542. In step 116F, a function to Base List is defined that representation of the embedding of permutations as a Quan iterates the extraction of digits from n using a list of bases ton. This is a significantly more efficient and reduced bs instead of a single base b. The function to Base List also memory representation for representing permutations as increments in with the number skip of the lists of digits indices in the Fibonacci lattice and their equivalence to the shorter than the very last one, to ensure that exactly the permutahedron as the preferred embodiment. However, “vectors of length matching the length of bs are extracted. given the Fibonacci mapping, a few points of review are in Additionally in step 116F a complementary function to to order based on the sorting networks because their connec US 2016/0328253 A1 Nov. 10, 2016 29 tion to permutations may not be immediately obvious and tangent plane to one of the vertices 133. Element 132 hence their use in representations of data and reversible illustrates a probability distribution about the point on the circuit emulation. tangent space. 0556 FIG.21: Bitonic Sorting Network Polynomial Con 0565 FIG. 28: Geometries of Classes of Probability straints and Implementation Density Distributions Any sorting network, item 117, given n inputs with m Regular patterns that represent reversible quantum circuits comparators as per Goeman's and represent the permutahe as permutation circuits are shown in item 135 whereas, for dron with complexity of 0(m) variables and constraints an arbitrary pattern of moves, the path shown in 136 is based on, item 118, using the set of constraints for each learned from incoming data by an iterative process using an comparator k=1,2,3 . . . m to indicate the relationships Estimation of Distribution Algorithm (EDA). In the case of between the two inputs and the two outputs of each com the Quanton, recording the learned path is performed using parator, where the operation is illustrated in item 119. a set of curves in n-dimensions (example, splines or Bezier 0557 FIG. 22: Bitonic Sorting Network Operation and curves) whose control points are stored where they corre Design spond to the point of the Fibonacci lattice. The general relationship can be seen as a permutation 0566. Now referring to FIG. 29: Embedding Permuta between the inputs and outputs as shown in item 120, and in tions onto an Orbitope with Probability Density, a combi the general case, when the comparator directions can them nation of both defined circuits, item 137 or learned paths selves be permuted, these can represent the straightforward 138, can be combined as a mixture of probabilistic density sorting network or any general permutation between inputs paths, 140. By embedding the Fibonacci lattice, 139, then and outputs, shown in item 121. the probabilities can be quantized and correspond to the 0558 FIG. 23: Bitonic Sorting Network Operation as a permutations (i.e. state space at the lattice point), 141 to Polytope Design produce the final Quanton structure, 142. By further generalizing from a comparator viewpoint and 0567. In building the Quanton structure of FIG. 29, item taking the viewpoint of arbitrary dimension, the concept is 142, the flowchart of FIG. 30: Flowchart for Quanton Data extended to the sorting orbitope in SY shown in item 122. Structure Construction is used. In general, and following Such comparator operators can be reversibly implemented TABLE 6 as indicated earlier, hypercubes and general as indicated in the teachings of P. S. Phaneendra, C. associahedra are represented in the reduced representation of Vudadha, V. Sreehari and M. B. Srinivas, “An Optimized the permutohedron in the preferred embodiment using the Design of Reversible Quantum Comparator.” 2014, 27th procedure detailed earlier herein. Therefore, choosing one of International Conference on VLSI Design and 2014, 13th items 143 or 144 or 145, the representation of the permu International Conference on Embedded Systems, Mumbai, tohedron in item 146 is reduced to the polytope of 147 from 2014, pp. 557-562. which the Sorting orbitope is generated using the method of 0559 FIG. 24 illustrates Braidings seen via Bitonic Sort Goemans in item 148. Finally, the embedding into the ing Network. According to one embodiment, hypersphere using the Fibonacci lattice is performed in item the relationship between permutational quantum computing, 149 to produce the resultant Quanton structure 157. As has as indicated by the teaching of Stephen Jordan referred to been detailed earlier, training data is presented in item 150 earlier in the present disclosure, is shown in terms of the and this data is filtered, as per item 151 and quantized into operation of braiding, item 123, which is equivalent the patterns as per item 152 following from the detail to be permutation operation, item 124. These are, therefore, presented later, in FIGS. 32.36 and 37. A probability density directly equivalent to the sorting orbitope design of FIG. 22 function 153 is selected per TABLE 5 as defined earlier, and item 122 for which reversible computation has been already this is mapped to the Quanton by the process shown in FIG. presented herein. 14 earlier. At this point, the Quanton has learned the distri 0560 FIG. 25: Permutation Matrix and Permutation Sign butions and can perform either deductive reasoning with the Inversion Matrix noisy data, item 154 using the filter, 155 and a projection 0561. The additional properties of the Quanton are that algorithm 156 that can, if needed, adduce the missing data values can range from negative to positive and that the (if partial noisy data is presented). permutation cycle representation can be viewed in terms of 0568. In order to make clear the specification of the its representation as a permutation matrix, item 125, and that patterns as permutations, FIG. 31: Re-Interpretation of Ver permutations on the Quanton have the additional property tices of Permutation Matrix Zonotope as Combinatorial that they can change the signs of the inputs, illustrate in item Patterns illustrates how, in the case of a binary matrix, a 126, that they operate on. pattern can be digitized. The equivalent for higher dimen 0562 FIG. 26: Permutations Matrix and Permutation sions is a tensor instead of a matrix. For the present Pattern Matrix purposes, however, the matrix will illustrate the essential 0563 As stated previously, the concept of the permuton method of the present disclosure. For example, the pattern of and permutrix was introduced and the purpose of items 127 the matrix, item 158, corresponds to a binary pattern, item and 128 is to illustrate the concept where the two different 159 which could be, for example, a 3x3 set of black-and patterns in the matrices are composites of the permuton and white pixels of an image plane. permutrix so that patternavoiding or pattern conserving. The 0569. The complete set of all patterns is shown in FIG. blocks indicated by item 130 show how arbitrary pattern 32: Enumeration for the 3-Permutation Orbitope Patterns. To permutation operations can also be represented. this set of patterns, each pattern is associated to the permu 0564 FIG. 27: Relationship Between Point and Tangent tation vector, simply shown as integers. So for example, 164 at a Point to Angular Parameters illustrates the matrix for the permutation with corresponding This figure shows an exemplary geometry and topology of integer 312 that represents permutation P=(3,1,2). Elements the Quanton data structure 130 and 131. Element 134 is a 160, 161, 162, 163 and 165 illustrate other examples. US 2016/0328253 A1 Nov. 10, 2016 30 0570. Each pattern, item 166 of FIG. 33: Encoding the the pattern mapped to the surface along with the probability 3-Permutation Orbitope Patterns in Rank Ordered Indices distribution and noise, 182, that is a Quanton. using the Lehmer Index, is converted to the Lehmer code 0575. It is to be noted that larger permutations contain the 167 that uses the Factoradic numbering system. Those patterns of Smaller permutations and that the patterns are skilled in the art are well familiar that this results in an better associated when they separated into nested structures ordering of the patterns by the Lehmer code as an index. to provide variability in the choice of probability measures 0571 When sampling any input, therefore, at the resolu at each level of nesting. The nesting is illustrated in FIG.38: tion of the permutation chosen (i.e. in this case a 3-element Hierarchical Structure of the Quanton with Nested Quantons permutation matrix of 6 patterns) any signal can be sampled where although hyperspheres have been used for illustration, with respect to some clock, or a logical Lamport clock to the situation also applies to any of the other choices of produce a sequence of ordered pattern indices as shown in regular manifold (such as hypertorii). The insight and the FIG. 34: Digitizing a signal by Pattern Based Sampling and reason for creating a hierarchy or a population of Quantons Showing the Numerical Encoding. The important part of 183 and 184 is to use estimation of distribution to evolve the FIG. 34 is to note that when patterns come in, as in items best solution generating Quanton. The lowest level Quanton 168, 169, 170 and 171 as shown in the signal stream, that corresponds to Quantons that are coupled to state transition these represent the indices to the matrix patterns 172 and functions and hence the probability density functions may be hence a code 173 for each pattern that best approximates the different compared to those at higher levels. A Quanton Qat input is written. Therefore, the compression of the input generation t is noted Qn(t) and corresponds to an orbitope signal by the number of patterns is proportionally com with a directional probability density over permuted states, pressed according to the size of the permutation and the representing Qbits (respectively Rebits). Each Quanton can sampling rate. Note the sequence of patterns could represent therefore be viewed as a distribution of bit strings of length the sequence of transitions of binary computer registers as in W., expressed as a permutation with a probability density function distribution. In this model, which is a genetic a CPU instead of as pixels in a black and white screen that evolutionary type of model, the Quanton is the genotype change over time, as in a video. which expresses a dominant phenotype, a permutation, 0572 Aspects of the present disclosure provide for a according to the probability density. Each Quanton, there noise function that, on the surface of the manifold, in the fore, ultimately produces a binary string that represents a example a sphere in FIG.35: Patterns Mapped to the Sphere, Solution to a given input data problem. is Zero noise at the Surface but is non-Zero and pure noise at 0576 FIG. 39: Estimation Distribution Algorithms Based the centroid of the Quanton 178. This means that measure on Quantons for Quantum-Like Computing. Given the pre ments whose values do not exactly fall on the surface of the ceding model of the Quanton, then the estimation of prob Quanton can be extrapolated within the threshold of noise to ability density distribution, which is a variant of the Esti the nearest candidate points on the Surface. In this case, the mation of Distribution Algorithm, is applied to adjust the Quanton is working in a noisy data environment to produce Quanton in a machine learning process given input data. the best nearest hypotheses to the true solution. Various 0577 Quanton populations and hierarchies correspond to noise functions can be used and are best left as user-defined quantum multisets. The Quanton populations are divided choices where a simple Shepard interpolation function that into m Subsets, item 185, each containing in Quantons, item varies the weighting of noise from Zero at the Surface to 1 at 186, per group and having the ability of synchronizing their the center is used. In this way, noisy images with noisy state transition functions (i.e. by just using the exact same similarities or patterns of noisy circuits can be emulated and function and parameters). In steps 187 and 188, the Quanton fast candidates can be generated to produce an answer: by estimate of distribution evolves in the following twin statistically, iterating the answer, weak signal biases are coupled ways which is given first, in population evolution expected to select the best candidate by simply observing the steps listed in items 185 through 191, called PARTA, and frequency of Success in k-trials. Finally, the noise can be secondarily, at the level of the individual Quanton in steps accounted for by using the Estimation of Distribution Algo from item 192 through 198 called PART-B: the unique rithm as shown later in the present disclosure is an integral aspect of this coupled evolution is that the state of the local part of the Quanton machine learning process. Quanton can become entangled or correlated with the global 0573 FIG. 36: Pattern at the center of the Sphere is a state. The EDA algorithm proceeds as follows where Mixed State while the Surface is Pure is intended, for a PARTA and PART-B evolve in tandem. quantum circuit, to relate the noise at the surface 179 being (0578 Referring to Items Items 185 Through 191, as Zero as a pure state while closer to the center 180, the state PARTA: is a fully mixed State that resembles noise: in this case, the 0579 (1) A first initializing generation, is iterated to a noise function is permutationally generated with some second generation; where, Gaussian noise injected from uncertainty measurements on 0580 (2) Every Quanton Qn is associated to data input the data. The procedure for choosing noise is that a statistical string, An, using the method of permutation represen moment (such as variance) is chosen and added to, for tation presented earlier; and, example, Scrambled Sobol sequences generated via permu 0581 (3) This data set is used to build the initial tation as per teachings of Michael Mascagni and Haohai Yu, probability distributions, item 187; then, "Scrambled Sobol Sequences via Permutation, Monte 0582 (4) The probability distribution is sampled, item Carlo Methods and Applications. Volume 15, Issue 4, Pages 188; and, 311-332, ISSN (Online) 1569-3961, ISSN (Print) 0929 0583 (5) The values of bit strings between An and Qn 9629, DOI: 10.1515/MCMA2009.017, January 2010. are compared for fitness 189: 0574. The overall situation is shown in FIG. 37: Patterns 0584 (6) If An is better than the previous generation, Mapped to the Quanton where the element 181 represents and if their bit values differ, a quantum gate gate US 2016/0328253 A1 Nov. 10, 2016 operator is applied on the corresponding Qubits of Qn 0594 (2) An orbitope representing the permutations is using the previously described permuational methods; instantiated, item 193, to represent an individual Quan 0585 (7) Therefore, after the gate operation the distri ton as part of a Quanton population; and, bution Qn is slightly moved toward a given Solution An 0595 (3) The permutation vector is embedded into a of the total solution space, item 190; and, manifold, item 194 (for example, a hypersphere or any of the geometries in TABLE 1) using a chosen dimen 0586 (8) Steps from item 186 are repeated until ter sion usually less than the size of the permutation space. mination when a threshold either in number of itera For example, the permutation vector for 3-elements, tions or fitness is reached, item 191. {1,2,3} produces points in the usual Euclidean 3-space. 0587. The initial set up of the Quanton is that X=(X1, .. But permutations of 4-elements, {1, 2, 3, 4} produces .., Xn) denotes a vector of discrete random variables in higher dimensional points of the permutohedron. which X=(X1, ..., Xn) represents a permutation (respec Therefore, if for example, a circle is chosen as the tively a braid) of length n. We use (lowercase) X=(x1, . . . . embedding space, the for 3-elements there will be six Xn) to denote an assignment of values to the variables. divisions, or for 4-elements, 24 divisions of the circle Finally, we define K that denotes a set of indices in {1, ... and the adjacency relationship can be, for example, , n}, as a permutation and Xk (respectively xk) a subset of transpositions of adjacent elements in the permutation. the variables of X (respectively x) determined by the indices Hence, the space of embedding need not be the same in K. For each X, it therefore takes values in {0, 1, . . . . size as the permutation space itself, and, 2g-1} where g is the number of generators for values (and 0596 (4) The mixture model 195 can be evolved and that these can originate from training data). adjusted by the methods described previously; and, 0588. The Boltzmann probability distribution is used in 0597 (5) The probability model adjusted by the action statistical physics to associate a probability with a system of a Quantum Gate operator, changing the probability state according to its energy: this process can be used to transitions between permutations (as gate operations) drive computation or as a Surrogate for estimation of distri in item 196. In classical evolutionary algorithms, bution for the Quanton. In driving computation, energy operators such as bit-string crossover or mutation are carries away information and the design is such that the used to explore a search space while the Quanton permutons serve the role of information sinks while the quantum-like analog to these operators is the quantum permutants become the information Sources. The Quanton gate; then, structure, therefore, uses the boltzman distribution, pFB(x), in 0598 (6) The fitness is assessed 197. The effect is that its computing evolutionary mode which models the braiding local Quantons will evolve such they become synchro (respectively permutation gate) operator and assigns a nized from local to global states and move collectively higher probability to braids that gives a more accurate closer toward the solution; until, approximation to the target gate, thus modeling the quantum 0599 (7) The solution threshold in either fitness or gate itself. number of iterations is achieved, item 198. 0589 The Boltzmann distribution p(x) is defined for the 0600 The best most fit state transition function is stored Quanton as: at every generation and is distributed to the group functions at intermittent intervals which represents a local synchroni Zation. From the set of all populations of groups of Quan tons, that is to say, nxm Quantons in total, then the best most fit transition function among all Quanton multisets is stored every generation and also periodically distributed to the group functions. This is effectively a phase of global Syn chronization. All Quantons start in an initial population Such 0590 Here g(x) is the objective function and T is the that their probability distribution corresponds to system temperature to smooth the probabilities. |C.?=||B-1/2 so that the two states “0” and “1” are 0591 Quantons with the highest probability, from the equiprobable in the case of a binary valued observation Estimation of Distribution Algorithm, correspond to the logic. braids that maximize the objective function. The Quanton is 0601 The Quanton model generalizes the two-qubit state Solving the problem of finding a product of generator model to any finite number of states. The smallest unit of matrices for braids that represent a target gate and uses an information is now a Quanton which is defined by a tuple of object function of braid length as the following: size n, being in the total numbers of different possible S states, defined with respect to the ordering of a set of probabilistic bits whose values are analogs to the quantum 1 - bits (qubits). fitness(x) = 1 --he e. -- t i is braid length, is a control parameter 0592 Referring Now to Items Items 192 Through 198, as PART-B: 0593 (1) A random population of Quantons is built, item 192, with varying sized permutations using the methods previously described in the present disclosure; and, US 2016/0328253 A1 Nov. 10, 2016 32 0602) Let the state, S={s1, s2, ss, sa, so that if SeO, 1} straints and other hard correlation relationships to there are 2' patterns but that there are (4)=64 combinatorial ensure synchronization with diversity; solution sets represented if S is a Quanton where paths 0610 (iii) The Quantons individually as well as their between permutations can return to their various origins (i.e. populations are scored using a model with a method that a permutation can have path back to itself). This mixed based on the fitness function that gives a numerical state probability distribution has individual states C, labeled ranking for each individual in the population, with the as pure states. higher the number the better. From this ranked popu 0603 The Quanton obeys the total probability conserva lation, a Subset of the most promising solutions are tion equation for each probability of each state as follows: selected by the selection operator. 0611 (iv) The algorithm then constructs a probabilistic P=Xplo-1 model which attempts to estimate the probability dis 0604 Quanton populations represent the concept of a tribution of the selected solutions. New solutions (off linear superposition of states probabilistically both in terms spring) are generated by sampling the distribution of the paths on an individual Quanton as well as in terms of encoded by this model. These new solutions are then a population of Quantons. Therefore, in learning distribu incorporated back into the old population, possibly tions of states, the algorithm in the preferred embodiment is replacing it entirely. Finally, iteration of these steps are the Estimation of Distribution Algorithm (EDA) which is repeated in continuous cycle until an expected or also known variously in the literature as Probabilistic Model pre-determined criterion is reached: this model departs Building Genetic Algorithm (PMBGA). Both terms are used from the usual evolutionary models to model quantum interchangeably in the present disclosure without distinc like computation becase a random selection of indi tion. viduals is tested against the incoming data using quan 0605. The Quantons are represented by distributions over tum gate operations and that the individuals may be permutations and by probabilistic transition paths between produced from a machine-learning process as opposed permutations. These correspond to a string of Qubits, or to a random process. Rebits as noted earlier, such that any Qubit (respectively 0612 (V) Finally, most implementations are based on a Rebit) can also represent a linear Superposition of Solutions probability vector that provides a fast and efficient since all the permutations (i.e. states) are all available in a model mainly due the assumption that all problem single Quanton. If the truth values for the observables are variables are independent: The Quanton can handle this binary, then the solutions are superpositions of binary sets, case as well as the case that variables are not indepen else, if another logic is used, the Superpositions are solutions dent (i.e. entangled in Some way) and in the classical in the order of the logic (e.g. 3-valued logics). case are related by some unforeseen hidden variables 0606 Traditional genetic algorithms based on bit-strings, (i.e. the permuton and permutrix correlation). where bits have only one of two possible states (0 or 1) have 0613. An estimated probabilistic model is built from significant and serious disadvantages at very high scale, selecting the current best solutions and this is used in Such as massive increases of dimensionality, long conver producing samples to guide the search process and update gence times, weak signal confusion, numerical imprecision the induced model. The main process that the Quantons are and complex codes. In contrast, Quantons as individuals and solving for is to capture the joint probabilities between as populations represent a Superposition of all combinatorial variables in a highly complicated data space. The main Solutions: this provides a Substantial efficiency and enhance bottleneck is, therefore, in estimating the joint probability ment in performance in combinatorial optimization by distribution associated with the data. genetic algorithms using Quantons while being a natural part 0614 The Quanton model uses the following sets of of the Quanton/quantum representation model. heuristic rules to alleviate the bottleneck: 0.615 (a) Independent Distribution Heuristic: Assume 0607 Now, referring to FIG. 39: Flowchart Schemata for that there is no dependency between the variables of the Quantons in Estimation of Distribution Algorithm, a few problem. The joint probability distribution is factorized clarifying notes of specific importance to the present dis as n independent univariate probability distributions closure are in order. First, it is commonly known to those and the Univariate Marginal Distribution Algorithm skilled in the art that Estimation of distribution algorithms (UMDA) is used using the teachings of H. Muhlenbein, (EDAs) are a type of evolutionary model based on estimat “The equation for response to selection and its use for ing a probability distribution of better candidate solutions to prediction'. Evolutionary Computation, 1997. Other a problem space. As shown earlier, the Quanton uses a choices with the same heuristic include Population tandem model of local (at the level of Quanton) and global Based Incremental Learning (PBIL) and compact (at the level of the population of Quantons) evolution: the genetic algorithm (cGA) as referred in the literature to main steps of the Quanton based PMBGA, therefore, those skilled in the art, for example, M. Pelikan and H. diverge significantly from the canonical PMBGA (respec Mühlenbein, “Marginal distributions in evolutionary tively EDA) as follows: algorithms, in Proceedings of the International Con 0608 (i) An initial population of Quantons is randomly ference on Genetic Algorithms Mendel, 1998. generated, where each Quanton can itself have its own 0616 (b) Pair Dependency Heuristic: Assume depen local probability density function, different to, and with dency between pairs of variables. Therefore, the joint a global probability density function or a mixture; probability distribution of the variables is factorized as 0609 (ii) Starting with a population generated by a the product of a univariate density function and (n-1) uniform distribution over all admissible solutions pair wise conditional density functions given a permu locally entangled Subpopulation of individuals can be tation between variables. For example, the Bivariate created by setting permutation pattern avoiding con Marginal Distribution Algorithm (BMDA) as cited in US 2016/0328253 A1 Nov. 10, 2016 33 M. Pelikan and H. Mühlenbein, “The Bivariate Mar From which we can derive a query and answer expression as ginal Distribution Algorithm, in Advances in Soft follows: Computing: Engineering Design and Manufacturing, pages 521-535, Springer, 1999 and also the Mutual Information Maximization for Input Clustering (MIMIC) as referenced in Romero, Leoncio A. et al. “A Comparison between Metaheuristics as Strategies for Minimizing Cyclic Instability in Ambient Intelli gence.” Sensors (Basel, Switzerland) 12.8 (2012): And where we represent entanglement as: 10990-11012. PMC. Web. 5 May 2016 or also Com bining Optimizers with Mutual Information Trees (CO MIT) algorithms as per the teachings in S. Baluja and ... 0)|O) ... 1)0) S. Davies. Combining multiple optimization runs with 1 22 optimal dependency trees. Technical Report CMU-CS 97-157. V 1 1 ... O)1) 0.617 (c) Multiple Dependency: Assume dependencies 1 ... 1)0) (strong correlations or entanglement) between multiple variables. In this case, the local and global Quanton Given a density matrix defined as 6 where p is the fraction evolution and population evolution as per the present of the ensemble in each pure state up): patent is the preferred embodiment. 0618. In the case of multiple dependencies, the Quanton simply written as: has special advantage and polynomial efficiency over other methods. At the foundation of the Quanton is the idea of the quantum system which is usually composed of a query And the answer 8: 8 =Tr(8) register, q) and an answer register, la), acting on Hilbert Accordingly, let k denote the overall number of solutions, Space: then 6' takes the form shown in Expression below: (H-H & H. 0619. In the Quanton, the query register and answer register are both n-qubit registers where possible values for the variables of a specific problem instance exist in the space The overall state is separable only when k=0, i.e. no solution of permutations of the assignments of values to variables. exists, or when k2n, each value belonging to 0,2'-1 is a For example, the SAT problem, known to be NP-complete Solution. Otherwise, the system is entangled. Thus, the usually has multiple dependencies. The Quanton quantum problem of determining whether a solution to a problem efficiency over straightforward classical computation is that, exists that can be reduced to the problem of determining q) is placed in a uniform Superposition of the computation whether the state is separable or entangled. In order to basis by applying the Hadamard transform H a total of n determine if the state is entangled, the following decision times to the initializing n-qubit state, 0), i.e. equation is used: 0.19) la) =|g) at Eft(q) ) which employs an auxiliary function for SP = p) (up 1 if (q) is a solution and q ea, b ?tabl (q) = { O if f(q) is not a solution and q ea, b This creates a Superposition of an exponential number of states, each of representing a possible path in the Quanton, This process entangles the quantum registers and using the but, due to the Quanton permutational representation, only auxiliary function, entanglement can be checked for in the employing a polynomial number of gates. A single qubit, is presence of a solution state for a given range. In contrast, the initialized to state |0) and this is the answer register. The absence of a solution means that a specific range is pruned state of the system is: from the search procedure. 0620 FIG. 40: Hardware Design for Quantons for A System On A Chip (SOC) Given the preceding information, th = H, & H. the Quanton is also suited for implementation in both classical or quantum hardware, whenever quantum hardware shonoyon becomes available. In an embodiment in classical hardware, a data sampler, item 199, feeds into a processor using a probability distribution, item G. The processor computes the posterior distribution using either a GPU (graphics process ing unit coprocessor) or DSP (digital signal processing) unit, item 200. This is combined with the permutation represen US 2016/0328253 A1 Nov. 10, 2016 34 tation of quantum gates in the form of Lehmer code “revis program being stored in any of the above-described non ers, item 201, that operate as adders and buffers to produce transitory electronic memories and/or a hard disk drive, CD, the final output code, item 202, using the compressed DVD, FLASH drive or any other known storage media. encoding described previously for signal encoding. Further, the computer-readable instructions may be provided 0621 FIG. 41: Overall Master Flow Chart for Quanton as a utility application, background daemon, or component PMBGA Consensus Output A population of Quantons, items of an operating system, or combination thereof, executing in 204 to 211 can evolve according to different evolutionary conjunction with a processor, Such as a Xenon processor regimes and these can serve as the metaheuristic estimators from Intel of America or an Opteron processor from AMD of distribution for learning the probability density distribu of America and an operating system, Such as Microsoft tion of a parent Quanton, item 203: in this way, very VISTA, UNIX, Solaris, LINUX, Apple, MAC-OSX and complex patterns, or temporal events, can be learned and other operating systems known to those skilled in the art. stored in a single master Quanton ready for recall and use in Even on a ONE CPU machine it acts in “parallel' due to the problem solving in a given domain, or, as an analog to permutation encoding. Solutions in other domains, as a consensus of the probabili 0627. In addition, the invention can be implemented ties (i.e. a mixture model), item 212. using a computer based system 1000 shown in FIG. 42. The 0622. Thus, the foregoing discussion discloses and computer 1000 includes a bus B or other communication describes merely exemplary embodiments of the present mechanism for communicating information, and a proces invention. As will be understood by those skilled in the art, sor/CPU 1004 coupled with the bus B for processing the the present invention may be embodied in other specific information. The computer 1000 also includes a main forms without departing from the spirit or essential charac memory/memory unit 1003. Such as a random access teristics thereof. Accordingly, the disclosure of the present memory (RAM) or other dynamic storage device (e.g., invention is intended to be illustrative, but not limiting of the dynamic RAM (DRAM), static RAM (SRAM), and syn Scope of the invention, as well as other claims. The disclo chronous DRAM (SDRAM)), coupled to the bus B for Sure, including any readily discernible variants of the teach storing information and instructions to be executed by ings herein, define, in part, the scope of the foregoing claim processor/CPU 1004. In addition, the memory unit 1003 terminology Such that no inventive subject matter is dedi may be used for storing temporary variables or other inter cated to the public. mediate information during the execution of instructions by 0623 Features of the disclosure can be implemented the CPU 1004. The computer 1000 may also further include using some form of computer processor. As stated previ a read only memory (ROM) or other static storage device ously, each of the functions of the above-described embodi (e.g., programmable ROM (PROM), erasable PROM ments may be implemented by one or more processing (EPROM), and electrically erasable PROM (EEPROM)) circuits. A processing circuit includes a programmed pro coupled to the bus B for storing static information and cessor (for example, processor 1004 in FIG. 42), as a instructions for the CPU 1004. processor includes circuitry. A processing circuit also 0628. The computer 1000 may also include a disk con includes devices such as an application-specific integrated troller coupled to the bus B to control one or more storage circuit (ASIC) and conventional circuit components devices for storing information and instructions, such as arranged to perform the recited functions. The circuitry may mass storage 1002, and drive device 1006 (e.g., floppy disk be particularly designed or programmed to implement the drive, read-only compact disc drive, read/write compact disc above described functions and features which improve the drive, compact disc jukebox, tape drive, and removable processing of the circuitry and allow data to be processed in magneto-optical drive). The storage devices may be added to ways not possible by a human or even a general purpose the computer 1000 using an appropriate device interface computer lacking the features of the present embodiments. (e.g., Small computer system interface (SCSI), integrated 0624 The present embodiments can emulate quantum device electronics (IDE), enhanced-IDE (E-IDE), direct computing on a digital computing device having circuitry or memory access (DMA), or ultra-DMA). can be implemented on a quantum computer or the like. 0629. The computer 1000 may also include special pur 0625. As one of ordinary skill in the art would recognize, pose logic devices (e.g., application specific integrated cir the computer processor can be implemented as discrete logic cuits (ASICs)) or configurable logic devices (e.g., simple gates, as an Application Specific Integrated Circuit (ASIC), programmable logic devices (SPLDS), complex program a Field Programmable Gate Array (FPGA) or other Complex mable logic devices (CPLDS), and field programmable gate Programmable Logic Device (CPLD). An FPGA or CPLD arrays (FPGAs)). implementation may be coded in VHDL. Verilog or any 0630. The computer 1000 may also include a display other hardware description language and the code may be controller coupled to the bus B to control a display, such as stored in an electronic memory directly within the FPGA or a cathode ray tube (CRT), for displaying information to a CPLD, or as a separate electronic memory. Further, the computer user. The computer system includes input devices, electronic memory may be non-volatile, such as ROM, Such as a keyboard and a pointing device, for interacting EPROM, EEPROM or FLASH memory. The electronic with a computer user and providing information to the memory may also be volatile, such as static or dynamic processor. The pointing device, for example, may be a RAM, and a processor, Such as a microcontroller or micro mouse, a trackball, or a pointing stick for communicating processor, may be provided to manage the electronic direction information and command selections to the pro memory as well as the interaction between the FPGA or cessor and for controlling cursor movement on the display. CPLD and the electronic memory. In addition, a printer may provide printed listings of data 0626. Alternatively, the computer processor may execute stored and/or generated by the computer system. a computer program including a set of computer-readable 06.31 The computer 1000 performs at least a portion of instructions that perform the functions described herein, the the processing steps of the invention in response to the CPU US 2016/0328253 A1 Nov. 10, 2016 1004 executing one or more sequences of one or more coupling to a network that is connected to, for example, a instructions contained in a memory, Such as the memory unit local area network (LAN), or to another communications 1003. Such instructions may be read into the memory unit network Such as the Internet. For example, the communica from another computer readable medium, Such as the mass tion interface 1005 may be a network interface card to attach storage 1002 or a removable media 1001. One or more to any packet Switched LAN. As another example, the processors in a multi-processing arrangement may also be communication interface 1005 may be an asymmetrical employed to execute the sequences of instructions contained digital subscriber line (ADSL) card, an integrated services in memory unit 1003. In alternative embodiments, hard digital network (ISDN) card or a modem to provide a data wired circuitry may be used in place of or in combination communication connection to a corresponding type of com with software instructions. Thus, embodiments are not lim munications line. Wireless links may also be implemented. ited to any specific combination of hardware circuitry and In any such implementation, the communication interface software. 1005 sends and receives electrical, electromagnetic or opti 0632. As stated above, the computer 1000 includes at cal signals that carry digital data streams representing vari least one computer readable medium 1001 or memory for ous types of information. holding instructions programmed according to the teachings 0638. The network typically provides data communica of the invention and for containing data structures, tables, tion through one or more networks to other data devices. For records, or other data described herein. Examples of com example, the network may provide a connection to another puter readable media are compact discs, hard disks, floppy computer through a local network (e.g., a LAN) or through disks, tape, magneto-optical disks, PROMs (EPROM, equipment operated by a service provider, which provides EEPROM, flash EPROM), DRAM, SRAM, SDRAM, or communication services through a communications net any other magnetic medium, compact discs (e.g., CD work. The local network and the communications network ROM), or any other medium from which a computer can use, for example, electrical, electromagnetic, or optical read. signals that carry digital data streams, and the associated 0633 Stored on any one or on a combination of computer physical layer (e.g., CAT5 cable, coaxial cable, optical fiber, readable media, the present invention includes software for etc). Moreover, the network may provide a connection to, controlling the main processing unit, for driving a device or and the computer 1000 may be, a mobile device such as a devices for implementing the invention, and for enabling the personal digital assistant (PDA) laptop computer, or cellular main processing unit to interact with a human user. Such telephone. software may include, but is not limited to, device drivers, 0639. In summary, the Quantum method and apparatus operating systems, development tools, and applications soft use a quantum-inspired computational data representation ware. Such computer readable media further includes the and algorithm called a Quanton. The Quanton is a Surrogate computer program product of the present invention for for a mixed State representation and algorithms for operating performing all or a portion (if processing is distributed) of on the Quanton to perform learning or inference to produce the processing performed in implementing the invention. results. The Quanton can be used as an encoding for a 0634. The computer code elements on the medium of the fictitious, virtual (e.g. phonon) or real (e.g. atomic) physical present invention may be any interpretable or executable particle with properties that can be graded between classical code mechanism, including but not limited to Scripts, inter like and quantum-like analogous behaviors by choosing the pretable programs, dynamic link libraries (DLLs), Java probability density function, the lattice that quantize the classes, and complete executable programs. Moreover, parts continuous space and the permutation encoding that repre of the processing of the present invention may be distributed sents a state space at each lattice index. for better performance, reliability, and/or cost. 0640 The Quanton can be defined by any set of self 0635. The term “computer readable medium' as used similar permutation operators, acting on the digits and herein refers to any medium that participates in providing places of any number, such as, for example, the permutation instructions to the CPU 1004 for execution. A computer model proposed by Palmer noted above for representing the readable medium may take many forms, including but not quantum complex space by a real binary expansion of a limited to, non-volatile media, and volatile media. Non Borel normal real number. Volatile media includes, for example, optical, magnetic 0641 For the purposes of explaining the embodiment, we disks, and magneto-optical disks. Such as the mass storage will use a Fibonacci lattice and a simple transposition 1002 or the removable media 1001. Volatile media includes permutation operator to generate the permutations. A simple dynamic memory, such as the memory unit 1003. probability density function, the Von Mises Bingham distri 0636 Various forms of computer readable media may be bution will be used and is allocated to the Fibonacci lattice. involved in carrying out one or more sequences of one or The points of the Fibonacci lattice, which are associated to more instructions to the CPU 1004 for execution. For the permutation, together represent a permutation polytope. example, the instructions may initially be carried on a We will embed this polytope into a hypersphere, therefore, magnetic disk of a remote computer. An input coupled to the which plays the role of a Riemann sphere, quantized by the bus B can receive the data and place the data on the bus B. lattice and its embedded permutations. The bus B carries the data to the memory unit 1003, from 0642. As a result of its granularity (i.e. quantization), the which the CPU 1004 retrieves and executes the instructions. value of the probability density distribution is defined to be The instructions received by the memory unit 1003 may only uniquely definable on a countable set of the vertices of optionally be stored on mass storage 1002 either before or the polytope embedded in the hypersphere or sphere (in the after execution by the CPU 1004. case of low dimensional permutations). The permutation 0637. The computer 1000 also includes a communication operator describes permutation-operator representations interface 1005 coupled to the bus B. The communication from which transformations of a probability density value interface 1004 provides a two-way data communication under rotations of the sphere can be defined, or, conversely, US 2016/0328253 A1 Nov. 10, 2016 36 that operations in the vector space representing probability the Quanton needed amounts to estimating the quantization densities produce permutation results noting that permuta limit of the Quanton itself as function of the distance tions can encode arbitrary data structures and models as between adjacent lattice points of the spherical Fibonacci described further in this disclosure. Using the permutation lattice which serves as the foundation for the spherical operator representation and the representation of arbitrary distribution of the vertices of the permutation polytope. data by permutation encodings as natural numbers, a con 0648. Effective data structure encoding is the main reason structive quantum-like probabilistic virtual machine is in enabling algorithmic speedups for example, in both defined. classical and quantum estimation of distribution and particle 0.643. In analogy to standard quantum theory, the Quan Swarm algorithms as well as many other fundamentally hard ton permutation operator can also serve to encode the optimization problems. Poor data structures imply that solu stochasticity of a complex phase function in the standard tions will need hacks and work-arounds to implement. quantum theoretic wave function: for example, in S2, quaternionic permutations can encode the spatial entangle 0649. The Quanton makes use of a succession of embed ment relationships in the wave function as noted in Palmer. dings. In other words it partially evaluates and therefore In higher dimensions, the general geometric algebra of compiles out the combinatorial complexity, without trading rotors extends this formalism. The Quanton provides finite space for speed (as is usually done) between factorially large samples of values of the probability density on the vertices, data spaces and a n-dimensional manifold that are precom and the vertex sets can also be used to define a probability piled into a probabilistic particle which serves as the repre measure consistent with quantum-theoretic probability and sentation for data. correlation if the model of Palmer is used (ref). 0650. The Quanton data encoding makes use of the 0644. The quantization of the Quanton in terms of its concepts of entanglement and Superposition to represent an vertex based probability density accounts for the Quanton entire joint probability density over states of Superposition, virtual machine not requiring fulfillment of the Bell and approximates all inference trajectories simultaneously inequalities, and hence rendering the Quanton a "quantum (e.g. markov chain monte carlo (MCMC))—a feat not inspired approximate probabilistic virtual machine in con possible with any current purely classical data encoding trast to a true quantum-virtual machine. However, standard models. A single Quanton data encoding, as single particle, quantum phenomena are emulated by the Quanton. In this can encode a massive amount of information and a multi respect, the two key properties of the Quanton are that plicity of Such Quanton encodings forms a quantum particle quantum-like state preparation corresponds to choice of ensemble for more complex representational data Surro probability density (or mixture model) and that the permu gates. tation operator serves to define the granularity of the obser 0651. As an example of a more complex data represen Vation state space (i.e. the model state space) from which tational Surrogate, for example, in case-based reasoning measurement probabilities and correlations can be derived (CBR) the dynamic case memory is the key to the reasoning by frequentism or Bayesianism (as have been encoded via process in which machine learning is also an inherent part of the machine learning EDA or by direct preparation). both the case matching as well as the case adaptation 0645 Temporal and spatial domains can be combined via process. In real-world applications, big data arrive as the use of time domain density functional theory (TDDFT) unstructured sets of high dimensional case vectors, requiring which is used as the quantum inspired approach for encod memory-based or instance-based reasoning and learning, ing dynamics when the data set requires time as a parameter. even though the situation consists of knowledge-poor shal TDDFT can be used to approximately simulate quantum low sets of cases—such as the difficult speech to-speech algorithms on a classical computer and with the Quanton machine translation problem. In our approach using the model, this can also be used for clustering or inference over Quanton data encoding, the case indexing problem is time. There will be systems that will be very hard to simulate addressed at a modeling level where matching can be using approximate functionals. Such as those that are in the performed efficiently by using the built-in quantum inspired complexity class QMA (Quantum Merlin Arthur) protocols parallelism through the distribution of probability density and may require exponentially scaling resources even on a over the permutations that represent the cases. real quantum computer http://arxiv.org/abs/0712.0483. 0652 The Quanton makes it possible to implement quan 0646 Finding functionals that carry out complex quan tum-Surrogates of inference procedures capable of perform tum computational tasks is extremely difficult and the pres ing at very high speed with quadratic, exponential and ent disclosure provides a classical approximation to approxi factorial speedup over the best classical approaches since it mate quantum-like computing by combining a number of provides what has been called a “backdoor. The Quanton distinct elements in a very unique way, based on estimation backdoor is the connection that is formed between embed of distribution algorithms (EDA), so as to result in func ding in a continuous manifold and its embedded probability tionals that are computationally useful within a specified densities between the manifold and the tangent space prob scope of problems. By scope of problem we mean their abilities to the same manifold: these are a sparse number of scale, or size, or complexity and constrained to within a points relative to the total problem space. Backdoors are domain of application (i.e. instead of general purpose com sparse numbers of parameters that can dramatically impact puting, we mean special purpose computing). very high dimensional data spaces and bring them down to 0647. The specified scale is determined by the size of the manageable levels because they act like shortcuts through permutations or depth of a structure as represented by the high combinatorial complexity. Therefore the typical targets Quanton. Within the scale, therefore, the Quanton will for the use of the Quanton data structure is in attacking compute at high speed, but, if the permutation scale is too NP-hard problems because a Quanton enables a generic way Small, then the computations degenerate back into the expo to approximate NP by P by providing backdoors between nential or factorial sizes. Therefore, estimating the size of continous probability density functionals and structure. US 2016/0328253 A1 Nov. 10, 2016 0653) One example is the NP-Hard maximum a priori only, and are not intended to limit the Scope of the inven (MAP) inference in (Bayesian) graphical models as well as tions. Indeed the novel methods and systems described inference over permutations, computing matrix permanents herein may be embodied in a variety of other forms; fur and seriation problems. The algorithms presented using the thermore, various omissions, Substitutions, and changes in data structures can quickly find an approximation-MAP the form of the methods and systems described herein may configuration with a quadratic speedup over classical, can be made without departing from the spirit of the inventions. find an approximation inference to the inference over per The accompanying claims and their equivalents are intended mutations, compute approximate near permanents and rap to cover such forms or modifications as would fall within the idly approximate the solution to the noisy seriation problem Scope and spirit of the inventions. Such that classical methods can select the precise solution, 1. A method of emulating a quantum like machine, the out of the approximate candidates, for each of these NP method being performed by circuitry, the method compris Hard problems. ing: 0654 Any one of, or, a mixture of, the Complex Normal Distribution, Kent, Gaussian, Bingham, Watson, Dirichlet determining a size of a largest permutation group that fits Process mixture models or von Mises-Fisher distribution, or a problem size based on a Landau number, their multivariate versions, for directional data presents a generating a closed geometrical Surface in a high-dimen tractable form and has all the modeling capability required sional space, the closed geometrical Surface corre for the Quanton. sponding the size of the largest permutation group; 0655 The present invention uses a generative model of embedding a lattice of vertices in the closed geometrical mixtures of distributions on a hypersphere that provide Surface; numerical approximations of the parameters in an Expecta assigning respective permutations to corresponding ver tion Maximization (EM) setting. This embodiment also tices of the lattice; allows us to present an explanation for choosing the right associating linear tangent spaces to the receptive vertices embedding dimension for spectral clustering as well as of the lattice; choosing the right embedding dimension from a distribution associating transition operators between respective per perspective. In text analytics, however, and in order to mutations of the vertices of the lattice, to correspond represent semantics (content structure) between related with quantum gate operations; documents, for clustering purposes, the topics can be rep associating permutations as Surrogates for computation; resented as permutations over concepts. To learn the distri distributing a nonlinear directional probability distribu butions over permutations, the Generalized Mallows Model tion function across the closed geometric Surface, the (GMM) concentrates probability mass on permutations nonlinear directional probability distribution function close to canonical permutations and, therefore, permutations representing respective likelihoods of the correspond from this distribution are likely to be similar, thus clustering similar documents. In the GMM model the number of ing transition operators; and parameters grows linearly with the number of concepts, thus updating the nonlinear directional probability distribution sidestepping tractability problems typically associated with function to modify the likelihoods of the transition the large discrete space of permutations. operators, thereby to generate an emulation of a quan 0656. While certain embodiments have been described, tum gate. these embodiments have been presented by way of example