The Oracle Separation of BQP and PH: a Recent Advancement in Quantum Complexity Theory
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Database Theory
DATABASE THEORY Lecture 4: Complexity of FO Query Answering Markus Krotzsch¨ TU Dresden, 21 April 2016 Overview 1. Introduction | Relational data model 2. First-order queries 3. Complexity of query answering 4. Complexity of FO query answering 5. Conjunctive queries 6. Tree-like conjunctive queries 7. Query optimisation 8. Conjunctive Query Optimisation / First-Order Expressiveness 9. First-Order Expressiveness / Introduction to Datalog 10. Expressive Power and Complexity of Datalog 11. Optimisation and Evaluation of Datalog 12. Evaluation of Datalog (2) 13. Graph Databases and Path Queries 14. Outlook: database theory in practice See course homepage [) link] for more information and materials Markus Krötzsch, 21 April 2016 Database Theory slide 2 of 41 How to Measure Query Answering Complexity Query answering as decision problem { consider Boolean queries Various notions of complexity: • Combined complexity (complexity w.r.t. size of query and database instance) • Data complexity (worst case complexity for any fixed query) • Query complexity (worst case complexity for any fixed database instance) Various common complexity classes: L ⊆ NL ⊆ P ⊆ NP ⊆ PSpace ⊆ ExpTime Markus Krötzsch, 21 April 2016 Database Theory slide 3 of 41 An Algorithm for Evaluating FO Queries function Eval(', I) 01 switch (') f I 02 case p(c1, ::: , cn): return hc1, ::: , cni 2 p 03 case : : return :Eval( , I) 04 case 1 ^ 2 : return Eval( 1, I) ^ Eval( 2, I) 05 case 9x. : 06 for c 2 ∆I f 07 if Eval( [x 7! c], I) then return true 08 g 09 return false 10 g Markus Krötzsch, 21 April 2016 Database Theory slide 4 of 41 FO Algorithm Worst-Case Runtime Let m be the size of ', and let n = jIj (total table sizes) • How many recursive calls of Eval are there? { one per subexpression: at most m • Maximum depth of recursion? { bounded by total number of calls: at most m • Maximum number of iterations of for loop? { j∆Ij ≤ n per recursion level { at most nm iterations I • Checking hc1, ::: , cni 2 p can be done in linear time w.r.t. -
Quantum Computing Overview
Quantum Computing at Microsoft Stephen Jordan • As of November 2018: 60 entries, 392 references • Major new primitives discovered only every few years. Simulation is a killer app for quantum computing with a solid theoretical foundation. Chemistry Materials Nuclear and Particle Physics Image: David Parker, U. Birmingham Image: CERN Many problems are out of reach even for exascale supercomputers but doable on quantum computers. Understanding biological Nitrogen fixation Intractable on classical supercomputers But a 200-qubit quantum computer will let us understand it Finding the ground state of Ferredoxin 퐹푒2푆2 Classical algorithm Quantum algorithm 2012 Quantum algorithm 2015 ! ~3000 ~1 INTRACTABLE YEARS DAY Research on quantum algorithms and software is essential! Quantum algorithm in high level language (Q#) Compiler Machine level instructions Microsoft Quantum Development Kit Quantum Visual Studio Local and cloud Extensive libraries, programming integration and quantum samples, and language debugging simulation documentation Developing quantum applications 1. Find quantum algorithm with quantum speedup 2. Confirm quantum speedup after implementing all oracles 3. Optimize code until runtime is short enough 4. Embed into specific hardware estimate runtime Simulating Quantum Field Theories: Classically Feynman diagrams Lattice methods Image: Encyclopedia of Physics Image: R. Babich et al. Break down at strong Good for binding energies. coupling or high precision Sign problem: real time dynamics, high fermion density. There’s room for exponential -
Interactive Proof Systems and Alternating Time-Space Complexity
Theoretical Computer Science 113 (1993) 55-73 55 Elsevier Interactive proof systems and alternating time-space complexity Lance Fortnow” and Carsten Lund** Department of Computer Science, Unicersity of Chicago. 1100 E. 58th Street, Chicago, IL 40637, USA Abstract Fortnow, L. and C. Lund, Interactive proof systems and alternating time-space complexity, Theoretical Computer Science 113 (1993) 55-73. We show a rough equivalence between alternating time-space complexity and a public-coin interactive proof system with the verifier having a polynomial-related time-space complexity. Special cases include the following: . All of NC has interactive proofs, with a log-space polynomial-time public-coin verifier vastly improving the best previous lower bound of LOGCFL for this model (Fortnow and Sipser, 1988). All languages in P have interactive proofs with a polynomial-time public-coin verifier using o(log’ n) space. l All exponential-time languages have interactive proof systems with public-coin polynomial-space exponential-time verifiers. To achieve better bounds, we show how to reduce a k-tape alternating Turing machine to a l-tape alternating Turing machine with only a constant factor increase in time and space. 1. Introduction In 1981, Chandra et al. [4] introduced alternating Turing machines, an extension of nondeterministic computation where the Turing machine can make both existential and universal moves. In 1985, Goldwasser et al. [lo] and Babai [l] introduced interactive proof systems, an extension of nondeterministic computation consisting of two players, an infinitely powerful prover and a probabilistic polynomial-time verifier. The prover will try to convince the verifier of the validity of some statement. -
Simulating Quantum Field Theory with a Quantum Computer
Simulating quantum field theory with a quantum computer John Preskill Lattice 2018 28 July 2018 This talk has two parts (1) Near-term prospects for quantum computing. (2) Opportunities in quantum simulation of quantum field theory. Exascale digital computers will advance our knowledge of QCD, but some challenges will remain, especially concerning real-time evolution and properties of nuclear matter and quark-gluon plasma at nonzero temperature and chemical potential. Digital computers may never be able to address these (and other) problems; quantum computers will solve them eventually, though I’m not sure when. The physics payoff may still be far away, but today’s research can hasten the arrival of a new era in which quantum simulation fuels progress in fundamental physics. Frontiers of Physics short distance long distance complexity Higgs boson Large scale structure “More is different” Neutrino masses Cosmic microwave Many-body entanglement background Supersymmetry Phases of quantum Dark matter matter Quantum gravity Dark energy Quantum computing String theory Gravitational waves Quantum spacetime particle collision molecular chemistry entangled electrons A quantum computer can simulate efficiently any physical process that occurs in Nature. (Maybe. We don’t actually know for sure.) superconductor black hole early universe Two fundamental ideas (1) Quantum complexity Why we think quantum computing is powerful. (2) Quantum error correction Why we think quantum computing is scalable. A complete description of a typical quantum state of just 300 qubits requires more bits than the number of atoms in the visible universe. Why we think quantum computing is powerful We know examples of problems that can be solved efficiently by a quantum computer, where we believe the problems are hard for classical computers. -
Complexity Theory Lecture 9 Co-NP Co-NP-Complete
Complexity Theory 1 Complexity Theory 2 co-NP Complexity Theory Lecture 9 As co-NP is the collection of complements of languages in NP, and P is closed under complementation, co-NP can also be characterised as the collection of languages of the form: ′ L = x y y <p( x ) R (x, y) { |∀ | | | | → } Anuj Dawar University of Cambridge Computer Laboratory NP – the collection of languages with succinct certificates of Easter Term 2010 membership. co-NP – the collection of languages with succinct certificates of http://www.cl.cam.ac.uk/teaching/0910/Complexity/ disqualification. Anuj Dawar May 14, 2010 Anuj Dawar May 14, 2010 Complexity Theory 3 Complexity Theory 4 NP co-NP co-NP-complete P VAL – the collection of Boolean expressions that are valid is co-NP-complete. Any language L that is the complement of an NP-complete language is co-NP-complete. Any of the situations is consistent with our present state of ¯ knowledge: Any reduction of a language L1 to L2 is also a reduction of L1–the complement of L1–to L¯2–the complement of L2. P = NP = co-NP • There is an easy reduction from the complement of SAT to VAL, P = NP co-NP = NP = co-NP • ∩ namely the map that takes an expression to its negation. P = NP co-NP = NP = co-NP • ∩ VAL P P = NP = co-NP ∈ ⇒ P = NP co-NP = NP = co-NP • ∩ VAL NP NP = co-NP ∈ ⇒ Anuj Dawar May 14, 2010 Anuj Dawar May 14, 2010 Complexity Theory 5 Complexity Theory 6 Prime Numbers Primality Consider the decision problem PRIME: Another way of putting this is that Composite is in NP. -
The Shrinking Property for NP and Conp✩
Theoretical Computer Science 412 (2011) 853–864 Contents lists available at ScienceDirect Theoretical Computer Science journal homepage: www.elsevier.com/locate/tcs The shrinking property for NP and coNPI Christian Glaßer a, Christian Reitwießner a,∗, Victor Selivanov b a Julius-Maximilians-Universität Würzburg, Germany b A.P. Ershov Institute of Informatics Systems, Siberian Division of the Russian Academy of Sciences, Russia article info a b s t r a c t Article history: We study the shrinking and separation properties (two notions well-known in descriptive Received 18 August 2009 set theory) for NP and coNP and show that under reasonable complexity-theoretic Received in revised form 11 November assumptions, both properties do not hold for NP and the shrinking property does not hold 2010 for coNP. In particular we obtain the following results. Accepted 15 November 2010 Communicated by A. Razborov P 1. NP and coNP do not have the shrinking property unless PH is finite. In general, Σn and P Πn do not have the shrinking property unless PH is finite. This solves an open question Keywords: posed by Selivanov (1994) [33]. Computational complexity 2. The separation property does not hold for NP unless UP ⊆ coNP. Polynomial hierarchy 3. The shrinking property does not hold for NP unless there exist NP-hard disjoint NP-pairs NP-pairs (existence of such pairs would contradict a conjecture of Even et al. (1984) [6]). P-separability Multivalued functions 4. The shrinking property does not hold for NP unless there exist complete disjoint NP- pairs. Moreover, we prove that the assumption NP 6D coNP is too weak to refute the shrinking property for NP in a relativizable way. -
On the Randomness Complexity of Interactive Proofs and Statistical Zero-Knowledge Proofs*
On the Randomness Complexity of Interactive Proofs and Statistical Zero-Knowledge Proofs* Benny Applebaum† Eyal Golombek* Abstract We study the randomness complexity of interactive proofs and zero-knowledge proofs. In particular, we ask whether it is possible to reduce the randomness complexity, R, of the verifier to be comparable with the number of bits, CV , that the verifier sends during the interaction. We show that such randomness sparsification is possible in several settings. Specifically, unconditional sparsification can be obtained in the non-uniform setting (where the verifier is modelled as a circuit), and in the uniform setting where the parties have access to a (reusable) common-random-string (CRS). We further show that constant-round uniform protocols can be sparsified without a CRS under a plausible worst-case complexity-theoretic assumption that was used previously in the context of derandomization. All the above sparsification results preserve statistical-zero knowledge provided that this property holds against a cheating verifier. We further show that randomness sparsification can be applied to honest-verifier statistical zero-knowledge (HVSZK) proofs at the expense of increasing the communica- tion from the prover by R−F bits, or, in the case of honest-verifier perfect zero-knowledge (HVPZK) by slowing down the simulation by a factor of 2R−F . Here F is a new measure of accessible bit complexity of an HVZK proof system that ranges from 0 to R, where a maximal grade of R is achieved when zero- knowledge holds against a “semi-malicious” verifier that maliciously selects its random tape and then plays honestly. -
Quantum Clustering Algorithms
Quantum Clustering Algorithms Esma A¨ımeur [email protected] Gilles Brassard [email protected] S´ebastien Gambs [email protected] Universit´ede Montr´eal, D´epartement d’informatique et de recherche op´erationnelle C.P. 6128, Succursale Centre-Ville, Montr´eal (Qu´ebec), H3C 3J7 Canada Abstract Multidisciplinary by nature, Quantum Information By the term “quantization”, we refer to the Processing (QIP) is at the crossroads of computer process of using quantum mechanics in order science, mathematics, physics and engineering. It con- to improve a classical algorithm, usually by cerns the implications of quantum mechanics for infor- making it go faster. In this paper, we initiate mation processing purposes (Nielsen & Chuang, 2000). the idea of quantizing clustering algorithms Quantum information is very different from its classi- by using variations on a celebrated quantum cal counterpart: it cannot be measured reliably and it algorithm due to Grover. After having intro- is disturbed by observation, but it can exist in a super- duced this novel approach to unsupervised position of classical states. Classical and quantum learning, we illustrate it with a quantized information can be used together to realize wonders version of three standard algorithms: divisive that are out of reach of classical information processing clustering, k-medians and an algorithm for alone, such as being able to factorize efficiently large the construction of a neighbourhood graph. numbers, with dramatic cryptographic consequences We obtain a significant speedup compared to (Shor, 1997), search in a unstructured database with a the classical approach. quadratic speedup compared to the best possible clas- sical algorithms (Grover, 1997) and allow two people to communicate in perfect secrecy under the nose of an 1. -
Week 1: an Overview of Circuit Complexity 1 Welcome 2
Topics in Circuit Complexity (CS354, Fall’11) Week 1: An Overview of Circuit Complexity Lecture Notes for 9/27 and 9/29 Ryan Williams 1 Welcome The area of circuit complexity has a long history, starting in the 1940’s. It is full of open problems and frontiers that seem insurmountable, yet the literature on circuit complexity is fairly large. There is much that we do know, although it is scattered across several textbooks and academic papers. I think now is a good time to look again at circuit complexity with fresh eyes, and try to see what can be done. 2 Preliminaries An n-bit Boolean function has domain f0; 1gn and co-domain f0; 1g. At a high level, the basic question asked in circuit complexity is: given a collection of “simple functions” and a target Boolean function f, how efficiently can f be computed (on all inputs) using the simple functions? Of course, efficiency can be measured in many ways. The most natural measure is that of the “size” of computation: how many copies of these simple functions are necessary to compute f? Let B be a set of Boolean functions, which we call a basis set. The fan-in of a function g 2 B is the number of inputs that g takes. (Typical choices are fan-in 2, or unbounded fan-in, meaning that g can take any number of inputs.) We define a circuit C with n inputs and size s over a basis B, as follows. C consists of a directed acyclic graph (DAG) of s + n + 2 nodes, with n sources and one sink (the sth node in some fixed topological order on the nodes). -
Free-Electron Qubits
Free-Electron Qubits Ori Reinhardt†, Chen Mechel†, Morgan Lynch, and Ido Kaminer Department of Electrical Engineering and Solid State Institute, Technion - Israel Institute of Technology, 32000 Haifa, Israel † equal contributors Free-electron interactions with laser-driven nanophotonic nearfields can quantize the electrons’ energy spectrum and provide control over this quantized degree of freedom. We propose to use such interactions to promote free electrons as carriers of quantum information and find how to create a qubit on a free electron. We find how to implement the qubit’s non-commutative spin-algebra, then control and measure the qubit state with a universal set of 1-qubit gates. These gates are within the current capabilities of femtosecond-pulsed laser-driven transmission electron microscopy. Pulsed laser driving promise configurability by the laser intensity, polarizability, pulse duration, and arrival times. Most platforms for quantum computation today rely on light-matter interactions of bound electrons such as in ion traps [1], superconducting circuits [2], and electron spin systems [3,4]. These form a natural choice for implementing 2-level systems with spin algebra. Instead of using bound electrons for quantum information processing, in this letter we propose using free electrons and manipulating them with femtosecond laser pulses in optical frequencies. Compared to bound electrons, free electron systems enable accessing high energy scales and short time scales. Moreover, they possess quantized degrees of freedom that can take unbounded values, such as orbital angular momentum (OAM), which has also been proposed for information encoding [5-10]. Analogously, photons also have the capability to encode information in their OAM [11,12]. -
Design of a Qubit and a Decoder in Quantum Computing Based on a Spin Field Effect
Design of a Qubit and a Decoder in Quantum Computing Based on a Spin Field Effect A. A. Suratgar1, S. Rafiei*2, A. A. Taherpour3, A. Babaei4 1 Assistant Professor, Electrical Engineering Department, Faculty of Engineering, Arak University, Arak, Iran. 1 Assistant Professor, Electrical Engineering Department, Amirkabir University of Technology, Tehran, Iran. 2 Young Researchers Club, Aligudarz Branch, Islamic Azad University, Aligudarz, Iran. *[email protected] 3 Professor, Chemistry Department, Faculty of Science, Islamic Azad University, Arak Branch, Arak, Iran. 4 faculty member of Islamic Azad University, Khomein Branch, Khomein, ABSTRACT In this paper we present a new method for designing a qubit and decoder in quantum computing based on the field effect in nuclear spin. In this method, the position of hydrogen has been studied in different external fields. The more we have different external field effects and electromagnetic radiation, the more we have different distribution ratios. Consequently, the quality of different distribution ratios has been applied to the suggested qubit and decoder model. We use the nuclear property of hydrogen in order to find a logical truth value. Computational results demonstrate the accuracy and efficiency that can be obtained with the use of these models. Keywords: quantum computing, qubit, decoder, gyromagnetic ratio, spin. 1. Introduction different hydrogen atoms in compound applied for the qubit and decoder designing. Up to now many papers deal with the possibility to realize a reversible computer based on the laws of 2. An overview on quantum concepts quantum mechanics [1]. In this chapter a short introduction is presented Modern quantum chemical methods provide into the interesting field of quantum in physics, powerful tools for theoretical modeling and Moore´s law and a summary of the quantum analysis of molecular electronic structures. -
A Theoretical Study of Quantum Memories in Ensemble-Based Media
A theoretical study of quantum memories in ensemble-based media Karl Bruno Surmacz St. Hugh's College, Oxford A thesis submitted to the Mathematical and Physical Sciences Division for the degree of Doctor of Philosophy in the University of Oxford Michaelmas Term, 2007 Atomic and Laser Physics, University of Oxford i A theoretical study of quantum memories in ensemble-based media Karl Bruno Surmacz, St. Hugh's College, Oxford Michaelmas Term 2007 Abstract The transfer of information from flying qubits to stationary qubits is a fundamental component of many quantum information processing and quantum communication schemes. The use of photons, which provide a fast and robust platform for encoding qubits, in such schemes relies on a quantum memory in which to store the photons, and retrieve them on-demand. Such a memory can consist of either a single absorber, or an ensemble of absorbers, with a ¤-type level structure, as well as other control ¯elds that a®ect the transfer of the quantum signal ¯eld to a material storage state. Ensembles have the advantage that the coupling of the signal ¯eld to the medium scales with the square root of the number of absorbers. In this thesis we theoretically study the use of ensembles of absorbers for a quantum memory. We characterize a general quantum memory in terms of its interaction with the signal and control ¯elds, and propose a ¯gure of merit that measures how well such a memory preserves entanglement. We derive an analytical expression for the entanglement ¯delity in terms of fluctuations in the stochastic Hamiltonian parameters, and show how this ¯gure could be measured experimentally.