A Mathematical Elements

Total Page:16

File Type:pdf, Size:1020Kb

A Mathematical Elements A Mathematical elements The language of quantum mechanics is rooted in linear algebra and functional analysis. It often expands when the foundations of the theory are reassessed, which they periodically have been. As a result, additional mathematical struc- tures not ordinarily covered in physics textbooks, such as the logic of linear subspaces and Liouville space, have come to play a role in quantum mechan- ics. Information theory makes use of statistical quantities, some of which are covered in the main text, as well as discrete mathematics, which are also not included in standard treatments of quantum theory. Here, before describ- ing the fundamental mathematical structure of quantum mechanics, Hilbert space, some basic mathematics of binary arithmetic, finite fields and random variables is given. After a presentation of the basic elements of Hilbert space theory, the Dirac notation typically used in the literature of quantum informa- tion science, and operators and transformations related to it, some elements of quantum probability and quantum logic are also described. A.1 Boolean algebra and Galois fields n A Boolean algebra Bn is an algebraic structure given by the collection of 2 subsets of the set I = {1, 2,...,n} and three operations under which it is closed: the two binary operations of union (∨) and intersection (∧), and a unary operation, complementation (¬). In addition to there being comple- ments (and hence the null set ∅ being an element), the following axioms hold. (i) Commutativity: S ∨ T = T ∨ S and S ∧ T = T ∧ S; (ii) Associativity: S ∨(T ∨U)=(S ∨T )∨U and S ∧(T ∧U)=(S ∧T )∧U; (iii) Distributivity: S ∧ (T ∨ U)=(S ∧ T ) ∨ (S ∧ U)andS ∨ (T ∧ U)= (S ∨ T ) ∧ (S ∨ U); (iv) ¬∅ = I, ¬I = ∅,S∧¬S = ∅,S∨¬S = I, ¬(¬S)=S , for all its elements S, T, U. The algebra B1 is the propositional calculus arising from the set I = {1}, which is also used in digital circuit theory, where ∅ 232 A Mathematical elements corresponds to FALSE, I to TRUE, ∨ to OR, ∧ to AND, and ¬ to NOT. The “basic” logic operation XOR corresponds to (S ∨ T ) ∧ (¬S ∨¬T ). A field is a set of elements, with two operations (called multiplication and addition) under which it is closed, that satisfies the axioms of associativity, commutativity and distributivity and in which there exist additive and mul- tiplicative identity elements and inverses. The order of a field is the number of its elements. Theorem (Galois): There exists a field of order q if and only if q is a prime power, that is, q = pn where p is prime and n is a positive integer; furthermore, if q is a prime power, then there is (up to a relabeling), only one field of that order. A field of order q is known as a Galois field and is denoted GF (q). If p is prime, then GF (p) is simply the set {0, 1, ..., p−1} with n mod-p arithmetic. GF (p ) is a linear space of dimension n over Zp (the set of integers 0 to p − 1) containing a copy of it as a subfield. For Z2, GF (2) is such a field over the set {0, 1} with XOR as addition and AND as multiplication. In computing, one is primarily interested in functions f : {0, 1}m →{0, 1}n; for example, two-bit logic gates g : GF (2) → GF (2). A.2 Random variables A random variable is a deterministic function from a given sample space S, that is, the set of all possible outcomes of a given experiment, the subsets of which are known as events, those containing only a single element being the elementary events, to the real numbers. The expectation value, E[Y (X)], of a function Y (X) of a random variable X is given by a linear operator such that n E[Y (X)] = Y (xi)P (X = xi) , (A.1) i=1 +∞ E[Y (X)] = Y (x)f(x)dx , (A.2) −∞ in the cases of discrete and continuous variables, respectively, where in the former P (X = xi)istheprobability andinthelatterf(x)istheprobability density (p.d.f.), that is, a collection of numbers between 0 and 1 summing to unity. For a random variable with a (differentiable) cumulative distribution function F (X), f(x) ≡ dF/dx; a random variable has the cumulative distri- bution function F (X) if the probability of an experiment with sample space S to yield x<X as an outcome is x F (x)=P (X<x)= f(x)dx . (A.3) −∞ A Markov chain is a sequence of random variables X1,X2,... such that a given random variable Xi is independent of all the variables X1,X2,...Xi−1, that is, is memoryless. A.3 Vector Spaces and Hilbert space 233 A.3 Vector Spaces and Hilbert space The central mathematical structure of quantum mechanics is Hilbert space, which is a specific sort of vector space. The Dirac notation commonly used to describe Hilbert-space calculations in the physics literature is introduced below, after this structure is described in standard mathematical notation. A vector space V over a field F is a set of vectors together with operations of scalar-multiplication and addition satisfying the following. For all elements a, b ∈ F and u, v, w ∈ V (with the zero vector denoted 0), scalar multiples and vector sums are elements of V such that: (i) There is a unique scalar zero element 0 ∈ F such that 0u = 0 for all u; (ii) 0 + u = u; (iii) a(u + v)=au + av and (a + b)u = au + bu;and (iv) u + v = v + u, and u +(v + w)=(u + v)+w. A (scalar) inner product (u, v) is assumed for all pairs u, v ∈ V such that: (i) (u, v)=(v, u)∗,where∗ indicates complex conjugation; (ii) (u, u) ≥ 0, with (u, u) = 0 if and only if u = 0; (iii) (u, v + w)=(u, v)+(u, w); and (iv) (u, av)=a(u, v). The norm of a vector, ||u|| ≡ (u, u), satisfies: (i) ||u|| ≥ 0 for all u ∈ V ,with||u|| = 0 if and only if u = 0; (ii) ||u + v||≤||u|| + ||v||, for all u, v ∈ V ;and (iii) ||au|| = |a|||u||, for all a ∈ F and all u ∈ V . The field F used in quantum mechanics is usually taken to be that of the complex numbers, C. The vectors for which ||u|| =1aretheunit vectors. Vectors u and v are orthogonal (u ⊥ v) if and only if (u, v)=0. A function of two vector arguments that is more general than the inner product, which serves similar purposes but is applicable in broader contexts, is the Hermitian form, h(u, v), which satisfies: (i) h(u, v)=h(v, u)∗; (ii) h(u, av)=ah(u, v), for all a ∈ F , u, v ∈ V ;and (iii) h(u + v, w)=h(u, w)+h(v, w), for all u, v, w ∈ V . A positive Hermitian form is an Hermitian form that also satisfies the con- dition h(u, u) ≥ 0 for every u ∈ V . An Hermitian form is positive-definite if h(u, u) = 0 implies u = 0;apositive-definite Hermitian form is an inner product. A basis for a vector space is a set of mutually orthogonal vectors such that every u ∈ V can be written as a linear combination of its elements; a basis is orthonormal if it is composed entirely of unit vectors. A set S of vectors in V is a subspace of V if S is a vector space in the same sense as V itself is. The one-dimensional subspaces of V are called rays. 234 A Mathematical elements A Hilbert space, H, is a complete complex vector space with an inner product for which (u, u) ≥ 0 for all u ∈H.AmapA : H→H, u → Au is a linear operator if, for all u, v ∈Hand scalars a ∈ F : (i) A(u + v)=Au + Av;and (ii) A(au)=a(Au). If, instead of (ii), one has A(au)=a∗(Au), then A is an anti-linear operator. (Together, the linear and anti-linear operators are fundamental to quantum mechanics; see Wigner’s theorem, below.) A vector space with an inner product, such as a Hilbert space, can be attributed a norm ||·||=(v, v)1/2, which provides a distance via d(v, w)= ||v − w||. Such a vector space is separable if there exists a countable subset in the space that is everywhere dense, that is, for every vector there is an element of the space within a distance of it for every positive real ; the space is complete if every Cauchy sequence—namely, every sequence such that for every >0thereisanumberN( ) such that ||vm − vn|| < if m, n > N( )— has a limit in the space. (For finite-dimensional spaces, one usually considers the norm topology, although weak topologies may be required to define needed limits and to give proper definitions of continuity.) A subspace of a Hilbert space H is a closed linear manifold, that is, a linear manifold containing its limit points; a linear manifold in H is a collection of vectors such that the scalar multiples and sums of all its vectors are in it. A bounded linear operator is a linear transformation L between normed vector spaces V and W such that the ratio of the norm of L(v) to the norm of v is bounded by the same number, for all non-zero vectors v ∈ V .The set of bounded linear operators on a Hilbert space H is designated B(H). The sum of two operators A and B, A + B, is another operator defined by (A + B)v = Av + Bv, for all v ∈H; multiplication of an operator by a scalar a is defined by (aA)v = a(Av), for all v ∈H; multiplication of two operators is defined by (AB)v = A(Bv), for all v ∈H.Thezero operator, O,and the unit operator, I, are defined by Ov = 0 and Iv = v, respectively.
Recommended publications
  • Pilot Quantum Error Correction for Global
    Pilot Quantum Error Correction for Global- Scale Quantum Communications Laszlo Gyongyosi*1,2, Member, IEEE, Sandor Imre1, Member, IEEE 1Quantum Technologies Laboratory, Department of Telecommunications Budapest University of Technology and Economics 2 Magyar tudosok krt, H-1111, Budapest, Hungary 2Information Systems Research Group, Mathematics and Natural Sciences Hungarian Academy of Sciences H-1518, Budapest, Hungary *[email protected] Real global-scale quantum communications and quantum key distribution systems cannot be implemented by the current fiber and free-space links. These links have high attenuation, low polarization-preserving capability or extreme sensitivity to the environment. A potential solution to the problem is the space-earth quantum channels. These channels have no absorption since the signal states are propagated in empty space, however a small fraction of these channels is in the atmosphere, which causes slight depolarizing effect. Furthermore, the relative motion of the ground station and the satellite causes a rotation in the polarization of the quantum states. In the current approaches to compensate for these types of polarization errors, high computational costs and extra physical apparatuses are required. Here we introduce a novel approach which breaks with the traditional views of currently developed quantum-error correction schemes. The proposed solution can be applied to fix the polarization errors which are critical in space-earth quantum communication systems. The channel coding scheme provides capacity-achieving communication over slightly depolarizing space-earth channels. I. Introduction Quantum error-correction schemes use different techniques to correct the various possible errors which occur in a quantum channel. In the first decade of the 21st century, many revolutionary properties of quantum channels were discovered [12-16], [19-22] however the efficient error- correction in quantum systems is still a challenge.
    [Show full text]
  • Analysis of Quantum Error-Correcting Codes: Symplectic Lattice Codes and Toric Codes
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Caltech Theses and Dissertations Analysis of quantum error-correcting codes: symplectic lattice codes and toric codes Thesis by James William Harrington Advisor John Preskill In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2004 (Defended May 17, 2004) ii c 2004 James William Harrington All rights Reserved iii Acknowledgements I can do all things through Christ, who strengthens me. Phillipians 4:13 (NKJV) I wish to acknowledge first of all my parents, brothers, and grandmother for all of their love, prayers, and support. Thanks to my advisor, John Preskill, for his generous support of my graduate studies, for introducing me to the studies of quantum error correction, and for encouraging me to pursue challenging questions in this fascinating field. Over the years I have benefited greatly from stimulating discussions on the subject of quantum information with Anura Abeyesinge, Charlene Ahn, Dave Ba- con, Dave Beckman, Charlie Bennett, Sergey Bravyi, Carl Caves, Isaac Chenchiah, Keng-Hwee Chiam, Richard Cleve, John Cortese, Sumit Daftuar, Ivan Deutsch, Andrew Doherty, Jon Dowling, Bryan Eastin, Steven van Enk, Chris Fuchs, Sho- hini Ghose, Daniel Gottesman, Ted Harder, Patrick Hayden, Richard Hughes, Deborah Jackson, Alexei Kitaev, Greg Kuperberg, Andrew Landahl, Chris Lee, Debbie Leung, Carlos Mochon, Michael Nielsen, Smith Nielsen, Harold Ollivier, Tobias Osborne, Michael Postol, Philippe Pouliot, Marco Pravia, John Preskill, Eric Rains, Robert Raussendorf, Joe Renes, Deborah Santamore, Yaoyun Shi, Pe- ter Shor, Marcus Silva, Graeme Smith, Jennifer Sokol, Federico Spedalieri, Rene Stock, Francis Su, Jacob Taylor, Ben Toner, Guifre Vidal, and Mas Yamada.
    [Show full text]
  • QIP 2010 Tutorial and Scientific Programmes
    QIP 2010 15th – 22nd January, Zürich, Switzerland Tutorial and Scientific Programmes asymptotically large number of channel uses. Such “regularized” formulas tell us Friday, 15th January very little. The purpose of this talk is to give an overview of what we know about 10:00 – 17:10 Jiannis Pachos (Univ. Leeds) this need for regularization, when and why it happens, and what it means. I will Why should anyone care about computing with anyons? focus on the quantum capacity of a quantum channel, which is the case we understand best. This is a short course in topological quantum computation. The topics to be covered include: 1. Introduction to anyons and topological models. 15:00 – 16:55 Daniel Nagaj (Slovak Academy of Sciences) 2. Quantum Double Models. These are stabilizer codes, that can be described Local Hamiltonians in quantum computation very much like quantum error correcting codes. They include the toric code This talk is about two Hamiltonian Complexity questions. First, how hard is it to and various extensions. compute the ground state properties of quantum systems with local Hamiltonians? 3. The Jones polynomials, their relation to anyons and their approximation by Second, which spin systems with time-independent (and perhaps, translationally- quantum algorithms. invariant) local interactions could be used for universal computation? 4. Overview of current state of topological quantum computation and open Aiming at a participant without previous understanding of complexity theory, we will discuss two locally-constrained quantum problems: k-local Hamiltonian and questions. quantum k-SAT. Learning the techniques of Kitaev and others along the way, our first goal is the understanding of QMA-completeness of these problems.
    [Show full text]
  • Reliably Distinguishing States in Qutrit Channels Using One-Way LOCC
    Reliably distinguishing states in qutrit channels using one-way LOCC Christopher King Department of Mathematics, Northeastern University, Boston MA 02115 Daniel Matysiak College of Computer and Information Science, Northeastern University, Boston MA 02115 July 15, 2018 Abstract We present numerical evidence showing that any three-dimensional subspace of C3 ⊗ Cn has an orthonormal basis which can be reliably dis- tinguished using one-way LOCC, where a measurement is made first on the 3-dimensional part and the result used to select an optimal measure- ment on the n-dimensional part. This conjecture has implications for the LOCC-assisted capacity of certain quantum channels, where coordinated measurements are made on the system and environment. By measuring first in the environment, the conjecture would imply that the environment- arXiv:quant-ph/0510004v1 1 Oct 2005 assisted classical capacity of any rank three channel is at least log 3. Sim- ilarly by measuring first on the system side, the conjecture would imply that the environment-assisting classical capacity of any qutrit channel is log 3. We also show that one-way LOCC is not symmetric, by providing an example of a qutrit channel whose environment-assisted classical capacity is less than log 3. 1 1 Introduction and statement of results The noise in a quantum channel arises from its interaction with the environment. This viewpoint is expressed concisely in the Lindblad-Stinespring representation [6, 8]: Φ(|ψihψ|)= Tr U(|ψihψ|⊗|ǫihǫ|)U ∗ (1) E Here E is the state space of the environment, which is assumed to be initially prepared in a pure state |ǫi.
    [Show full text]
  • Potential Capacities of Quantum Channels 2
    WINTER AND YANG: POTENTIAL CAPACITIES OF QUANTUM CHANNELS 1 Potential Capacities of Quantum Channels Andreas Winter and Dong Yang Abstract—We introduce potential capacities of quantum chan- Entangled inputs between different quantum channels open the nels in an operational way and provide upper bounds for these door to all kinds of effects that are impossible in classical quantities, which quantify the ultimate limit of usefulness of a information theory. An extreme phenomenon is superactivation channel for a given task in the best possible context. Unfortunately, except for a few isolated cases, potential capac- [13]; there exist two quantum channels that cannot transmit ities seem to be as hard to compute as their “plain” analogues. quantum information when they are used individually, but can We thus study upper bounds on some potential capacities: For transmit at positive rate when they are used together. the classical capacity, we give an upper bound in terms of the The phenomenon of superactivation, and more broadly of entanglement of formation. To establish a bound for the quantum super-additivity, implies that the capacity of a quantum channel and private capacity, we first “lift” the channel to a Hadamard channel and then prove that the quantum and private capacity does not adequately characterize the channel, since the utility of a Hadamard channel is strongly additive, implying that for of the channel depends on what other contextual channels are these channels, potential and plain capacity are equal. Employing available. So it is natural to ask the following question: What these upper bounds we show that if a channel is noisy, however is the maximum possible capability of a channel to transmit close it is to the noiseless channel, then it cannot be activated information when it is used in combination with any other into the noiseless channel by any other contextual channel; this conclusion holds for all the three capacities.
    [Show full text]
  • Linear Operators
    C H A P T E R 10 Linear Operators Recall that a linear transformation T ∞ L(V) of a vector space into itself is called a (linear) operator. In this chapter we shall elaborate somewhat on the theory of operators. In so doing, we will define several important types of operators, and we will also prove some important diagonalization theorems. Much of this material is directly useful in physics and engineering as well as in mathematics. While some of this chapter overlaps with Chapter 8, we assume that the reader has studied at least Section 8.1. 10.1 LINEAR FUNCTIONALS AND ADJOINTS Recall that in Theorem 9.3 we showed that for a finite-dimensional real inner product space V, the mapping u ’ Lu = Óu, Ô was an isomorphism of V onto V*. This mapping had the property that Lauv = Óau, vÔ = aÓu, vÔ = aLuv, and hence Lau = aLu for all u ∞ V and a ∞ ®. However, if V is a complex space with a Hermitian inner product, then Lauv = Óau, vÔ = a*Óu, vÔ = a*Luv, and hence Lau = a*Lu which is not even linear (this was the definition of an anti- linear (or conjugate linear) transformation given in Section 9.2). Fortunately, there is a closely related result that holds even for complex vector spaces. Let V be finite-dimensional over ç, and assume that V has an inner prod- uct Ó , Ô defined on it (this is just a positive definite Hermitian form on V). Thus for any X, Y ∞ V we have ÓX, YÔ ∞ ç.
    [Show full text]
  • On Some Extensions of Bernstein's Inequality for Self-Adjoint Operators
    On Some Extensions of Bernstein's Inequality for Self-Adjoint Operators Stanislav Minsker∗ e-mail: [email protected] Abstract: We present some extensions of Bernstein's concentration inequality for random matrices. This inequality has become a useful and powerful tool for many problems in statis- tics, signal processing and theoretical computer science. The main feature of our bounds is that, unlike the majority of previous related results, they do not depend on the dimension d of the ambient space. Instead, the dimension factor is replaced by the “effective rank" associated with the underlying distribution that is bounded from above by d. In particular, this makes an extension to the infinite-dimensional setting possible. Our inequalities refine earlier results in this direction obtained by D. Hsu, S. M. Kakade and T. Zhang. Keywords and phrases: Bernstein's inequality, Random Matrix, Effective rank, Concen- tration inequality, Large deviations. 1. Introduction Theoretical analysis of many problems in applied mathematics can often be reduced to problems about random matrices. Examples include numerical linear algebra (randomized matrix decom- positions, approximate matrix multiplication), statistics and machine learning (low-rank matrix recovery, covariance estimation), mathematical signal processing, among others. n P Often, resulting questions can be reduced to estimates for the expectation E Xi − EXi or i=1 n P n probability P Xi − EXi > t , where fXigi=1 is a finite sequence of random matrices and i=1 k · k is the operator norm. Some of the early work on the subject includes the papers on noncom- mutative Khintchine inequalities [LPP91, LP86]; these tools were used by M.
    [Show full text]
  • MATRICES WHOSE HERMITIAN PART IS POSITIVE DEFINITE Thesis by Charles Royal Johnson in Partial Fulfillment of the Requirements Fo
    MATRICES WHOSE HERMITIAN PART IS POSITIVE DEFINITE Thesis by Charles Royal Johnson In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1972 (Submitted March 31, 1972) ii ACKNOWLEDGMENTS I am most thankful to my adviser Professor Olga Taus sky Todd for the inspiration she gave me during my graduate career as well as the painstaking time and effort she lent to this thesis. I am also particularly grateful to Professor Charles De Prima and Professor Ky Fan for the helpful discussions of my work which I had with them at various times. For their financial support of my graduate tenure I wish to thank the National Science Foundation and Ford Foundation as well as the California Institute of Technology. It has been important to me that Caltech has been a most pleasant place to work. I have enjoyed living with the men of Fleming House for two years, and in the Department of Mathematics the faculty members have always been generous with their time and the secretaries pleasant to work around. iii ABSTRACT We are concerned with the class Iln of n><n complex matrices A for which the Hermitian part H(A) = A2A * is positive definite. Various connections are established with other classes such as the stable, D-stable and dominant diagonal matrices. For instance it is proved that if there exist positive diagonal matrices D, E such that DAE is either row dominant or column dominant and has positive diag­ onal entries, then there is a positive diagonal F such that FA E Iln.
    [Show full text]
  • ECE 487 Lecture 3 : Foundations of Quantum Mechanics II
    ECE 487 Lecture 12 : Tools to Understand Nanotechnology III Class Outline: •Inverse Operator •Unitary Operators •Hermitian Operators Things you should know when you leave… Key Questions • What is an inverse operator? • What is the definition of a unitary operator? • What is a Hermetian operator and what do we use them for? M. J. Gilbert ECE 487 – Lecture 1 2 02/24/1 1 Tools to Understand Nanotechnology- III Let’s begin with a problem… Prove that the sum of the modulus squared of the matrix elements of a linear operator  is independent of the complete orthonormal basis used to represent the operator. M. J. Gilbert ECE 487 – Lecture 1 2 02/24/1 1 Tools to Understand Nanotechnology- III Last time we considered operators and how to represent them… •Most of the time we spent trying to understand the most simple of the operator classes we consider…the identity operator. •But he was only one of four operator classes to consider. 1. The identity operator…important for operator algebra. 2. Inverse operators…finding these often solves a physical problem mathematically and the are also important in operator algebra. 3. Unitary operators…these are very useful for changing the basis for representing the vectors and describing the evolution of quantum mechanical systems. 4. Hermitian operators…these are used to represent measurable quantities in quantum mechanics and they have some very powerful mathematical properties. M. J. Gilbert ECE 487 – Lecture 1 2 02/24/1 1 Tools to Understand Nanotechnology- III So, let’s move on to start discussing the inverse operator… Consider an operator  operating on some arbitrary function |f>.
    [Show full text]
  • An Introduction to Functional Analysis for Science and Engineering David A
    An introduction to functional analysis for science and engineering David A. B. Miller Stanford University This article is a tutorial introduction to the functional analysis mathematics needed in many physical problems, such as in waves in continuous media. This mathematics takes us beyond that of finite matrices, allowing us to work meaningfully with infinite sets of continuous functions. It resolves important issues, such as whether, why and how we can practically reduce such problems to finite matrix approximations. This branch of mathematics is well developed and its results are widely used in many areas of science and engineering. It is, however, difficult to find a readable introduction that both is deep enough to give a solid and reliable grounding but yet is efficient and comprehensible. To keep this introduction accessible and compact, I have selected only the topics necessary for the most important results, but the argument is mathematically complete and self-contained. Specifically, the article starts from elementary ideas of sets and sequences of real numbers. It then develops spaces of vectors or functions, introducing the concepts of norms and metrics that allow us to consider the idea of convergence of vectors and of functions. Adding the inner product, it introduces Hilbert spaces, and proceeds to the key forms of operators that map vectors or functions to other vectors or functions in the same or a different Hilbert space. This leads to the central concept of compact operators, which allows us to resolve many difficulties of working with infinite sets of vectors or functions. We then introduce Hilbert-Schmidt operators, which are compact operators encountered extensively in physical problems, such as those involving waves.
    [Show full text]
  • Bulk-Boundary Correspondence for Disordered Free-Fermion Topological
    Bulk-boundary correspondence for disordered free-fermion topological phases Alexander Alldridge1, Christopher Max1,2, Martin R. Zirnbauer2 1 Mathematisches Institut, E-mail: [email protected] 2 Institut für theoretische Physik, E-mail: [email protected], [email protected] Received: 2019-07-12 Abstract: Guided by the many-particle quantum theory of interacting systems, we de- velop a uniform classification scheme for topological phases of disordered gapped free fermions, encompassing all symmetry classes of the Tenfold Way. We apply this scheme to give a mathematically rigorous proof of bulk-boundary correspondence. To that end, we construct real C∗-algebras harbouring the bulk and boundary data of disordered free-fermion ground states. These we connect by a natural bulk-to-boundary short ex- act sequence, realising the bulk system as a quotient of the half-space theory modulo boundary contributions. To every ground state, we attach two classes in different pic- tures of real operator K-theory (or KR-theory): a bulk class, using Van Daele’s picture, along with a boundary class, using Kasparov’s Fredholm picture. We then show that the connecting map for the bulk-to-boundary sequence maps these KR-theory classes to each other. This implies bulk-boundary correspondence, in the presence of disorder, for both the “strong” and the “weak” invariants. 1. Introduction The research field of topological quantum matter stands out by its considerable scope, pairing strong impact on experimental physics with mathematical structure and depth. On the experimental side, physicists are searching for novel materials with technolog- ical potential, e.g., for future quantum computers; on the theoretical side, mathemati- cians have been inspired to revisit and advance areas such as topological field theory and various types of generalised cohomology and homotopy theory.
    [Show full text]
  • Nearly Tight Trotterization of Interacting Electrons
    Nearly tight Trotterization of interacting electrons Yuan Su1,2, Hsin-Yuan Huang1,3, and Earl T. Campbell4 1Institute for Quantum Information and Matter, Caltech, Pasadena, CA 91125, USA 2Google Research, Venice, CA 90291, USA 3Department of Computing and Mathematical Sciences, Caltech, Pasadena, CA 91125, USA 4AWS Center for Quantum Computing, Pasadena, CA 91125, USA We consider simulating quantum systems on digital quantum computers. We show that the performance of quantum simulation can be improved by simultaneously exploiting commutativity of the target Hamiltonian, sparsity of interactions, and prior knowledge of the initial state. We achieve this us- ing Trotterization for a class of interacting electrons that encompasses various physical systems, including the plane-wave-basis electronic structure and the Fermi-Hubbard model. We estimate the simulation error by taking the tran- sition amplitude of nested commutators of the Hamiltonian terms within the η-electron manifold. We develop multiple techniques for bounding the transi- tion amplitude and expectation of general fermionic operators, which may be 5/3 of independent interest. We show that it suffices to use n n4/3η2/3 no(1) η2/3 + gates to simulate electronic structure in the plane-wave basis with n spin or- bitals and η electrons, improving the best previous result in second quantiza- tion up to a negligible factor while outperforming the first-quantized simulation when n = η2−o(1). We also obtain an improvement for simulating the Fermi- Hubbard model. We construct concrete examples for which our bounds are almost saturated, giving a nearly tight Trotterization of interacting electrons. arXiv:2012.09194v2 [quant-ph] 30 Jun 2021 Accepted in Quantum 2021-06-03, click title to verify.
    [Show full text]