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Randomized Linear Algebra Approaches to Estimate the Von Neumann Entropy of Density Matrices

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Randomized Linear Algebra Approaches to Estimate the Von Neumann Entropy of Density Matrices Randomized Linear Algebra Approaches to Estimate the Von Neumann Entropy of Density Matrices Eugenia-Maria Kontopoulou Ananth Grama Wojciech Szpankowski Petros Drineas Purdue University Purdue University Purdue University Purdue University Computer Science Computer Science Computer Science Computer Science West Lafayette, IN, USA West Lafayette, IN, USA West Lafayette, IN, USA West Lafayette, IN, USA Email: [email protected] Email: [email protected] Email: [email protected] Email: [email protected] Abstract—The von Neumann entropy, named after John von Motivated by the above discussion, we seek numerical Neumann, is the extension of classical entropy concepts to the algorithms that approximate the von Neumann entropy of large field of quantum mechanics and, from a numerical perspective, density matrices, e.g., symmetric positive definite matrices can be computed simply by computing all the eigenvalues of a 3 density matrix, an operation that could be prohibitively expensive with unit trace, faster than the trivial O(n ) approach. Our for large-scale density matrices. We present and analyze two algorithms build upon recent developments in the field of randomized algorithms to approximate the von Neumann entropy Randomized Numerical Linear Algebra (RandNLA), an in- of density matrices: our algorithms leverage recent developments terdisciplinary research area that exploits randomization as a in the Randomized Numerical Linear Algebra (RandNLA) liter- computational resource to develop improved algorithms for ature, such as randomized trace estimators, provable bounds for the power method, and the use of Taylor series and Chebyschev large-scale linear algebra problems. Indeed, our work here polynomials to approximate matrix functions. Both algorithms focuses at the intersection of RandNLA and information come with provable accuracy guarantees and our experimental theory, delivering novel randomized linear algebra algorithms evaluations support our theoretical findings showing considerable and related quality-of-approximation results for a fundamental speedup with small accuracy loss. information-theoretic metric. I. INTRODUCTION A. Background Entropy is a fundamental quantity in many areas of science We will focus on finite-dimensional function (state) spaces. and engineering. The von Neumann entropy, named after John In this setting, the density matrix R represents the statistical von Neumann, is the extension of classical entropy concepts mixture of pure states, and from a linear algebraic perspective to the field of quantum mechanics, and its foundations can be can be written as traced to von Neumann’s work on Mathematische Grundlagen 1 T n×n der Quantenmechanik . In his work, Von Neumann introduced R = ΨΣpΨ 2 R ; (2) the notion of the density matrix, which facilitated the extension where Ψ 2 n×k is the orthonormal matrix whose columns of the tools of classical statistical mechanics to the quantum R k×k are the k pure states and Σp 2 R is a diagonal matrix domain in order to develop a theory of quantum measurements. whose entries are the (positive) pi’s. Let h(x) = x ln x for From a mathematical perspective (see Section I-A for de- any x > 0 and let h(0) = 0. Then, using the cyclical property tails) the density matrix R is a symmetric positive semidefinite of the trace and the definition of h(x) eqn. (1) becomes n×n matrix in with unit trace. Let pi, i = 1 : : : n be the X T R − pi ln pi = −tr Ψh(Σp)Ψ = −tr (h(R)) = −tr (R ln R) : R R eigenvalues of , in decreasing order; then, the entropy of i;pi>0 (3) is defined as n X The second equality follows from the definition of matrix H(R) = − p ln p : (1) i i functions [3]. i=1 We conclude the section by noting that our algorithms will The above definition is a proper extension of both the Gibbs use two tools that appeared in prior work. The first tool is entropy and the Shannon entropy to the quantum case and the power method, with a provable analysis that first appeared implies an obvious algorithm to compute H(R) by first in [4]. The second tool is a provably accurate trace estimation computing the eigendecomposition of R; known algorithms algorithm for symmetric positive semidefinite matrices that 3 for this task necessitate O(n ) time [1]. Clearly, as n grows, appeared in [5]. such running times are impractical. For example, [2] describes an entangled two-photon state generated by spontaneous para- B. Our contributions metric down-conversion, which can result in a density matrix We present and analyze two randomized algorithms to ap- with n ≈ 108. proximate the von Neumann entropy of density matrices, lever- 1Originally published in German in 1932; published in English under the aging two different polynomial approximations of the matrix title Mathematical Foundations of Quantum Mechanics in 1955. function H(R) = −tr (R ln R): the first approximation uses a Taylor series expansion while the second approximation uses of ill-conditioned matrices. However, the dependency of the Chebyschev polynomials. Both algorithms return, with high algorithm of Theorem 35 of [8] on terms like (log n/)6 or probability, relative-error approximations to the true entropy n1=3nnz(A)+npnnz(A) could blow up the running time of of the input density matrix, under certain assumptions. More the proposed algorithm for reasonably conditioned matrices. specifically, in both cases, we need to assume that the input We also note the recent work in [10], which used Taylor density matrix has n non-zero eigenvalues, or, equivalently, approximations to matrix functions to estimate the log determi- that the probabilities pi, i = 1 : : : n, corresponding to the nant of symmetric positive definite matrices (see also Section underlying n pure states are non-zero. The running time of 1.2 of [10] for an overview of prior work on approximating both algorithms is proportional to the sparsity of the input matrix functions via Taylor series). Finally, the work of [11] density matrix and depends (see Theorems 1 and 3 for precise used a Chebyschev polynomial approximation to estimate the statements) on, roughly, the ratio of the largest to the smallest log determinant of a matrix and is reminiscent of our approach probability p1=pn, as well as the desired accuracy. in Section III and, of course, the work of [2]. From a technical perspective, the theoretical analysis of both II. AN APPROACH VIA TAYLOR SERIES algorithms proceeds by combining the power of polynomial approximations, either using Taylor series or Chebyschev Our first approach to approximate the von Neumann entropy polynomials, to matrix functions, combined with randomized of a density matrix uses a Taylor series expansion to approxi- trace estimators. A provably accurate variant of the power mate the logarithm of a matrix, combined with a relative-error method is used to estimate the largest probability p1. If this trace estimator for symmetric positive semi-definite matrices estimate is significantly smaller than one, it can improve the and the power method to upper bound the largest singular running times of the proposed algorithms (see discussion after value of a matrix. Theorem 1). We note that while the power method introduces A. Algorithm and Main Theorem an additional nnz(R) log(n) term in the running time, in practice, its computational overhead is negligible. Our main result is an analysis of Algorithm 1 (see below) Finally, in Section IV, we present a highlight of our that guarantees relative error approximation to the entropy experimental evaluations of our algorithms on large-scale of the density matrix R, under the assumption that R = Pn T n×n synthetic density matrices, generated using Matlab’s QETLAB i=1 pi i i 2 R has n pure states with 0 < ` ≤ pi toolbox [6]. For a 30,000 by 30,000 matrix that was used in our for all i = 1 : : : n. The following theorem is our main quality- evaluations, the exact computation of the entropy takes hours, Algorithm 1 A Taylor series approach to estimate the entropy. whereas our algorithms return approximations with relative n×n errors well below 0:5% in only a few minutes. 1: INPUT: R 2 R , accuracy parameter " > 0, failure We conclude by noting that a longer version of this short probability δ, and integer m > 0. abstract is available in [7]. Indeed, we will often reference [7] 2: Estimate p~1 using the power method of [4] with t = for omitted proofs, omitted experimental evaluations, and O(ln n) and q = O(ln(1/δ)). detailed discussions that are omitted from this version due 3: Set u = minf1; 6~p1g. 2 to space considerations. 4: Set s = 20 ln(2/δ)=" . n 5: Let g1; g2;:::; gs 2 R be i.i.d. random Gaussian vec- C. Prior work tors. To the best of our knowledge, prior to this work, the only 6: OUTPUT: return s m > −1 k non-trivial algorithm to approximate the von Neumann entropy 1 X X g R(In − u R) gi Hb (R) = ln u−1 + i : of a density matrix appeared in [2]. Their approach is based s k on the Chebyschev polynomial expansion that we also analyze i=1 k=1 in Section III. However, our analysis is somewhat different and, overall, much tighter, leveraging a provably accurate of-approximation result for Algorithm 1. variant of the power method as well as provably accurate trace estimators to derive a relative error approximation to the Theorem 1. Let R be a density matrix such that all proba- entropy of a density matrix, under appropriate assumptions. A bilities pi, i = 1 : : : n satisfy 0 < ` ≤ pi. Let u be computed brief technical comparison between our results in Section III as in Algorithm 1 and let Hb (R) be the output of Algorithm 1 and the work of [2] appears in Section III-A, while a detailed on inputs R, m, and < 1; Then, with probability at least 1 − 2δ, comparison can be found in Section 3.3 of [7].
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