Quantum algorithm for estimating volumes of convex bodies

Estimating the volume of a convex body is a central problem in convex geometry and can be viewed as a continuous version of counting. We present a quantum algorithm that estimates the volume of an $n$-dimensional convex body within multiplicative error $\epsilon$ using $\tilde{O}(n^{3}+n^{2.5}/\epsilon)$ queries to a membership oracle and $\tilde{O}(n^{5}+n^{4.5}/\epsilon)$ additional arithmetic operations. For comparison, the best known classical algorithm uses $\tilde{O}(n^{4}+n^{3}/\epsilon^{2})$ queries and $\tilde{O}(n^{6}+n^{5}/\epsilon^{2})$ additional arithmetic operations. To the best of our knowledge, this is the first quantum speedup for volume estimation. Our algorithm is based on a refined framework for speeding up simulated annealing algorithms that might be of independent interest. This framework applies in the setting of "Chebyshev cooling", where the solution is expressed as a telescoping product of ratios, each having bounded variance. We develop several novel techniques when implementing our framework, including a theory of continuous-space quantum walks with rigorous bounds on discretization error. To complement our quantum algorithms, we also prove that volume estimation requires $\Omega(\sqrt n+1/\epsilon)$ quantum membership queries, which rules out the possibility of exponential quantum speedup in $n$ and shows optimality of our algorithm in $1/\epsilon$ up to poly-logarithmic factors.Comment: 61 pages, 8 figures. v2: Quantum query complexity improved to $\tilde{O}(n^{3}+n^{2.5}/\epsilon)$ and number of additional arithmetic operations improved to $\tilde{O}(n^{5}+n^{4.5}/\epsilon)$. v3: Improved Section 4.3.3 on nondestructive mean estimation and Section 6 on quantum lower bounds; various minor change

Entanglement in bipartite quantum systems: Euclidean volume ratios and detectability by Bell inequalities

Euclidean volume ratios between quantum states with positive partial transpose and all quantum states in bipartite systems are investigated. These ratios allow a quantitative exploration of the typicality of entanglement and of its detectability by Bell inequalities. For this purpose a new numerical approach is developed. It is based on the Peres-Horodecki criterion, on a characterization of the convex set of quantum states by inequalities resulting from Newton identities and from Descartes' rule of signs, and on a numerical approach involving the multiphase Monte Carlo method and the hit-and-run algorithm. This approach confirms not only recent analytical and numerical results on two-qubit, qubit--qutrit, and qubit--four-level qudit states but also allows for a numerically reliable numerical treatment of so far unexplored qutrit--qutrit states. Based on this numerical approach with the help of the Clauser-Horne-Shimony-Holt inequality and the Collins-Gisin inequality the degree of detectability of entanglement is investigated for two-qubit quantum states. It is investigated quantitatively to which extent a combined test of both Bell inequalities can increase the detectability of entanglement beyond what is achievable by each of these inequalities separately.Comment: 29 pages, 4 figure

Two-Qubit Separability Probabilities and Beta Functions

Due to recent important work of Zyczkowski and Sommers (quant-ph/0302197 and quant-ph/0304041), exact formulas are available (both in terms of the Hilbert-Schmidt and Bures metrics) for the (n^2-1)-dimensional and (n(n-1)/2-1)-dimensional volumes of the complex and real n x n density matrices. However, no comparable formulas are available for the volumes (and, hence, probabilities) of various separable subsets of them. We seek to clarify this situation for the Hilbert-Schmidt metric for the simplest possible case of n=4, that is, the two-qubit systems. Making use of the density matrix (rho) parameterization of Bloore (J. Phys. A 9, 2059 [1976]), we are able to reduce each of the real and complex volume problems to the calculation of a one-dimensional integral, the single relevant variable being a certain ratio of diagonal entries, nu = (rho_{11} rho_{44})/{rho_{22} rho_{33})$. The associated integrand in each case is the product of a known (highly oscillatory near nu=1) jacobian and a certain unknown univariate function, which our extensive numerical (quasi-Monte Carlo) computations indicate is very closely proportional to an (incomplete) beta function B_{nu}(a,b), with a=1/2, b=sqrt{3}in the real case, and a=2 sqrt{6}/5, b =3/sqrt{2} in the complex case. Assuming the full applicability of these specific incomplete beta functions, we undertake separable volume calculations.Comment: 17 pages, 4 figures, paper is substantially rewritten and reorganized, with the quasi-Monte Carlo integration sample size being greatly increase

Simpler (classical) and faster (quantum) algorithms for Gibbs partition functions

We consider the problem of approximating the partition function of a classical Hamiltonian using simulated annealing. This requires the computation of a cooling schedule, and the ability to estimate the mean of the Gibbs distributions at the corresponding inverse temperatures. We propose classical and quantum algorithms for these two tasks, achieving two goals: (i) we simplify the seminal work of \v{S}tefankovi\v{c}, Vempala and Vigoda (\emph{J.~ACM}, 56(3), 2009), improving their running time and almost matching that of the current classical state of the art; (ii) we quantize our new simple algorithm, improving upon the best known algorithm for computing partition functions of many problems, due to Harrow and Wei (SODA 2020). A key ingredient of our method is the paired-product estimator of Huber (\emph{Ann.\ Appl.\ Probab.}, 25(2),~2015). The proposed quantum algorithm has two advantages over the classical algorithm: it has quadratically faster dependence on the spectral gap of the Markov chains as well as the precision, and it computes a shorter cooling schedule, which matches the length conjectured to be optimal by \v{S}tefankovi\v{c}, Vempala and Vigoda.Comment: Comments welcom

Applied Harmonic Analysis and Sparse Approximation

Efficiently analyzing functions, in particular multivariate functions, is a key problem in applied mathematics. The area of applied harmonic analysis has a significant impact on this problem by providing methodologies both for theoretical questions and for a wide range of applications in technology and science, such as image processing. Approximation theory, in particular the branch of the theory of sparse approximations, is closely intertwined with this area with a lot of recent exciting developments in the intersection of both. Research topics typically also involve related areas such as convex optimization, probability theory, and Banach space geometry. The workshop was the continuation of a first event in 2012 and intended to bring together world leading experts in these areas, to report on recent developments, and to foster new developments and collaborations

Convex Geometry and its Applications (hybrid meeting)

The geometry of convex domains in Euclidean space plays a central role in several branches of mathematics: functional and harmonic analysis, the theory of PDE, linear programming and, increasingly, in the study of algorithms in computer science. The purpose of this meeting was to bring together researchers from the analytic, geometric and probabilistic groups who have contributed to these developments

Quantum algorithms for machine learning and optimization

The theories of optimization and machine learning answer foundational questions in computer science and lead to new algorithms for practical applications. While these topics have been extensively studied in the context of classical computing, their quantum counterparts are far from well-understood. In this thesis, we explore algorithms that bridge the gap between the fields of quantum computing and machine learning. First, we consider general optimization problems with only function evaluations. For two core problems, namely general convex optimization and volume estimation of convex bodies, we give quantum algorithms as well as quantum lower bounds that constitute the quantum speedups of both problems to be polynomial compared to their classical counterparts. We then consider machine learning and optimization problems with input data stored explicitly as matrices. We first look at semidefinite programs and provide quantum algorithms with polynomial speedup compared to the classical state-of-the-art. We then move to machine learning and give the optimal quantum algorithms for linear and kernel-based classifications. To complement with our quantum algorithms, we also introduce a framework for quantum-inspired classical algorithms, showing that for low-rank matrix arithmetics there can only be polynomial quantum speedup. Finally, we study statistical problems on quantum computers, with the focus on testing properties of probability distributions. We show that for testing various properties including L1-distance, L2-distance, Shannon and Renyi entropies, etc., there are polynomial quantum speedups compared to their classical counterparts. We also extend these results to testing properties of quantum states