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arXiv:2408.00235

Solving cluster moment relaxation with hierarchical matrix

Yi Wang, Huang, Rizheng, Yuehaw Khoo

Published Aug 1, 2024• 0 citations
arXivHTMLPDF

Abstract

Convex relaxation methods are powerful tools for studying the lowest energy of many-body problems. By relaxing the representability conditions for marginals to a set of local constraints, along with a global semidefinite constraint, a polynomial-time solvable semidefinite program (SDP) that provides a lower bound for the energy can be derived. In this paper, we propose accelerating the solution of such an SDP relaxation by imposing a hierarchical structure on the positive semidefinite (PSD) primal and dual variables. Furthermore, these matrices can be updated efficiently using the algebra of the compressed representations within an augmented Lagrangian method. We achieve quadratic and even near-linear time per-iteration complexity. Through experimentation on the quantum transverse field Ising model, we showcase the capability of our approach to provide a sufficiently accurate lower bound for the exact ground-state energy.

DOI
10.48550/arxiv.2408.00235

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Convex relaxation methods are powerful tools for studying the lowest energy of many-body problems. By relaxing the representability conditions for marginals to a set of local constraints, along with a global semidefinite constraint, a polynomial-time solvable semidefinite program (SDP) that provides a lower bound for the energy can be derived. In this paper, we propose accelerating the solution of such an SDP relaxation by imposing a hierarchical structure on the positive semidefinite (PSD) primal and dual variables. Furthermore, these matrices can be updated efficiently using the algebra of the compressed representations within an augmented Lagrangian method. We achieve quadratic and even near-linear time per-iteration complexity. Through experimentation on the quantum transverse field Ising model, we showcase the capability of our approach to provide a sufficiently accurate lower bound for the exact ground-state energy.
We study the use of von Neumann entropy constraints for obtaining lower bounds on the ground energy of quantum many-body systems. Known methods for obtaining certificates on the ground energy typically use consistency of local observables and are expressed as semidefinite programming relaxations. The local marginals defined by such a relaxation do not necessarily satisfy entropy inequalities that follow from the existence of a global state. Here, we propose to add such entropy constraints that lead to tighter convex relaxations for the ground energy problem. We give analytical and numerical results illustrating the advantages of such entropy constraints. We also show limitations of the entropy constraints we construct: they are implied by doubling the number of sites in the relaxation and as a result they can at best lead to a quadratic improvement in terms of the matrix sizes of the variables. We explain the relation to a method for approximating the free energy known as the Markov Entropy Decomposition method.
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We give a quantum algorithm for solving semidefinite programs (SDPs). It has worst-case running time n12m12s2poly(log⁡(n),log⁡(m),R,r,1/δ)n^{\frac{1}{2}} m^{\frac{1}{2}} s^2 \text{poly}(\log(n), \log(m), R, r, 1/δ)n21​m21​s2poly(log(n),log(m),R,r,1/δ), with nnn and sss the dimension and row-sparsity of the input matrices, respectively, mmm the number of constraints, δδδ the accuracy of the solution, and R,rR, rR,r a upper bounds on the size of the optimal primal and dual solutions. This gives a square-root unconditional speed-up over any classical method for solving SDPs both in nnn and mmm. We prove the algorithm cannot be substantially improved (in terms of nnn and mmm) giving a Ω(n12+m12)Ω(n^{\frac{1}{2}}+m^{\frac{1}{2}})Ω(n21​+m21​) quantum lower bound for solving semidefinite programs with constant s,R,rs, R, rs,R,r and δδδ. The quantum algorithm is constructed by a combination of quantum Gibbs sampling and the multiplicative weight method. In particular it is based on a classical algorithm of Arora and Kale for approximately solving SDPs. We present a modification of their algorithm to eliminate the need for solving an inner linear program which may be of independent interest.
In computer vision, many problems can be formulated as binary quadratic programs (BQPs), which are in general NP hard. Finding a solution when the problem is of large size to be of practical interest typically requires relaxation. Semidefinite relaxation usually yields tight bounds, but its computational complexity is high. In this work, we present a semidefinite programming (SDP) formulation for BQPs, with two desirable properties. First, it produces similar bounds to the standard SDP formulation. Second, compared with the conventional SDP formulation, the proposed SDP formulation leads to a considerably more efficient and scalable dual optimization approach. We then propose two solvers, namely, quasi-Newton and smoothing Newton methods, for the simplified dual problem. Both of them are significantly more efficient than standard interior-point methods. Empirically the smoothing Newton solver is faster than the quasi-Newton solver for dense or medium-sized problems, while the quasi-Newton solver is preferable for large sparse/structured problems.
We give a quantum algorithm for solving semidefinite programs (SDPs). It has worst-case running time n1/2m1/2s2poly(log(n), log(m), R, r, 1/δ), with n and s the dimension and row-sparsity of the input matrices, respectively, m the number of constraints, δ the accuracy of the solution, and R, r upper bounds on the size of the optimal primal and dual solutions, respectively. This gives a square-root unconditional speed-up over any classical method for solving SDPs both in n and m. We prove the algorithm cannot be substantially improved (in terms of n and m) giving a Ω(n1/2+ m2) quantum lower bound for solving semidefinite programs with constant s, R, r and δ. The quantum algorithm is constructed by a combination of quantum Gibbs sampling and the multiplicative weight method. In particular it is based on a classical algorithm of Arora and Kale for approximately solving SDPs. We present a modification of their algorithm to eliminate the need for solving an inner linear program which may be of independent interest.
A hierarchy of semidefinite programming (SDP) relaxations approximates the global optimum of polynomial optimization problems of noncommuting variables. Generating the relaxation, however, is a computationally demanding task, and only problems of commuting variables have efficient generators. We develop an implementation for problems of noncommuting variables that creates the relaxation to be solved by SDPA--a high-performance solver that runs in a distributed environment. We further exploit the inherent sparsity of optimization problems in quantum physics to reduce the complexity of the resulting relaxations. Constrained problems with a relaxation of order two may contain up to a hundred variables. The implementation is available in Python. The tool helps solve such as finding the ground state energy or testing quantum correlations.