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arXiv:2307.15688

An SU(2)-symmetric Semidefinite Programming Hierarchy for Quantum Max Cut

Jun Takahashi, Sankara Sai Chaithanya Rayudu, Cunlu Zhou, Robbie King, Kevin Thompson, Ojas Parekh

Published Jul 28, 2023• 3 citations
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Abstract

Understanding and approximating extremal energy states of local Hamiltonians is a central problem in quantum physics and complexity theory. Recent work has focused on developing approximation algorithms for local Hamiltonians, and in particular the ``Quantum Max Cut'' (QMax-Cut) problem, which is closely related to the antiferromagnetic Heisenberg model. In this work, we introduce a family of semidefinite programming (SDP) relaxations based on the Navascues-Pironio-Acin (NPA) hierarchy which is tailored for QMaxCut by taking into account its SU(2) symmetry. We show that the hierarchy converges to the optimal QMaxCut value at a finite level, which is based on a new characterization of the algebra of SWAP operators. We give several analytic proofs and computational results showing exactness/inexactness of our hierarchy at the lowest level on several important families of graphs. We also discuss relationships between SDP approaches for QMaxCut and frustration-freeness in condensed matter physics and numerically demonstrate that the SDP-solvability practically becomes an efficiently-computable generalization of frustration-freeness. Furthermore, by numerical demonstration we show the potential of SDP algorithms to perform as an approximate method to compute physical quantities and capture physical features of some Heisenberg-type statistical mechanics models even away from the frustration-free regions.

DOI
10.48550/arxiv.2307.15688

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Understanding and approximating extremal energy states of local Hamiltonians is a central problem in quantum physics and complexity theory. Recent work has focused on developing approximation algorithms for local Hamiltonians, and in particular the ``Quantum Max Cut'' (QMax-Cut) problem, which is closely related to the antiferromagnetic Heisenberg model. In this work, we introduce a family of semidefinite programming (SDP) relaxations based on the Navascues-Pironio-Acin (NPA) hierarchy which is tailored for QMaxCut by taking into account its SU(2) symmetry. We show that the hierarchy converges to the optimal QMaxCut value at a finite level, which is based on a new characterization of the algebra of SWAP operators. We give several analytic proofs and computational results showing exactness/inexactness of our hierarchy at the lowest level on several important families of graphs. We also discuss relationships between SDP approaches for QMaxCut and frustration-freeness in condensed matter physics and numerically demonstrate that the SDP-solvability practically becomes an efficiently-computable generalization of frustration-freeness. Furthermore, by numerical demonstration we show the potential of SDP algorithms to perform as an approximate method to compute physical quantities and capture physical features of some Heisenberg-type statistical mechanics models even away from the frustration-free regions.
Quantum Max Cut (QMC) problem for systems of qubits is an example of a 2-local Hamiltonian problem, and a prominent paradigm in computational complexity theory. This paper investigates the algebraic structure of a higher-dimensional analog of the QMC problem for systems of qudits. The Quantum Max d -Cut ( d -QMC) problem asks for the largest eigenvalue of a Hamiltonian on a graph with n vertices whose edges correspond to swap operators acting on ( C d ) ⊗ n . The algebra generated by the swap operators is identified as a quotient of a free algebra modulo symmetric group relations and a single additional relation of degree d . This presentation leads to a tailored hierarchy of semidefinite programs, leveraging noncommutative polynomial optimization (NPO) methods, that converges to the solution of the d -QMC problem. For a large class of complete bipartite graphs, exact solutions for the d -QMC problem are derived using the representation theory of symmetric groups and Littlewood-Richardson coefficients. Lastly, the paper addresses a refined d -QMC problem focused on finding the largest eigenvalue within each isotypic component (irreducible block) of the graph Hamiltonian. It is shown that the spectrum of the star graph Hamiltonian distinguishes between isotypic components of the 3 -QMC problem. For general d , low-degree relations for separating isotypic components are presented, enabling adaptation of the global NPO hierarchy to efficiently compute the largest eigenvalue in each isotypic component.
The Quantum Max Cut (QMC) problem has emerged as a test-problem for designing approximation algorithms for local Hamiltonian problems. In this paper we attack this problem using the algebraic structure of QMC, in particular the relationship between the quantum max cut Hamiltonian and the representation theory of the symmetric group.The first major contribution of this paper is an extension of non-commutative Sum of Squares (ncSoS) optimization techniques to give a new hierarchy of relaxations to Quantum Max Cut. The hierarchy we present is based on optimizations over polynomials in the qubit swap operators. This is in contrast to the "standard" quantum Lasserre Hierarchy, which is based on polynomials expressed in terms of the Pauli matrices. To prove correctness of this hierarchy, we exploit a finite presentation of the algebra generated by the qubit swap operators. This presentation allows for the use of computer algebraic techniques to manipulate and simplify polynomials written in terms of the swap operators, and may be of independent interest. Surprisingly, we find that level-2 of this new hierarchy is numerically exact (up to tolerance 10−7 ) on all QMC instances with uniform edge weights on graphs with at most 8 vertices.The second major contribution of this paper is a polynomial-time algorithm that computes (in exact arithmetic) the maximum eigenvalue of the QMC Hamiltonian for certain graphs, including graphs that can be "decomposed" as a signed combination of cliques. A special case of the latter are complete bipartite graphs with uniform edge-weights, for which exact solutions are known from the work of Lieb and Mattis \cite{lieb1962ordering}. Our methods, which use representation theory of the symmetric group, can be seen as a generalization of the Lieb-Mattis result.
The Quantum Max Cut (QMC) problem has emerged as a test-problem for designing approximation algorithms for local Hamiltonian problems. In this paper we attack this problem using the algebraic structure of QMC, in particular the relationship between the quantum max cut Hamiltonian and the representation theory of the symmetric group. The first major contribution of this paper is an extension of non-commutative Sum of Squares (ncSoS) optimization techniques to give a new hierarchy of relaxations to Quantum Max Cut. The hierarchy we present is based on optimizations over polynomials in the qubit swap operators. This is in contrast to the "standard" quantum Lasserre Hierarchy, which is based on polynomials expressed in terms of the Pauli matrices. To prove correctness of this hierarchy, we exploit a finite presentation of the algebra generated by the qubit swap operators. This presentation allows for the use of computer algebraic techniques to manipulate and simplify polynomials written in terms of the swap operators, and may be of independent interest. Surprisingly, we find that level-2 of this new hierarchy is numerically exact (up to tolerance 10^(-7)) on all QMC instances with uniform edge weights on graphs with at most 8 vertices. The second major contribution of this paper is a polynomial-time algorithm that computes (in exact arithmetic) the maximum eigenvalue of the QMC Hamiltonian for certain graphs, including graphs that can be "decomposed" as a signed combination of cliques. A special case of the latter are complete bipartite graphs with uniform edge-weights, for which exact solutions are known from the work of Lieb and Mattis. Our methods, which use representation theory of the symmetric group, can be seen as a generalization of the Lieb-Mattis result.
The kkk-local Hamiltonian problem is a central model for quantum many-body systems and Hamiltonian complexity. Semidefinite programming and noncommutative sum-of-squares hierarchies provide systematic certificates for ground-state energies, but existing finite-convergence results give no quantitative guarantee on the accuracy of the low hierarchy levels accessible in computation. We prove explicit finite-level convergence rates for these hierarchies in the Pauli setting. For kkk-local Hamiltonians whose Pauli expansion contains only even-weight terms, we show that both the NPA-type lower-bound hierarchy and the upper-bound hierarchy on the spectral minimum have error at most C(k)ξd+1n,4/nC(k)ξ^{n,4}_{d+1}/nC(k)ξd+1n,4​/n, where ξd+1n,4ξ^{n,4}_{d+1}ξd+1n,4​ is the smallest root of a Krawtchouk polynomial and C(k)C(k)C(k) is independent of the number of qubits nnn and the hierarchy level ddd. General kkk-local Hamiltonians reduce to this even-weight case by adding one ancilla qubit while preserving the spectrum. The proof constructs almost-reproducing kernels for the Pauli algebra and relates their spectra to Krawtchouk polynomials, giving a noncommutative analogue of recent kernel-based convergence analyses for commutative polynomial optimization. These results provide the first quantitative finite-level accuracy guarantees for noncommutative semidefinite relaxations of Pauli Hamiltonians.
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The Navascués-Pironio-Acín (NPA) hierarchy provides a convergent sequence of semidefinite programming (SDP) relaxations for noncommutative polynomial optimisation, ubiquitous in quantum physics. However, its practical applicability is limited by the combinatorial growth in operator moments required at each level. Since not all moments contribute equally to bound tightness, selecting moments within a fixed computational budget is a relevant problem. We reframe moment selection as combinatorial subset selection and show it is governed by strong higher-order synergistic interactions among moments, quantified through a marginal synergy diagnostic adapted from complex systems theory. We develop and compare three optimisation methods: Parallel Tempering (PT), an RBM-based reinforcement learning policy, and Bayesian Optimisation (BO). On the I3322I_{3322}I3322​ Bell inequality benchmark, all three substantially outperform greedy approaches at costs around two orders of magnitude below brute force, with the RBM achieving the closest approach to optimal throughout the hard transition regime. We apply the framework to the 174 Bell inequalities in the (4,4,2,2)(4,4,2,2)(4,4,2,2) scenario, finding heterogeneous convergence behaviour across inequalities, and to the one-dimensional Heisenberg spin chain, demonstrating that physically motivated monomial bases are internally compressible and are not globally optimal in general. A budget-aware search over a broader pool improves certified bounds on long-range correlations by nearly two orders of magnitude. These results establish a scalable framework for moment selection in noncommutative polynomial optimisation, with broad applications across quantum physics and quantum information.