Moment/Sum-of-Hermitian-Squares relaxations for noncommutative polynomial optimization problems have become an important tool for analyzing problems within quantum theory. Despite their widespread success, little is known about their rate of convergence and, consequently, their accuracy. In this work, we develop explicit convergence rates for relaxations of noncommutative polynomial optimization problems generated from the Pauli algebra -- covering applications to the ground state energy problem for n-qubit systems. In particular, we show that the rate of convergence can be bounded in terms of the smallest roots of a family of orthogonal polynomials known as Krawtchouk polynomials. Our result represents the first quantitative analysis of the rate of convergence for relaxations of noncommutative polynomial optimization problems.
A central question in quantum many-body physics is estimating the ground energy of a
k-local Hamiltonian system. In this work, we present a spectral technique for certifying a lower bound on the ground energy of a random
n-qubit Hamiltonian system defined as the sum of signed
k-local Pauli operators. In particular, we prove that for any constant
ℓ, there exists an efficiently computable length
nO(ℓ) certificate that is always a lower bound on the ground energy with the promise that, with high probability over the random Hamiltonian distribution, the certificate value is an
ε-good approximation of the true ground energy when the number of terms is sufficiently large. Second, we show by construction that this technique, while successful on average over random Hamiltonian systems, can fail to produce good certificates on worst-case instances. Our spectral technique for producing these certificates comes from extending classical results on
k-XOR refutations to
k-local Pauli Hamiltonians by crafting a quantum variant of the Kikuchi matrix for CSP refutations. To show the limitations of this technique, we prove non-commutative Sum-of-Squares lower bounds for worst-case signed
k-local Pauli operators. More generally, we explore how the non-commutative Sum-of-Squares relaxation can be understood as augmenting the standard Sum-of-Squares relaxation with the commutation relations between the Pauli operators. We instantiate the resulting framework with a modification to prior quantum code-based NLTS Hamiltonians that yields stronger complexity guarantees for the low-energy space; our Hamiltonian family satisfies simultaneously (1)
Ω(logn)-circuit depth lower bounds for all low-energy states, (2) constant-gap NP-hardness to approximate the ground energy, and (3) a constant-gap non-commutative Sum-of-Squares integrality gap up to
Ω(n)-levels.
A fundamental challenge in quantum physics is determining the ground-state properties of many-body systems. Whereas standard variational approaches posit a wave-function ansatz and minimize over the possible states expressible by that ansatz, the problem can alternatively be formulated as a noncommutative polynomial optimization problem and treated through a hierarchy of semidefinite programming relaxations. In contrast to variational calculations, these relaxations provide lower bounds on ground-state energies and both lower and upper bounds on observable expectation values. However, this approach typically suffers from severe scalability issues, limiting its applicability to small-to-medium-scale systems. In this article, we demonstrate that systematically leveraging the inherent structures of the system can substantially mitigate these scalability challenges and thus permits computing meaningful bounds for quantum spin systems on square lattices of size up to
16×16.
Convex relaxation hierarchies provide lower bounds to the ground state energy of quantum many-body systems that can be computed in polynomial time on a classical computer, at any fixed hierarchy level. However, scaling these methods to large systems and accurate approximations remains challenging due to the computational cost of traditional solvers and the scarcity of efficiency guarantees. In this work, we develop local relaxation hierarchies and efficient, highly parallelisable message passing algorithms for estimating the relaxed ground state energies. We show that the first level of the hierarchy---based on local consistency of pairwise reduced density matrices---is exact for commuting Hamiltonians on trees. We further establish that another hierarchy, based on consistent intervals, converges exponentially fast in the interval size to the ground state energy for weak perturbations of separable Hamiltonians on a chain, thereby providing an efficient classical algorithm for these systems. Then, we introduce two variants of message passing algorithms that run in
O(n/ε2) and
O(n/ε) time for any fixed level of the local hierarchy on bounded-degree graphs, where
ε is the precision for the relaxed ground state energy per site. This assumes that the optimal messages have
O(1) norm---a condition we observe in practical settings in our experiments. These algorithms are based on the subgradient method and the Nesterov-type accelerated gradient descent method applied to an entropy-smoothed objective. Finally, we benchmark the message passing algorithms across different quantum Hamiltonians, lattice geometries, and relaxation levels, validating the theoretical predictions and their potential to surpass standard convex optimisation solvers for this problem. We release the resulting library at github.com/rick1924/gse-message-passing.
We give a convergent hierarchy of SDP certificates for lower bounds on the spectral gap of a local qubit Hamiltonian
H. Our approach is based on the NPA hierarchy applied to a polynomially-sized system of constraints defining the universal enveloping algebra of the Lie algebra
su(2n) along with additional constraints which force the representation to be
∧2(C2n), one which the induced Hamiltonian has ground energy
λ0(H)+λ1(H). Combined with an upper bound on the ground state energy, which can be obtained either using a hierarchy introduced by Fawzi, Fawzi, and Scalet or a noncommutative analog developed by Klep, Magron, Massé, and Volčič of a hierarchy introduced by Lasserre, we obtain lower bounds on the spectral gap of
H without assuming frustration-freeness or geometric locality. We prove that the resulting certificates have polynomial size at fixed degree and converge at level
n. As illustrations of the certificates which can be obtained, we show that for a commuting 1-local Hamiltonian the hierarchy certifies the optimal bound on its spectral gap and prove that similar hierarchies require degree growing with
n to certify the same bound. We then extend this to prove a lower bound on the spectral gap of the transverse-field
XY model, which is not frustration-free, on arbitrary regular graphs without any assumption of geometric locality.
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.