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arXiv:2405.04884 · Communications Physics

Cost of locally approximating high-dimensional ground states of contextual quantum models

Kaiyan Yang, Yanzheng Zhu, Xiao Zeng, Zuoheng Zou, Man‐Hong Yung, Zizhu Wang

Published May 17, 2025• 2 citations
arXivHTMLPDF

Abstract

Contextuality, one of the strongest forms of quantum correlations, delineates the boundary between the quantum world and the classical one. Recent advances show that some translation-invariant contextuality witnesses are maximally violated by ground states and local observables of infinite one-dimensional translation-invariant Hamiltonians. However, these models all have local Hilbert space dimension larger than two, making the study of their ground states behavior difficult on current qubit-based platforms. In this work, we focus on the cost of simulating their 3-site reduced density matrices using qubit-based parameterized quantum circuits. The local approximations are purified then encoded into permutation-symmetric qubit states. By developing a universal set of permutation-symmetry preserving qubit-based gates, we assess the accuracy of simulating the purified local ground states against fixed classical and quantum resources. Results reveal that more contextual ground states with lower energy density are easier to simulate under identical resources. The authors encode qutrit contextual-Hamiltonian ground states into permutation-symmetric qubits with symmetry-preserving gates and use parameterized quantum circuits to variationally simulate their purified 3-site reduced states. They find that stronger contextuality reduces simulation cost, although all contextual states cost more than non-contextual ones.

DOI
10.1038/s42005-025-02127-9

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Contextuality, one of the strongest forms of quantum correlations, delineates the boundary between the quantum world and the classical one. Recent advances show that some translation-invariant contextuality witnesses are maximally violated by ground states and local observables of infinite one-dimensional translation-invariant Hamiltonians. However, these models all have local Hilbert space dimension larger than two, making the study of their ground states behavior difficult on current qubit-based platforms. In this work, we focus on the cost of simulating their 3-site reduced density matrices using qubit-based parameterized quantum circuits. The local approximations are purified then encoded into permutation-symmetric qubit states. By developing a universal set of permutation-symmetry preserving qubit-based gates, we assess the accuracy of simulating the purified local ground states against fixed classical and quantum resources. Results reveal that more contextual ground states with lower energy density are easier to simulate under identical resources. The authors encode qutrit contextual-Hamiltonian ground states into permutation-symmetric qubits with symmetry-preserving gates and use parameterized quantum circuits to variationally simulate their purified 3-site reduced states. They find that stronger contextuality reduces simulation cost, although all contextual states cost more than non-contextual ones.
Bell- and contextuality-type inequalities have become practical probes of many-body quantum correlations, often involving only few-body correlators and quantities with a direct Hamiltonian interpretation such as an energy density. Here we investigate the mechanism by which translation-invariant Hamiltonians generated from representative families of contextuality witnesses organize their ground-state structure in infinite one-dimensional systems. For the witness families considered, maximal quantum violation is realized by ground-state sectors with commensurate enlarged unit cells: the Hamiltonians are invariant under one-site translations, while the optimal ground states are ppp-periodic with p>1p>1p>1. At the corresponding classical-bound points, the ground-state sectors are highly degenerate and support many commensurate periods. Along the interpolation paths analyzed in this work, entering the contextual regime is accompanied by the lifting of this classical period degeneracy in favor of a quantum-selected period. We also identify finite periodic-boundary-condition benchmarks at the selected periods: for each model studied, the finite-ring witness reproduces the same classical bound and quantum value as the corresponding infinite-chain witness, and in several cases the resulting finite inequalities are tight. These reductions turn the infinite-chain contextuality certification into compact energy-estimation benchmarks requiring only local correlator measurements. We establish the mechanism analytically in representative two- and three-body witness models and corroborate it more broadly using translation-invariant semidefinite-program relaxations together with variational matrix-product-state calculations.
Abstract In recent years there has been a growing interest in treating many-body systems as Bell scenarios, where lattice sites play the role of distant parties and only near-neighbor statistics are accessible. We investigate contextuality arising from three Bell scenarios in infinite, translation-invariant 1D models: nearest-neighbor with two dichotomic observables per site; nearest- and next-to-nearest neighbor with two dichotomic observables per site, and nearest-neighbor with three dichotomic observables per site. For the first scenario, we give strong evidence that it cannot exhibit contextuality, not even in non-signaling physical theories beyond quantum mechanics. For the second one, we identify several low-dimensional models that reach the ultimate quantum limits, paving the way for self-testing ground states of quantum many-body systems. For the last scenario, which generalizes the Heisenberg model, we give strong evidence that, in order to exhibit contextuality, the dimension of the local quantum system must be at least 3.
We introduce a hierarchy of semidefinite relaxations of the set of quantum correlations in generalized contextuality scenarios.This constitutes a simple and versatile tool for bounding the magnitude of quantum contextuality.To illustrate its utility, we use it to determine the maximal quantum violation of several noncontextuality inequalities whose maximum violations were previously unknown.We then go further and use it to prove that certain preparation-contextual correlations cannot be explained with pure states, thereby showing that mixed states are an indispensable resource for contextuality.In the second part of the paper, we turn our attention to the simulation of preparation-contextual correlations in general operational theories.We introduce the information cost of simulating preparation contextuality, which quantifies the additional, otherwise forbidden, information required to simulate contextual correlations in either classical or quantum models.In both cases, we show that the simulation cost can be efficiently bounded using a variant of our hierarchy of semidefinite relaxations, and we calculate it exactly in the simplest contextuality scenario of parity-oblivious multiplexing.
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In this work we extend the notion of universal quantum Hamiltonians to the setting of translationally-invariant systems. We present a construction that allows a two-dimensional spin lattice with nearest-neighbour interactions, open boundaries, and translational symmetry to simulate any local target Hamiltonian---i.e. to reproduce the whole of the target system within its low-energy subspace to arbitrarily-high precision. Since this implies the capability to simulate non-translationally-invariant many-body systems with translationally-invariant couplings, any effect such as characteristics commonly associated to systems with external disorder, e.g. many-body localization, can also occur within the low-energy Hilbert space sector of translationally-invariant systems. Then we sketch a variant of the universal lattice construction optimized for simulating translationally-invariant target Hamiltonians. Finally we prove that qubit Hamiltonians consisting of Heisenberg or XY interactions of varying interaction strengths restricted to the edges of a connected translationally-invariant graph embedded in RD\mathbb{R}^DRD are universal, and can efficiently simulate any geometrically local Hamiltonian in RD\mathbb{R}^DRD.
We study local marginals in one-dimensional translation-invariant quantum systems that may hide finite-period structure. Given an nnn-site reduced density matrix, we ask whether it can be obtained by repeating a finite ppp-site block state along the chain and averaging over the ppp lattice translations. This defines a symmetrized block-product periodic marginal problem, which provides a route both to diagnosing hidden periodic order from local data and to upper bounding ground-state energy densities of infinite translation-invariant local Hamiltonians. We develop two complementary methods. The first is a semidefinite-programming relaxation based on block permutation symmetry and positive partial transpose constraints, which outer-approximates the convex hull of such marginals and yields certified infeasibility tests. The second is a symmetrized matrix product state ansatz, which constructs explicit block-product periodic states and gives variational upper bounds. We benchmark the framework on the Majumdar-Ghosh model, transverse-field Ising, XX, XXZ, and contextuality-related spin models. The results show that the method captures the expected finite-period structure in exactly solvable cases and gives systematically improving variational energies as the period and bond dimension increase. We also formulate a periodic-NPA relaxation for translation-invariant contextuality witnesses and recover the known quantum limits in the tested examples.