Learning quantum many-body systems from a few copies

Rouzé, Cambyse; França, Daniel Stilck

doi:10.48550/arxiv.2107.03333

Cited by 7 publications

(13 citation statements)

References 9 publications

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“…• Recently a number of works have narrowed down the sample and computational complexity of the problem of thermal state tomography [12,23,155,186]. The basic question is: can we learn the Hamiltonian from a small number of simple (local) measurements of few copies of e −βH /Z?…”

Section: Conclusion and Open Questionsmentioning

confidence: 99%

Quantum many-body systems in thermal equilibrium

Alhambra¹

2022

Preprint

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The thermal or equilibrium ensemble is one of the most ubiquitous states of matter. For models comprised of many locally interacting quantum particles, it describes a wide range of physical situations, relevant to condensed matter physics, high energy physics, quantum chemistry and quantum computing, among others. We give a pedagogical overview of some of the most important universal features about the physics and complexity of these states, which have the locality of the Hamiltonian at its core. We focus on mathematically rigorous statements, many of them inspired by ideas and tools from quantum information theory. These include bounds on their correlations, the form of the subsystems, various statistical properties, and the performance of classical and quantum algorithms. We also include a summary of a few of the most important technical tools, as well as some self-contained proofs.Parts of these notes were the basis for a lecture series within the "Quantum Thermo-

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Section: Conclusion and Open Questionsmentioning

confidence: 99%

Quantum many-body systems in thermal equilibrium

Alhambra¹

2022

Preprint

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“…Next, a state σ ∈ D(H Λ ) satisfies a transportation-cost inequality with constant C > 0 if for any ρ ∈ D(H Λ ), Transportation-cost inequalities in the form of (6) were recently used to produce sample efficient Gibbs state learning algorithms in [59]. They also imply Gaussian concentration inequalities for the statistics of quasi-local observables in the state σ and can be used to prove an exponential improvement over the weak Eigenstate Thermalization Hypothesis [52].…”

Section: Applicationsmentioning

confidence: 99%

Entropy decay for Davies semigroups of a one dimensional quantum lattice

Bardet¹,

Capel²,

Li³

et al. 2021

Preprint

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Given a finite-range, translation-invariant commuting system Hamiltonians on a spin chain, we show that the Davies semigroup describing the reduced dynamics resulting from the joint Hamiltonian evolution of a spin chain weakly coupled to a large heat bath thermalizes rapidly at any temperature. More precisely, we prove that the relative entropy between any evolved state and the equilibrium Gibbs state contracts exponentially fast with an exponent that scales logarithmically with the length of the chain. Our theorem extends a seminal result of Holley and Stroock [40] to the quantum setting, up to a logarithmic overhead, as well as provides an exponential improvement over the non-closure of the gap proved by Brandao and Kastoryano [43]. This has wide-ranging applications to the study of many-body in and out-of-equilibrium quantum systems. Our proof relies upon a recently derived strong decay of correlations for Gibbs states of one dimensional, translation-invariant local Hamiltonians, and tools from the theory of operator spaces.

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“…Devising a protocol that allocates the budget M to minimise the estimation error is far from trivial. Several approaches have been proposed, including joint measurements of pairwise commuting operators [17], randomized and derandomized measurements [19,21,25,26], grouping by weights [20], neural networks [22], minimizing the number of measurement groups [16,18,25,27,[29][30][31][32], and a method based on maximum entropy and optimal transport [33]. In this work, we introduce a measurement scheme for estimating observables that adaptively allocates the measurement budget based on previously collected data, allows for both non-bitwise commutation between PS and overlap in their grouping, and assesses both the average and variance of the observable O in Eq.…”

Section: Introductionmentioning

confidence: 99%

Adaptive estimation of quantum observables

Shlosberg

Jena

Mukhopadhyay

et al. 2023

Quantum

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The accurate estimation of quantum observables is a critical task in science. With progress on the hardware, measuring a quantum system will become increasingly demanding, particularly for variational protocols that require extensive sampling. Here, we introduce a measurement scheme that adaptively modifies the estimator based on previously obtained data. Our algorithm, which we call AEQuO, continuously monitors both the estimated average and the associated error of the considered observable, and determines the next measurement step based on this information. We allow both for overlap and non-bitwise commutation relations in the subsets of Pauli operators that are simultaneously probed, thereby maximizing the amount of gathered information. AEQuO comes in two variants: a greedy bucket-filling algorithm with good performance for small problem instances, and a machine learning-based algorithm with more favorable scaling for larger instances. The measurement configuration determined by these subroutines is further post-processed in order to lower the error on the estimator. We test our protocol on chemistry Hamiltonians, for which AEQuO provides error estimates that improve on all state-of-the-art methods based on various grouping techniques or randomized measurements, thus greatly lowering the toll of measurements in current and future quantum applications.

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Learning quantum many-body systems from a few copies

Cited by 7 publications

References 9 publications

Quantum many-body systems in thermal equilibrium

Quantum many-body systems in thermal equilibrium

Entropy decay for Davies semigroups of a one dimensional quantum lattice

Adaptive estimation of quantum observables

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