Sample-efficient learning of interacting quantum systems

Anshu, Anurag; Arunachalam, Srinivasan; Kuwahara, Tomotaka; Soleimanifar, Mehdi

doi:10.1038/s41567-021-01232-0

Cited by 105 publications

(94 citation statements)

References 35 publications

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“…Our upper bound improves the sample complexity of [AAKS21] and indeed matches the sample complexity of the classical algorithm in the high-temperature regime, where the 2 O(β) term can be dropped since it is constant. Furthermore, our algorithm has optimal time complexity.…”

Section: Introductionsupporting

confidence: 57%

“…2 )-strongly convex in the high-temperature regime; this is the main quantity bounded by [AAKS21] to achieve their sample complexity result, and our analysis improves this strong convexity parameter to within a constant factor of its true value.…”

Section: Introductionmentioning

confidence: 71%

“…Notice that this definition has no geometric constraints. Instead, it generalizes geometrically local Hamiltonians in fixed-dimensional Euclidean spaces, which is the class of physically motivated Hamiltonians with geometric constraints considered in prior work [AAKS21]. In such Hamiltonians, each operator E a is only supported on a constant number of qubits that are adjacent in the underlying geometry (e.g., a 2-dimensional grid).…”

Section: Introductionmentioning

confidence: 99%

“…The algorithm in [BAL19] allows us to learn Hamiltonians from stationary states of Hamiltonian dynamics (which include Gibbs states) or from the dynamics itself by measuring local observables and solving a system of linear equations; however, it was unclear how the algorithm would perform in worst cases. More recently, Anshu, Arunachalam, Kuwahara, and Soleimanifar [AAKS21] studied the sample complexity of the problem, first to rigorously establish sample complexity upper bounds in the full range of parameters. They showed that a geometrically local Hamiltonian in a constant-dimensional space can be learned to ∞ error ε using…”

Section: Introductionmentioning

confidence: 99%

“…The question of time complexity is not explicitly addressed in [AAKS21], but they note that the problem can be solved in polynomial time in the high-temperature regime, by combining their algorithm with the polynomial-time algorithm for computing partition functions at high temperatures due to [KKB20]. We discuss this approach further in the "Comparison with previous quantum algorithms" section, but in brief, this approach leaves significant room for improvement in both sample complexity and time complexity.…”

Section: Introductionmentioning

confidence: 99%

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Optimal learning of quantum Hamiltonians from high-temperature Gibbs states

Haah¹,

Kothari²,

Tang³

2021

Preprint

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We study the problem of learning a Hamiltonian H to precision ε, supposing we are given copies of its Gibbs state ρ = exp(−βH)/ Tr(exp(−βH)) at a known inverse temperature β. Anshu, Arunachalam, Kuwahara, and Soleimanifar [AAKS21] recently studied the sample complexity (number of copies of ρ needed) of this problem for geometrically local N -qubit Hamiltonians. In the high-temperature (low β) regime, their algorithm has sample complexity poly(N, 1/β, 1/ε) and can be implemented with polynomial, but suboptimal, time complexity.In this paper, we study the same question for a more general class of Hamiltonians. We show how to learn the coefficients of a Hamiltonian to error ε with sample complexity S = O(log N/(βε) 2 ) and time complexity linear in the sample size, O(SN ). Furthermore, we prove a matching lower bound showing that our algorithm's sample complexity is optimal, and hence our time complexity is also optimal.In the appendix, we show that virtually the same algorithm can be used to learn H from a real-time evolution unitary e −itH in a small t regime with similar sample and time complexity.

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Section: Introductionsupporting

confidence: 57%

Section: Introductionmentioning

confidence: 71%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Optimal learning of quantum Hamiltonians from high-temperature Gibbs states

Haah¹,

Kothari²,

Tang³

2021

Preprint

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Entanglement Hamiltonians: From Field Theory to Lattice Models and Experiments

et al. 2022

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Results about entanglement (or modular) Hamiltonians of quantum many-body systems in field theory and statistical mechanics models, and recent applications in the context of quantum information and quantum simulation, are reviewed. In the first part of the review, what is known about entanglement Hamiltonians of ground states (vacua) in quantum field theory is summarized, based on the Bisognano-Wichmann theorem and its extension to conformal field theory. This is complemented with a more rigorous mathematical discussion of the Bisognano-Wichmann theorem, within the framework of Tomita-Takesaki theorem of modular groups. The second part of the review is devoted to lattice models. There, exactly soluble cases are first considered and then the discussion is extended to non-integrable models, whose entanglement Hamiltonian is often well captured by the lattice version of the Bisognano-Wichmann theorem. In the last part of the review, recently developed applications in quantum information processing that rely upon the specific properties of entanglement Hamiltonians in many-body systems are summarized. These include protocols to measure entanglement spectra, and schemes to perform state tomography.

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Fat shattering, joint measurability, and PAC learnability of POVM hypothesis classes

Magner,

Padakandla

2024

Quantum Inf Process

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Sample-efficient learning of interacting quantum systems

Cited by 105 publications

References 35 publications

Optimal learning of quantum Hamiltonians from high-temperature Gibbs states

Optimal learning of quantum Hamiltonians from high-temperature Gibbs states

Entanglement Hamiltonians: From Field Theory to Lattice Models and Experiments

Fat shattering, joint measurability, and PAC learnability of POVM hypothesis classes

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