Efficient algorithms for approximating quantum partition functions

Mann, Ryan L.; Helmuth, Tyler

doi:10.1063/5.0013689

Cited by 10 publications

(2 citation statements)

References 21 publications

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“…The foundational tool of our approach is cluster expansion, which allows quantities like the log-partition function and marginals of hightemperature Gibbs states to be expressed as exponentially decaying Taylor series. This tool has been used to show a variety of efficient algorithms around Gibbs states and real-time evolution, including the computation of partition functions [MH21], learning of high-temperature Gibbs states [HKT22], and sampling from the measurement distribution of a high-temperature Gibbs state [YL23]. Among these, the latter sampling result of Yin and Lucas bears the most similarity to our result, using a sampling-to-counting reduction with cluster expansion to give a classical algorithm to sample from the measurement probabilities of a Gibbs state in, say, the computational basis.…”

Section: Related Workmentioning

confidence: 99%

A New Approach to Learning Linear Dynamical Systems

Bakshi

Liu

Moitra

et al. 2023

Proceedings of the 55th Annual ACM Symposium on Theory of Computing

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We show that thermal states of local Hamiltonians are separable above a constant temperature. Specifically, for a local Hamiltonian H on a graph with degree d, its Gibbs state at inverse temperature β, denoted by ρ = e −βH / tr(e −βH ), is a classical distribution over product states for all β < 1/(cd), where c is a constant. This sudden death of thermal entanglement upends conventional wisdom about the presence of short-range quantum correlations in Gibbs states.Moreover, we show that we can efficiently sample from the distribution over product states. In particular, for any β < 1/(cd 3 ), we can prepare a state ε-close to ρ in trace distance with a depth-one quantum circuit and poly(n) log(1/ε) classical overhead. 1 A priori the task of preparing a Gibbs state is a natural candidate for achieving super-polynomial quantum speedups, but our results rule out this possibility above a fixed constant temperature. 1 In independent and concurrent work, Rouzé, França, and Alhambra [RFA24] obtain an efficient quantum algorithm for preparing high-temperature Gibbs states via a dissipative evolution.

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Section: Related Workmentioning

confidence: 99%

A New Approach to Learning Linear Dynamical Systems

Bakshi

Liu

Moitra

et al. 2023

Proceedings of the 55th Annual ACM Symposium on Theory of Computing

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“…For discrete classical statistical mechanics systems at high temperatures this is a well-studied question, and we have a relatively complete understanding for some models, e.g., the hard-core model [1][2][3][4], although important problems remain open [5]. Our understanding of approximation algorithms for quantum spin systems at high temperatures is less advanced, but has received a good deal of recent attention [6][7][8][9][10][11][12].…”

Section: Introductionmentioning

confidence: 99%

Efficient Algorithms for Approximating Quantum Partition Functions at Low Temperature

Helmuth¹,

Mann²

2022

Preprint

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We establish an efficient approximation algorithm for the partition functions of a class of quantum spin systems at low temperature, which can be viewed as stable quantum perturbations of classical spin systems. Our algorithm is based on combining the contour representation of quantum spin systems of this type due to Borgs, Kotecký, and Ueltschi with the algorithmic framework developed by Helmuth, Perkins, and Regts, and Borgs et al.

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Certified algorithms for equilibrium states of local quantum Hamiltonians

Fawzi,

Scalet

2024

Nat Commun

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Predicting observables in equilibrium states is a central yet notoriously hard question in quantum many-body systems. In the physically relevant thermodynamic limit, certain mathematical formulations of this task have even been shown to result in undecidable problems. Using a finite-size scaling of algorithms devised for finite systems often fails due to the lack of certified convergence bounds for this limit. In this work, we design certified algorithms for computing expectation values of observables in the equilibrium states of local quantum Hamiltonians, both at zero and positive temperature. Importantly, our algorithms output rigorous lower and upper bounds on these values. This allows us to show that expectation values of local observables can be approximated in finite time, contrasting related undecidability results. When the Hamiltonian is commuting on a 2-dimensional lattice, we prove fast convergence of the hierarchy at high temperature and as a result for a desired precision ε, local observables can be approximated by a convex optimization program of quasi-polynomial size in 1/ε.

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Efficient algorithms for approximating quantum partition functions

Cited by 10 publications

References 21 publications

A New Approach to Learning Linear Dynamical Systems

A New Approach to Learning Linear Dynamical Systems

Efficient Algorithms for Approximating Quantum Partition Functions at Low Temperature

Certified algorithms for equilibrium states of local quantum Hamiltonians

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