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arXiv:1807.08424

Polynomial-time Classical Simulation for One-dimensional Quantum Gibbs States

Tomotaka Kuwahara, Keiji Saito

Published Jul 23, 2018• 2 citations
arXivHTMLPDF

Abstract

This paper discusses a classical simulation to compute the partition function (or free energy) of generic one-dimensional quantum many-body systems. Many numerical methods have previously been developed to approximately solve one-dimensional quantum systems. However, there exists no exact proof that arbitrary one-dimensional quantum Gibbs states can be efficiently solved by a classical computer. Therefore, the aim of this paper is to prove this with the clustering properties for arbitrary finite temperatures $β^{-1}$. We explicitly show an efficient algorithm that approximates the partition function up to an error $ε$ with a computational cost that scales as $n\cdot {\rm poly}(1/ε)$, where the degree of the polynomial depends on $β$ as $e^{O(β)}$. Extending the analysis to higher dimensions at high temperatures, we obtain a weaker result for the computational cost n\cdot (1/ε)^{\log^{D-1} (1/ε)}, where $D$ is the lattice dimension.

DOI
10.48550/arxiv.1807.08424

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This paper discusses a classical simulation to compute the partition function (or free energy) of generic one-dimensional quantum many-body systems. Many numerical methods have previously been developed to approximately solve one-dimensional quantum systems. However, there exists no exact proof that arbitrary one-dimensional quantum Gibbs states can be efficiently solved by a classical computer. Therefore, the aim of this paper is to prove this with the clustering properties for arbitrary finite temperatures β−1β^{-1}β−1. We explicitly show an efficient algorithm that approximates the partition function up to an error εεε with a computational cost that scales as n⋅poly(1/ε)n\cdot {\rm poly}(1/ε)n⋅poly(1/ε), where the degree of the polynomial depends on βββ as eO(β)e^{O(β)}eO(β). Extending the analysis to higher dimensions at high temperatures, we obtain a weaker result for the computational cost n\cdot (1/ε)^{\log^{D-1} (1/ε)}, where DDD is the lattice dimension.
The computational complexity of simulating quantum many-body systems generally scales exponentially with the number of particles. This enormous computational cost prohibits first principles simulations of many important problems throughout science, ranging from simulating quantum chemistry to discovering the thermodynamic phase diagram of quantum materials or high-density neutron stars. We present a classical algorithm that samples from a high-temperature quantum Gibbs state in a computational (product state) basis. The runtime grows polynomially with the number of particles, while error vanishes polynomially. This algorithm provides an alternative strategy to existing quantum Monte Carlo methods for overcoming the sign problem. Our result implies that measurement-based quantum computation on a Gibbs state can provide exponential speed up only at sufficiently low temperature, and further constrains what tasks can be exponentially faster on quantum computers.
We introduce a classical algorithm to approximate the free energy of local, translation-invariant, one-dimensional quantum systems in the thermodynamic limit of infinite chain size. While the ground state problem (i.e., the free energy at temperature T=0 ) for these systems is expected to be computationally hard even for quantum computers, our algorithm runs for any fixed temperature T>0 in subpolynomial time, i.e., in time O((1ε)c) for any constant c>0 where ε is the additive approximation error. Previously, the best known algorithm had a runtime that is polynomial in 1ε . Our algorithm is also particularly simple as it reduces to the computation of the spectral radius of a linear map. This linear map has an interpretation as a noncommutative transfer matrix and has been studied previously to prove results on the analyticity of the free energy and the decay of correlations. We also show that the corresponding eigenvector of this map gives an approximation of the marginal of the Gibbs state and thereby allows for the computation of various thermodynamic properties of the quantum system.
We introduce a classical algorithm to approximate the free energy of local, translation-invariant, one-dimensional quantum systems in the thermodynamic limit of infinite chain size. While the ground state problem (i.e., the free energy at temperature T=0T = 0T=0) for these systems is expected to be computationally hard even for quantum computers, our algorithm runs for any fixed temperature T>0T > 0T>0 in subpolynomial time, i.e., in time O((1ε)c)O((\frac{1}{\varepsilon})^{c})O((ε1​)c) for any constant c>0c > 0c>0 where ε\varepsilonε is the additive approximation error. Previously, the best known algorithm had a runtime that is polynomial in 1ε\frac{1}{\varepsilon}ε1​. Our algorithm is also particularly simple as it reduces to the computation of the spectral radius of a linear map. This linear map has an interpretation as a noncommutative transfer matrix and has been studied previously to prove results on the analyticity of the free energy and the decay of correlations. We also show that the corresponding eigenvector of this map gives an approximation of the marginal of the Gibbs state and thereby allows for the computation of various thermodynamic properties of the quantum system.
It is of great interest to understand the thermalization of open quantum many-body systems, and how quantum computers are able to efficiently simulate that process. A recently introduced dissipative evolution, inspired by existing models of open system thermalization, has been shown to be efficiently implementable on a quantum computer. Here, we prove that, at high enough temperatures, this evolution reaches the Gibbs state in time scaling logarithmically with system size. The result holds for Hamiltonians that satisfy the Lieb-Robinson bound, such as local Hamiltonians on a lattice, and includes long-range systems. To the best of our knowledge, these are the first results rigorously establishing the rapid mixing property of high-temperature quantum Gibbs samplers, which is known to give the fastest possible speed for thermalization in the many-body setting. We then apply our result to the problem of estimating partition functions at high temperature, showing an improved performance over previous classical and quantum algorithms.
The sign problem remains a major obstacle to extending the success of Monte Carlo methods to generic quantum many-body systems. A powerful alternative relies on cluster expansions, which provide convergent representations of free energies and related quantities, with applications ranging from tensor network contractions to quantum dynamics and circuits. Within their convergence regime, these expansions often yield deterministic polynomial-time algorithms. However, such methods require exhaustive enumeration of connected clusters, leading to a computational overhead of (Nε−1)O(log⁡Δ)(N\varepsilon^{-1})^{\mathcal{O}(\logΔ)}(Nε−1)O(logΔ), where NNN is the system size, ε\varepsilonε the desired precision, and ΔΔΔ the interaction-graph degree. This becomes particularly severe in long-range systems, where Δ≡NΔ\equiv NΔ≡N, resulting in quasi-polynomial runtimes. Here, we introduce a Markov chain algorithm that samples cluster expansions directly in the space of connected clusters, replacing deterministic enumeration by Monte Carlo sampling. Because sampling occurs in an abstract cluster space rather than a physical configuration space, the conventional exponential sign problem is absent within the convergence regime. For general polymer models, we establish an FPRAS with runtime O~(N4ε−3)\tilde{\mathcal{O}}(N^4\varepsilon^{-3})O~(N4ε−3). For quantum partition functions at temperatures β≤βsampling≡O(1)β\leqβ_{\mathrm{sampling}}\equiv\mathcal{O}(1)β≤βsampling​≡O(1), this improves to O~(N3+1/Dε−3)\tilde{\mathcal{O}}(N^{3+1/D}\varepsilon^{-3})O~(N3+1/Dε−3), where DDD is the lattice dimension. The framework directly applies to long-range quantum systems with interactions ∝1/rα\propto1/r^α∝1/rα, including 222-local models with α>2Dα>2Dα>2D.
Quantum impurity models are paradigmatic models of interacting quantum matter, as well as key computational primitives for modern electronic-structure methods. They describe a small subsystem of interacting fermions coupled to a large, noninteracting bath. We perform a comprehensive study of the computational complexity of simulating impurity models, delineating the boundary between classical and quantum tractability for this class of problems. Our main finding is that static properties of quantum impurity models can be calculated efficiently on a classical computer. Specifically, we give classical algorithms that (1) estimate the ground-state energy to additive precision δδδ in time poly(n,δ−1)\mathrm{poly}(n,δ^{-1})poly(n,δ−1), and (2) estimate the partition function at inverse temperature βββ to relative precision δδδ in time poly(n,β,δ−1)\mathrm{poly}(n,β,δ^{-1})poly(n,β,δ−1), where nnn is the system size. These results improve the previous best-known complexity for ground-state energy estimation from quasipolynomial to polynomial time, while establishing for the first time rigorous polynomial-time guarantees for simulating impurity models in thermal equilibrium. On the other hand, we find that simulating dynamical properties of impurity models is hard for classical computers but easy on a quantum computer. As a canonical example, we show that computing their nonequilibrium Green's functions captures the full power of quantum computation, even at finite temperature. Taken together, our results rule out superpolynomial quantum speedups for computing static properties, but provide an avenue for quantum advantage in simulating impurity physics out of equilibrium.