A subpolynomial-time algorithm for the free energy of one-dimensional quantum systems in the thermodynamic limit

Abstract

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 $O((\frac{1}{\epsilon }{)}^c)$ for any constant $c>0$ where $\epsilon$ is the additive approximation error. Previously, the best known algorithm had a runtime that is polynomial in $\frac{1}{\epsilon }$. 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.

correction to inverse temperature dependence

References in corpus (14)

• The density-matrix renormalization group in the age of matrix product states
• Computational Studies of Quantum Spin Systems
• Quantum Hamiltonian Complexity
• Wang-Landau sampling for quantum systems: algorithms to overcome tunneling problems and calculate the free energy
• Quantum Belief Propagation
• Clustering of conditional mutual information for quantum Gibbs states above a threshold temperature
• Improved thermal area law and quasi-linear time algorithm for quantum Gibbs states
• Hilbert's projective metric in quantum information theory
• Large deviations in quantum lattice systems: one-phase region
• Classical algorithms, correlation decay, and complex zeros of partition functions of quantum many-body systems
• Exponential decay of mutual information for Gibbs states of local Hamiltonians
• Quantum many-body systems in thermal equilibrium
• The Complexity of Translationally-Invariant Spin Chains with Low Local Dimension
• Efficient Algorithms for Approximating Quantum Partition Functions

Cited by in corpus (6)

• Quantum many-body systems in thermal equilibrium
• Certified algorithms for equilibrium states of local quantum Hamiltonians
• Clustering theorem in 1D long-range interacting systems at arbitrary temperatures
• Provably Efficient Simulation of 1D Long-Range Interacting Systems at Any Temperature
• Conditional Independence of 1D Gibbs States with Applications to Efficient Learning
• A Faster Algorithm for the Free Energy in One-Dimensional Quantum Systems