Quantum Belief Propagation

Abstract

We present an accurate numerical algorithm, called quantum belief propagation (QBP), for simulation of one-dimensional quantum systems at non-zero temperature. The algorithm exploits the fact that quantum effects are short-range in these systems at non-zero temperature, decaying on a length scale inversely proportional to the temperature. We compare to exact results on a spin-1/2 Heisenberg chain. Even a very modest calculation, requiring diagonalizing only 10-by-10 matrices, reproduces the peak susceptibility with a relative error of less than $10^{-5}$, while more elaborate calculations further reduce the error.

4 pages, 1 figure; revised time estimates due to improved implementation. Typographical corrections to Eq. 7 made; thanks to David Poulin for pointing out the mistake

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• Clustering theorem in 1D long-range interacting systems at arbitrary temperatures
• Perturbation theory of KMS states
• Locality estimes for complex time evolution in 1D
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• Provably Efficient Simulation of 1D Long-Range Interacting Systems at Any Temperature
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• Statistical mechanics of classical and quantum computational complexity
• Stability of thermal equilibrium in long-range quantum systems
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• Gibbs state sampling via cluster expansions
• Constrained free energy minimization for the design of thermal states and stabilizer thermodynamic systems
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• Locally accurate tensor networks for thermal states and time evolution
• Quantum Belief Propagation Algorithm versus Suzuki-Trotter approach in the one-dimensional Heisenberg chains