Finite Correlation Length Implies Efficient Preparation of Quantum Thermal States

Abstract

Preparing quantum thermal states on a quantum computer is in general a difficult task. We provide a procedure to prepare a thermal state on a quantum computer with a logarithmic depth circuit of local quantum channels assuming that the thermal state correlations satisfy the following two properties: (i) the correlations between two regions are exponentially decaying in the distance between the regions, and (ii) the thermal state is an approximate Markov state for shielded regions. We require both properties to hold for the thermal state of the Hamiltonian on any induced subgraph of the original lattice. Assumption (ii) is satisfied for all commuting Gibbs states, while assumption (i) is satisfied for every model above a critical temperature. Both assumptions are satisfied in one spatial dimension. Moreover, both assumptions are expected to hold above the thermal phase transition for models without any topological order at finite temperature. As a building block, we show that exponential decay of correlation (for thermal states of Hamiltonians on all induced subgraphs) is sufficient to efficiently estimate the expectation value of a local observable. Our proof uses quantum belief propagation, a recent strengthening of strong sub-additivity, and naturally breaks down for states with topological order. arXiv:1609.07877v2 [quant-ph]

Locality of edge states and entanglement spectrum from strong subadditivity

“…Topological entanglement entropy has been linked to several other aspects of topological order. If TEE is zero, then the state can be created by a constant-depth local circuit, and thus in a topologically trivial phase [12][13][14]. Also, TEE upper bounds the logarithmic of the topological degeneracy of the model [15].…”

Approximate Quantum Markov Chains

“…The variational formula for the relative entropy (see (2.78)) shows that for t ∈ [0, 1] and H 1 , H 2 ∈ H(A) we have log tr e 29) where the final step uses (2.78).…”

“…To summarize, one-dimensional systems that satisfy the locality assumption (5.110) can be efficiently represented by a finite sequence of recovery maps given by Theorem 5.5. Theorem 5.5 has been successfully applied in other areas such as high energy physics [43,45,117], solid state physics [29,160,179], quantum error correction [66,118], quantum information theory [7,22,33,90,97,101], and foundations of quantum mechanics [109].…”

Approximate Quantum Markov Chains

“…To summarize, onedimensional systems that satisfy the locality assumption (5.113) can be efficiently represented by a finite sequence of recovery maps given by Theorem 5.5. Theorem 5.5 has been successfully applied in other areas such as high energy physics [43,45,115], solid state physics [29,141,160], quantum error correction [66,116], quantum information theory [7,22,33,88,95,99], and foundations of quantum mechanics [107].…”