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 subgraph) 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.

• The paper presents an efficient scheme using logarithmic-depth circuits to approximate quantum Gibbs states under finite correlation length conditions.
• It leverages the exponential decay of correlations and the approximate Markov property to stabilize local observable measurements.
• The method bridges classical Gibbs sampling algorithms with quantum simulations, promising practical thermal state preparation on near-term quantum computers.

Efficient Preparation of Quantum Thermal States with Finite Correlation Length

The paper "Finite correlation length implies efficient preparation of quantum thermal states" by Fernando G.S.L. Brandão and Michael J. Kastoryano addresses the computational complexity of preparing quantum thermal states on a quantum computer. The work presents a methodology to efficiently approximate quantum Gibbs states using a logarithmic depth circuit comprised of local quantum channels, contingent upon certain assumptions about the correlations in the thermal state. Crucial to this method is the finite correlation length condition, characterized by two primary properties: the exponential decay of correlations and the approximate Markov property for shielded regions.

Main Assumptions and Theoretical Framework

The procedure presented in the paper assumes that:

1. Exponential Decay of Correlations: The correlations between two regions decrease exponentially with the distance separating these regions.
2. Approximate Markov Property: The thermal state's correlations exhibit a pseudo-local behavior such that regions can be shielded effectively by local boundaries, a characteristic satisfied by commuting Hamiltonians known as Gibbs states.

Both assumptions are universally applicable in one-dimensional systems and are generally expected to be valid above the thermal phase transition in higher dimensional systems devoid of topological order. The authors provide a significant extension of existing classical Gibbs sampling algorithms to the quantum domain by considering these assumptions in the quantum context.

Methodology and Results

The authors develop a scheme for preparing quantum Gibbs states that involves decomposing the space into smaller regions, leveraging the local indistinguishability of quantum observables in these regions. Specifically, the method is underpinned by the framework of quantum belief propagation and a notable recent advancement in the strengthening of strong subadditivity of Von Neumann entropy. By ensuring that expectation values for local observables are not sensitive to boundary fluctuations, the authors validate the composite approximation of the Gibbs state.

A key result is that the thermal state of local Hamiltonians can be reconstructed efficiently under these assumptions. The analytical proof establishes that the effective simulation algorithm forms a circuit of local quantum channels approximating the Gibbs state with exponential accuracy in the circuit depth. The computations perform well even in higher-dimensional systems as long as the assumptions on correlation properties are met.

Implications and Future Directions

This paper's findings hold substantial implications for the simulation of quantum systems on near-term quantum computers. By providing a tractable approach to quantum Gibbs state preparation, this research bridges a crucial gap in realizing practical quantum simulations, particularly in contexts requiring the sampling of thermal states.

From a theoretical standpoint, the paper suggests potential extensions to systems defined by local frustration-free Hamiltonians and raises intriguing questions about topological order at finite temperatures. The authors conjecture that their preparation algorithm can efficiently converge beyond commuting Hamiltonians, contingent upon further exploration of the Markov properties in non-commuting models.

In summary, Brandão and Kastoryano provide a significant contribution to quantum algorithm development through their efficient thermal state preparation method, grounded in finite correlation length and locality principles. The work is poised to inform future investigations into quantum state preparation and the operational capacity of quantum computers in simulating complex quantum systems.