Quantum

Efficient Quantum Monte Carlo through Cluster Expansions

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Briefings

A new Markov chain algorithm replaces deterministic cluster enumeration with Monte Carlo sampling in the space of connected clusters, eliminating the sign problem for quantum Monte Carlo methods. The framework establishes an FPRAS with runtime O(N^4 epsilon^-3) for general polymer models and improves to O(N^3+1/D epsilon^-3) for quantum partition functions at temperatures beta less than or equal to O(1). This approach applies to long-range quantum systems with interactions scaling as 1/r^alpha, including 2-local models where alpha exceeds 2D.

Key points

• The method introduces a Markov chain algorithm that samples connected clusters directly in an abstract cluster space, avoiding the exponential sign problem present in physical configuration space.
• For general polymer models, an FPRAS is established with a runtime of O(N^4 epsilon^-3), where N is system size, epsilon is precision, and the algorithm applies to long-range interactions scaling as 1/r^alpha.

Context and limitations

The framework is limited to the convergence regime of cluster expansions, where deterministic polynomial-time algorithms are possible. It does not address the computational overhead of exhaustive enumeration in non-convergent regimes or systems with interaction degrees exceeding the sampling temperature threshold.

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