Quantum enhanced Markov chains require fine-tuned quenches

TL;DR

This paper investigates quantum-enhanced Markov chain Monte Carlo algorithms, revealing that their performance depends on a delicate balance in quantum dynamics and eigenstate properties, with no advantage in ergodic regimes.

Contribution

It provides bounds on the Markov chain gap, linking quantum dynamics to eigenstate properties, and identifies optimal regimes for spectral gap scaling in specific models.

Findings

The Markov chain gap is bounded by the inverse participation ratio.

No advantage in ergodic systems due to eigenstate delocalization.

Optimal spectral gap scaling depends on system eigenstates.

Abstract

Quantum-enhanced Markov chain Monte Carlo, an algorithm in which configurations are proposed through a measured quantum quench and accepted or rejected by a classical algorithm, has been proposed as a possible method for robust quantum speedup on imperfect quantum devices. While this procedure is resilient to noise and control imperfections, the potential for quantum advantage is unclear. By upper-bounding the gap of the Markov chain, we identify competing factors that limit the algorithm's performance. One needs the quantum dynamics to efficiently delocalize the system over a range of classical states, however, it is also detrimental to introduce too much entropy through the quench. Specifically, we show that in the long-time limit, the gap of the Markov chain is bounded by the inverse participation ratio of the classical states in the eigenstate basis, showing there is no advantage when quenching to an ergodic system. For the paradigmatic Sherrington-Kirkpatrick and 3-spin model, we identify the regime of optimal spectral gap scaling and link it to the system's eigenstate properties.