近日，美国纽约大学的Alev Orfi和Dries Sels合作并取得一项新进展。经过不懈努力，他们对量子增强马尔可夫链高效混合的障碍进行了研究。相关研究成果已于2024年11月25日在国际知名学术期刊《物理评论A》上发表。

通过为马尔可夫链的能隙设定上界，该研究团队确定了限制算法性能的竞争因素。一方面，需要量子动力学有效地使系统在一系列经典状态上非局域化；但另一方面，通过淬火引入过多的熵也是有害的。

具体来说，研究人员证明了在长时间极限下，马尔可夫链的能隙受限于经典状态在本征态基下的逆参与率，这表明淬火至遍历系统时并无优势。对于典型的谢林顿-柯克帕特里克模型和三自旋模型，研究人员确定了最优谱隙定标的区间，并将其与系统的本征态性质联系起来。

据悉，量子增强的马尔可夫链蒙特卡洛算法是一种通过测量的量子淬火提出配置，并由经典算法接受或拒绝的算法，已被提出作为一种可能的方法，用于在不完美的量子器件上实现稳健的量子加速。虽然该过程对噪声和控制不完美具有抵抗力，但量子优势的潜力尚不明确。

附：英文原文

Title: Barriers to efficient mixing of quantum-enhanced Markov chains

Author: Alev Orfi, Dries Sels

Issue&Volume: 2024/11/25

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, but 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 three-spin model, we identify the regime of optimal spectral gap scaling and link it to the system's eigenstate properties.

DOI: 10.1103/PhysRevA.110.052434

Source: https://journals.aps.org/pra/abstract/10.1103/PhysRevA.110.052434