[论文解读] Quantum Thermal State Preparation

Chi-Fang, Michael J. Kastoryano+2 more|arXiv (Cornell University)|Mar 31, 2023

Quantum Computing Algorithms and Architecture被引用 19

一句话总结

本文提出基于 Lindbladians 的简单连续时间量子 Gibbs 采样器，用以高效逼近量子 Gibbs 状态和纯化 Gibbs 状态，并给出可证明的保证以及一个非渐近的世俗近似与近似详细平衡的框架。

ABSTRACT

Preparing ground states and thermal states is essential for simulating quantum systems on quantum computers. Despite the hope for practical quantum advantage in quantum simulation, popular state preparation approaches have been challenged. Monte Carlo-style quantum Gibbs samplers have emerged as an alternative, but prior proposals have been unsatisfactory due to technical obstacles rooted in energy-time uncertainty. We introduce simple continuous-time quantum Gibbs samplers that overcome these obstacles by efficiently simulating Nature-inspired quantum master equations (Lindbladians). In addition, we construct the first provably accurate and efficient algorithm for preparing certain purified Gibbs states (called thermal field double states in high-energy physics) of rapidly thermalizing systems; this algorithm also benefits from a quantum walk speedup. Our algorithms' costs have a provable dependence on temperature, accuracy, and the mixing time (or spectral gap) of the relevant Lindbladian. We complete the first rigorous proof of finite-time thermalization for physically derived Lindbladians by developing a general analytic framework for nonasymptotic secular approximation and approximate detailed balance. Given the success of classical Markov chain Monte Carlo (MCMC) algorithms and the ubiquity of thermodynamics, we anticipate that quantum Gibbs sampling will become indispensable in quantum computing.

研究动机与目标

• Motivate and formalize the problem of preparing quantum Gibbs and ground states on quantum computers.
• Introduce a physically-inspired, robust Lindbladian-based approach to Gibbs sampling that overcomes energy-time uncertainty obstacles.
• Provide provable guarantees on accuracy, temperature dependence, and mixing (spectral gap) for the implemented samplers.
• Extend to purified Gibbs states (thermal field double states) with a quantum walk speedup.
• Establish a general analytic framework for nonasymptotic secular approximation and approximate detailed balance.

提出的方法

• Construct a Lindbladian whose fixed point approximates the quantum Gibbs state using a smoothed Davies-like generator and weighted operator Fourier transforms.
• Define jump operators Aa and their Fourier-transformed counterparts Âa(ω) with a Gaussian time filter to manage energy resolution.
• Enforce approximate detailed balance to relate energy uncertainty to the fixed-point accuracy of the Gibbs state.
• Analyze the algorithmic cost in terms of the Lindbladian mixing time or spectral gap and the temperature and accuracy parameters.
• Show a purified Gibbs-state preparation approach that yields a quadratic Szegedy-type speedup via coherent quantum walk techniques.

实验结果

研究问题

• RQ1 Can a Lindbladian-based Gibbs sampler achieve an approximately Gibbs fixed point for general noncommuting Hamiltonians without unphysical rounding promises?
• RQ2 How does energy-time uncertainty impact finite-time thermalization, and can approximate detailed balance compensate for this in nonasymptotic analysis?
• RQ3 What are the resource requirements (time, gates) for achieving a specified accuracy at a given temperature, in terms of the Lindbladian’s mixing time or spectral gap?
• RQ4 Is it possible to efficiently prepare purified Gibbs states (thermal field double states) with provable guarantees and speedups?
• RQ5 Can a general analytic framework for secular approximation and approximate detailed balance be extended to physically derived Lindbladians and system-bath models?

主要发现

• Introduces continuous-time quantum Gibbs samplers that overcome energy-time uncertainty obstacles by operating as Lindbladians.
• Provides the first provably accurate and efficient algorithm for preparing certain purified Gibbs states (thermal field double states) with a quantum walk speedup.
• Proves finite-time thermalization for physically derived Lindbladians within a general analytic framework for nonasymptotic secular approximation and approximate detailed balance.
• Demonstrates that the algorithmic cost depends on temperature, accuracy, and the Lindbladian’s mixing time (or spectral gap).
• connects Davies-type open-system dynamics to quantum Gibbs sampling with rigorous guarantees and without relying on unphysical assumptions (e.g., rounding promises).
• Offers a comprehensive comparison to prior approaches and explains why Lindbladian-based Gibbs sampling is a robust route for practical quantum thermal state preparation.