发表机构
Centre for Quantum Software and Information, University of Technology Sydney; School of Mathematical and Physical Sciences, Macquarie University; ContinoQuantum; Joint Center for Quantum Information and Computer Science, University of Maryland; Diraq; Beijing International Center for Mathematical Research, Peking University; Molecular Quantum Solutions ApS; Sydney Quantum Academy（悉尼科技大学量子软件与信息中心; 麦考瑞大学数理科学学院; ContinoQuantum; 马里兰大学量子信息与计算机科学联合中心; Diraq; 北京大学北京国际数学研究中心; Molecular Quantum Solutions ApS; 悉尼量子学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出离散绝热框架，用于经典和量子吉布斯态制备，推导跟踪定理和调度方案，并证明量子行走的平方根谱放大可带来间隙依赖的二次改进，使量子绝热成本降至 $O(\delta_{\min}^{-1/2})$。
AI 中文摘要
我们为经典和量子优化算法中的吉布斯态制备建立了一个离散绝热框架。对于有限时间非齐次可逆马尔可夫链，我们推导了一个跟踪定理，该定理通过结合两种效应来界定演化分布与瞬时平稳分布之间的距离：马尔可夫动力学对先前累积误差的收缩，以及当平稳分布从一步到下一步变化时产生的新误差。对于吉布斯路径，这导致了显式的步复杂度界限，以及由马尔可夫链谱间隙和能量涨落尺度 $V(\beta)=\sqrt{\operatorname{Var}_{\pi_\beta}(E)}$ 控制的间隙自适应和涨落自适应调度。在弱阻尼区域，所得调度形式为 $\dot{\beta}\propto\delta(\beta)/V(\beta)$。然后，我们将离散量子绝热定理应用于温度依赖的 Szegedy 量子行走。量子行走的本征相位间隙满足 $\Delta_w=\Theta(\sqrt{\delta})$，其中 $\delta$ 是底层马尔可夫链的谱间隙。通过分析进入定理的有限差分项，我们表明这种平方根谱放大仅在适当间隙自适应调度下转化为间隙依赖的二次改进。在相应的正则性条件下，量子绝热成本为 $O(\delta_{\min}^{-1/2})$，而经典弛豫尺度为 $O(\delta_{\min}^{-1})$。
英文摘要
We develop a discrete adiabatic framework for Gibbs-state preparation in classical and quantum optimisation algorithms. For finite time-inhomogeneous reversible Markov chains, we derive a tracking theorem that bounds the distance from the evolving distribution to the instantaneous stationary state by combining two effects: contraction of previously accumulated error by the Markov dynamics and the new error generated as the stationary distribution changes from one step to the next. For Gibbs paths, this leads to explicit step-complexity bounds and to gap- and fluctuation-adapted schedules controlled by the Markov-chain spectral gap and the energy-fluctuation scale $V(β)=\sqrt{\operatorname{Var}_{π_β}(E)}$. In the weak-damping regime, the resulting schedule takes the form $\dotβ\proptoδ(β)/V(β)$. We then apply the discrete quantum adiabatic theorem to temperature-dependent Szegedy quantum walks. The quantum-walk eigenphase gap satisfies $Δ_w=Θ(\sqrtδ)$, where $δ$ is the spectral gap of the underlying Markov chain. By analysing the finite-difference terms entering the theorem, we show that this square-root spectral amplification translates into a quadratic improvement in the gap dependence only for suitably gap-adapted schedules. Under the corresponding regularity conditions, the quantum adiabatic cost scales as $O(δ_{\min}^{-1/2})$, compared with the classical $O(δ_{\min}^{-1})$ relaxation scale.
Comments50 pages, 12 figures
