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量子模擬: 量子隨機行走法則 與 量子退火式最佳化演算

張凱鈞

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

不同於一般電腦的數位位元資訊只有兩種可能數值0與1，量子電腦利用的是量子位元，係一個二維希爾伯特(Hilbert)空間中的單位向量，其表示方法為0與1的線性疊加態。量子模擬利用量子物理的基本線性疊加原理，來得到更有效率的方法處理計算科學的問題。本論文討論兩種量子演算法，量子隨機行走法則與量子退火最佳化演算法，在本論文中分成兩大部分，在第一部分中，我們研究在各種圖像中的量子隨機行走法則。研究隨機漫步有助於我們了解各種自然界中的隨機過程，如擴散作用與布朗運動。隨機漫步也已經被運用在許多的電腦演算法中，如搜尋演算法或者最佳化演算法。量子的隨機漫步係建立於量子力學的波函數，也是古典隨機漫步原理的延伸。但古典與量子隨機漫步卻有著非常不同的特性，比如量子隨機漫步傳播的速度大於古典的隨機漫步，且量子隨機漫步並不會像古典隨機漫步一樣會趨向穩定態。量子隨機漫步的時間演進係一由么正(unitary)算符所規範的么正過程，根據定義的不同，我們區分離散時間與連續時間兩種量子隨機漫步。在本論文中，我們研究與比較古典與量子的隨機漫步，分析在圖像以及無序環境上的模擬行走。第二部分，我們利用量子退火式演算解最佳化問題。退火係一材料從高溫控制降溫速度使其保持平衡態最後達成完美結晶結構的物理過程。與傳統的退火演算法利用熱擾動的方法不同，量子退火演算法利用的是量子擾動，使系統在其各種可能的解之間穿隧(tunneling)，以有效的達到最佳解。在本論文中，我們利用建立於路徑積分的蒙地卡羅(Monte Carlo)量子退火演算法，找出自旋玻璃的基態能量。我們以離散的虛數時間方法進行標準的量子蒙地卡羅以及連續的虛數時間方法進行路徑積分的蒙地卡羅，將這兩種量子方法與退火演算法結果的做分析比較。

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In simple ferromagnetic quantum Ising models characterized by an effective double-well energy landscape the characteristic tunneling time of path-integral Monte Carlo (PIMC) simulations has been shown to scale as the incoherent quantum-tunneling time, i.e., as $1/{\mathrm{\ensuremath{\Delta}}}^{2}$, where $\mathrm{\ensuremath{\Delta}}$ is the tunneling gap. Since incoherent quantum tunneling is employed by quantum annealers (QAs) to solve optimization problems, this result suggests that there is no quantum advantage in using QAs with respect to quantum Monte Carlo (QMC) simulations. A counterexample is the recently introduced shamrock model (Andriyash and Amin, arXiv:1703.09277), where topological obstructions cause an exponential slowdown of the PIMC tunneling dynamics with respect to incoherent quantum tunneling, leaving open the possibility for potential quantum speedup, even for stoquastic models. In this work we investigate the tunneling time of projective QMC simulations based on the diffusion Monte Carlo (DMC) algorithm without guiding functions, showing that it scales as $1/\mathrm{\ensuremath{\Delta}}$, i.e., even more favorably than the incoherent quantum-tunneling time, both in a simple ferromagnetic system and in the more challenging shamrock model. However, a careful comparison between the DMC ground-state energies and the exact solution available for the transverse-field Ising chain indicates an exponential scaling of the computational cost required to keep a fixed relative error as the system size increases.
We investigate operator dynamics and entanglement growth in dual-unitary circuits, a class of locally scrambled quantum systems that enables efficient simulation beyond the exponential complexity of the Hilbert space. By mapping the operator evolution to a classical Markov process, we perform Monte Carlo simulations to access the time evolution of local operator density and entanglement with polynomial computational cost. Our results reveal that the operator density converges exponentially to a steady-state value, with analytical bounds that match our simulations. Additionally, we observe a volume-law scaling of operator entanglement across different subregions, and identify a critical transition from maximal to sub-maximal entanglement growth, governed by the circuit’s gate parameter. This transition, confirmed by both mean-field theory and Monte Carlo simulations, provides new insights into operator entanglement dynamics in quantum many-body systems. Our work offers a scalable computational framework for studying long-time operator evolution and entanglement, paving the way for deeper exploration of quantum information dynamics.
本文仅以两条基本公理为出发点——物质场空间分布非均匀性、物质场随时间连续演化且满足局域物质守恒,不引入任何量子公设与先天概率假设,对量子力学中的叠加态、不确定关系、波函数坍缩及量子隧穿等核心现象进行系统的场论分析。本文将量子叠加态描述为非均匀物质场多稳态构型的统计系综,将不确定关系归因于场非均匀度与动量场的动力学耦合,将波函数坍缩解释为场的稳态弛豫过程,将量子隧穿描述为非均匀势场下物质场的跨壁垒演化。全文所有推导均基于连续性方程与场演化基本规律,为量子现象提供了一套基于经典场动力学的确定性统一描述框架。