[论文解读] Transition in Splitting Probabilities of Quantum Walks

Prashant Singh, David A. Kessler+1 more|arXiv (Cornell University)|Jan 22, 2026

Quantum Computing Algorithms and Architecture被引用 0

一句话总结

论文展示了对带有两个目标的被监测的连续时间量子行走在采样时间控制下的分裂概率的尖锐、类似相变的变化，并通过基于奇偶性的辅助态构造将双目标问题映射为两个单目标问题。

ABSTRACT

We investigate the splitting probability of a monitored continuous-time quantum walk with two targets and show that, in stark contrast to a classical random walk, it exhibits a nonanalytic, phase-transition-like behavior controlled by the sampling time at the targets. For large systems and sampling times smaller than a critical value $τ_c = 2π/ΔE$, where $ΔE$ is the energy bandwidth, the splitting probability is universal and equal to $1/2$, independent of the initial condition and the sampling time. Above the critical sampling, a nonuniversal regime emerges in which the splitting probability deviates from $1/2$ and develops a fluctuating pattern of pronounced peaks and dips dependent on both the sampling time and the initial condition. These results follow from a nontrivial mapping of the splitting problem onto a pair of single-target detection problems enabled by the superposition principle.

研究动机与目标

• Motivate understanding of how measurements affect competing outcomes in quantum walks with two absorbing targets.
• Develop an exact theoretical framework for splitting probabilities under periodic target measurements.
• Reveal nonanalytic, phase-transition-like behavior governed by the sampling time and energy-band structure.
• Show how a dual-target problem maps onto two single-target detection problems via quantum superposition.

提出的方法

• Define a continuous-time quantum walk with two targets and periodic measurements at interval τ.
• Derive the splitting probability via first-detection amplitudes ϕ(α)n and survival operator S.
• Introduce auxiliary orthogonal states |d±> to map dual-target amplitudes to two single-target amplitudes χ(±)n (Eq. 4).
• Exploit parity symmetry [H,P]=0 to simplify cross-terms and enable the mapping for generic n.
• Specialize to a parity-symmetric tight-binding chain to obtain closed-form expressions for PL and PR (Eq. 6) and the interference term ξ(x0,N,τ).
• Analyze the spectral origin via survival-operator eigenvalues λ− and show a transition when τ crosses τc = ∆τc = 2π/∆E.

实验结果

研究问题

• RQ1 测量在两个边界处如何影响量子行走中的分裂概率？
• RQ2 双目标问题是否可以映射为两个单目标问题以便分析？
• RQ3 在何种条件下分裂概率偏离1/2并呈现非普适、波动行为？
• RQ4 作为采样时间的函数，分裂概率观察到的转变背后的光谱机制是什么？
• RQ5 系统尺寸和能带宽度如何影响转变和近似效应？

主要发现

• 分裂概率在采样时间τ的控制下呈现类似相变的变化，对于小τ(≤τc)呈现普适的1/2值；而τ>τc时进入非普适、波动的区域。
• 对于具有对称奇偶性的哈密顿量，双目标幅度可精确映射为两个单目标检测幅度，从而实现精确计算。
• 在共振的τ值上可能出现暗态，导致总检测概率低于1，并在PL和PR上产生不连续性。
• 近端效应在某些区间可能破缺，例如PL(x0) > 1/2，与经典预期相反。
• 在研究的紧束缚模型中，转变点为τc = π/2，与能带宽度(∆E = 4)相关。
• 对于较大N，当τ ≤ τc时，由特征值贡献的相长性抵消，ξ(x0,N,τ)消失，导致PL ≈ PR ≈ 1/2；而τ > τc时ξ不再为零，呈现非普适行为。