Uniformity Bias in Ground-State Sampling Induced by Replica Alignment in Quantum Monte Carlo for Quantum Annealing

TL;DR

This work probes why quantum annealing with a transverse field may unfairly sample degenerate ground states and whether discrete-time quantum Monte Carlo (QMC) can faithfully reproduce QA dynamics. By comparing final ground-state distributions from the QMC master equation $P_{ m QMC}$ with those from the Schrödinger dynamics $P_{ m SD}$, the authors demonstrate that QMC tends to produce more uniform ground-state probabilities, with the uniformity bias growing as annealing proceeds toward $s\to1$. The authors identify replica alignment along the Trotter axis as the core mechanism: ferromagnetic coupling between replicas suppresses kink configurations between adjacent replicas, an effect amplified by small effective time discretization $a=\frac{\beta}{M}(1-s)$ and modulated by the chosen transition rule (Metropolis or heat-bath). They show that the relative weight of kink-free configurations scales as $\exp(-2\beta J^{\star}K)$, and that increasing the Trotter number $M$ (toward the continuous-time limit) can reduce this bias. The results highlight a fundamental limitation of discrete-time QMC for simulating QA and suggest directions to mitigate or strategically control ground-state sampling through $M$ and transition-rule choices.

Abstract

Quantum annealing (QA) with a transverse field often fails to sample degenerate ground states fairly, limiting applicability to problems requiring diverse optimal solutions. Although Quantum Monte Carlo (QMC) is widely used to simulate QA, its ability to reproduce such unfair ground-state sampling remains unclear because stochastic and coherent quantum dynamics differ fundamentally. We quantitatively evaluate how accurately QMC reproduces the sampling bias in QA by comparing the final ground-state distributions from the QMC master equation and the Schrödinger equation. We find QMC tends to produce uniform ground-state probabilities, unlike QA's biased distribution, and that this uniformity bias strengthens as annealing proceeds. Our analysis reveals that this bias originates from replica alignment -- the dominance of configurations in which all Trotter replicas coincide -- caused by the energetic suppression and entropic reduction of kink configurations (replica mismatches). These findings clarify a fundamental limitation of discrete-time QMC in faithfully simulating QA dynamics, highlighting the importance of replica correlations and transition rules in achieving realistic ground-state sampling.

Figures (5)

• Figure 1: (Color online) (a) Simulation error $D(P_{\mathrm{QMC}}, P_{\mathrm{SD}})$ under Metropolis dynamics for each Trotter number and annealing time. Larger values indicate lower simulation accuracy. (b) Deviation from uniformity $D(P_{\mathrm{QMC}}, P_{\mathrm{uniform}})$ with the Metropolis method. Smaller values indicate more uniform sampling over ground states.
• Figure 2: (Color online) Time evolution of the probabilities of each ground state ($t$: MC steps; not physical time). We present the results for the Schrödinger dynamics, Metropolis, and heat-bath methods. Ground states are labeled in order, starting from $|\uparrow\cdots\uparrow\uparrow\rangle$.
• Figure 3: (Color online) Time evolution of the probability of each kink-number sector in QMC under Metropolis dynamics.
• Figure 4: (Color online) The Trotter number dependence of the expected number of kinks per one site at each time $s=0.1, 0.4, 0.7, 1.0$. The measured values obtained from the QMC's master equation and the equilibrium values exactly derived from the partition function are plotted respectively.
• Figure 5: (Color online) Dependence of simulation error $D(P_{\mathrm{QMC}}, P_{\mathrm{SD}})$ on annealing time and Trotter number for different transition rules. The heat-bath method achieves a minimum error at larger $(M,\tau)$ values than the Metropolis method.