Stoquasticity is not enough: towards a sharper diagnostic for Quantum Monte Carlo simulability

TL;DR

This work reframes the quantum Monte Carlo sign problem through Vanishing Geometric Phase (VGP), a geometric criterion applied to the computational state graph in Permutation Matrix Representation (PMR). It shows that VGP encompasses a broader class of sign-problem-free Hamiltonians than stoquastic ones and develops cycle-weight diagnostics that can efficiently identify VGP in certain sparse and periodically structured systems. The authors introduce quantitative measures (e.g., $f_{VGP}$, $f_\eta$, and off-diagonal interference metrics) to assess sign-problem severity and analyze how these metrics scale with system size $N$ and inverse temperature $\beta$, including concentration results for random basis transformations. They demonstrate that random basis searches are unlikely to cure the sign problem and provide explicit examples where VGP enables sign-problem-free behavior without readily identifiable stoquasticity. The work thus provides both a conceptual framework and practical tools for diagnosing and understanding QMC simulability, with implications for extending QMC applicability and guiding mitigation strategies.

Abstract

Quantum Monte Carlo (QMC) methods are powerful tools for simulating quantum many-body systems, yet their applicability is limited by the infamous sign problem. We approach this challenge through the lens of Vanishing Geometric Phases (VGP) \cite{Hen_2021}, introducing it as a `geometric' criterion for diagnosing QMC simulability. We characterize the class of VGP Hamiltonians, and analyze the complexity of recognizing this class, identifying both hard and efficiently identifiable cases. We further highlight the practical advantage of the VGP criterion by exhibiting specific Hamiltonians that are readily identified as sign-problem-free through VGP, yet whose stoquasticity is difficult to ascertain. These examples underscore the efficiency and sharpness of VGP as a diagnostic tool compared to stoquasticity-based heuristics. Beyond classification, we propose a family of VGP-inspired diagnostics that serve as quantitative indicators of sign problem severity. While exact evaluation of these quantities is generically intractable, we demonstrate their mathematical power in performing scaling analysis for the average sign under unitary transformations. Our results provide both a conceptual foundation and practical tools for understanding and mitigating the sign problem.

Figures (10)

• Figure 1: Fundamental cycle of length $5$, denoted by $\mathcal{C}_{i_5},$ produced by taking the string of $\Pi_{k=1}^5 P_{i_k}$ on state $\ket{z_1}$. The weight of the off-diagonals of this cycle is $W^{(z_1)}_{\mathcal{C}_{i_5}} = d^{(i_1)}_{(z_1)_{i_1}} d^{(i_2)}_{(z_2)_{i_2}} \ldots d^{(i_5)}_{(z_4)_{i_4}}$.
• Figure 2: Diagram showing classes of Hamiltonian that are sign-problem free.
• Figure 3: Figure of the ladder triangular lattice with a defect. The defect edge is highlighted in the figure.
• Figure 4: $\langle \text{sgn} \rangle$ vs. $N$ for $N_d=1$ at $\beta=0.1$. The $\langle q \rangle_{\text{QMC}}$, even though in this case, is increases as a function fo $N$ (linearly), it remains less than $3$ all of the relevant system size simulations.
• Figure 5: $\langle q \rangle_{\text{QMC}}$ vs. $N$ for $N_d=1$ at $\beta=0.5$. The $\langle q \rangle_{\text{QMC}}$ increases as a function for $N$ (linearly), however, it exceeds the length of the smallest fundamental cycles. This means that fundamental cycles of large length, and concatenation of smaller fundamental cycles (of average length $\langle q \rangle$) contribute to the partition function.

Theorems & Definitions (32)

• Definition 1
• Definition 2: Closed walks on $G_H$
• Definition 3: Fundamental cycles
• Definition 4: Fundamental generators
• Definition 5: Graph Diameter
• Lemma 1: Scaling of $\mathrm{diam}(G_H)$ for physical Hamiltonians
• proof
• Theorem 1: Largest fundamental cycles on $G_H$
• proof
• Definition 6: VGP indicator