Mitigating the sign problem by quantum computing

TL;DR

This work critically examines a quantum-computing stochastic series expansion method and shows that it does not strictly resolve the sign problem for Hamiltonians with non-commuting terms, but provides a practical mitigation strategy that suppresses the occurrence of negative weights.

Abstract

The notorious sign problem severely limits the applicability of quantum Monte Carlo (QMC) simulations, as statistical errors grow exponentially with system size and inverse temperature. A recent proposal of a quantum-computing stochastic series expansion (qc-SSE) method suggested that the problem could be avoided by introducing constant energy shifts into the Hamiltonian. Here we critically examine this framework and show that it does not strictly resolve the sign problem for Hamiltonians with non-commuting terms. Instead, it provides a practical mitigation strategy that suppresses the occurrence of negative weights. Using the antiferromagnetic anisotropic XY chain as a test case, we analyze the dependence of the average sign on system size, temperature, anisotropy, and shift parameters. An operator contraction method is introduced to improve efficiency. Our results demonstrate that moderate shifts optimally balance sign mitigation and statistical accuracy, while large shifts amplify errors, leaving the sign problem unresolved but alleviated.

Figures (5)

• Figure 1: (a) Average sign $\langle \textrm{sgn} \rangle$ and (b) absolute percentage error $|\delta E / E_\mathrm{ED}|$ as functions of $M = M_x = M_z$ for system size $N=3$ and temperature $T=2$. The inset of (a) shows the average operator string length $\langle n\rangle$, while the inset of (b) shows the absolute energy difference $\Delta E = |E - E_\mathrm{ED}|$. Panels (c) and (d) present the corresponding results at temperature $T=1$.
• Figure 2: (a) Average sign $\langle \textrm{sgn} \rangle$ and (b) absolute percentage error $|\delta E / E_\mathrm{ED}|$ as functions of $M_x$ with $M_z$ fixed at 1 for system size $N=3$ and temperature $T=2$. The inset of (a) shows the average operator string length $\langle n\rangle$, while the inset of (b) shows the absolute energy difference $\Delta E = |E - E_\mathrm{ED}|$.
• Figure 3: (a) Size dependence of average sign $\langle \textrm{sgn} \rangle$ with fixed $T=2$; (b) temperature dependence of average sign $\langle \textrm{sgn} \rangle$ with fixed $N=3$. Here we take $M_x=M_z=1$. The inset of (a) shows the average operator string length $\langle n\rangle_{|W|}$.
• Figure 4: Anisotropy dependence of average sign $\langle \textrm{sgn} \rangle$ for $N=3$ and $M_z=M_x=1$ at $T=2$. The inset shows the average operator string length.
• Figure 5: Average sign $\langle \textrm{sgn} \rangle$ of classical SSE on z-basis and qc-SSE as a function of $T$ for $\Delta=1$ and $N=3$.