Addressing general measurements in quantum Monte Carlo

Abstract

Quantum Monte Carlo is one of the most promising approaches for dealing with large-scale quantum many-body systems. It has played an extremely important role in understanding strongly correlated physics. However, two fundamental problems, namely the sign problem and general measurement issues, have seriously hampered its scope of application. We propose a universal scheme to tackle the problems of general measurement. The target observables are expressed as the ratio of two types of partition functions $\langle \mathrm{O} \rangle=\bar{Z}/Z$, where $\bar{Z}=\mathrm{tr} (\mathrm{Oe^{-βH}})$ and $Z=\mathrm{tr} (\mathrm{e^{-βH}})$. These two partition functions can be estimated separately within the reweight-annealing frame, and then be connected by an easily solvable reference point. We have successfully applied this scheme to XXZ model and transverse field Ising model, from 1D to 2D systems, from two-body to multi-body correlations and even non-local disorder operators, and from equal-time to imaginary-time correlations. The reweighting path is not limited to physical parameters, but also works for space and time. Essentially, this scheme solves the long-standing problem of calculating the overlap between different distribution functions in mathematical statistics, which can be widely used in statistical problems, such as quantum many-body computation, big data and machine learning.

Figures (21)

• Figure 1: Schematic of the bipartite reweight-annealing process. Direct evaluation of the ratio $\bar{Z}(J_0)/Z(J_0)$ is generally infeasible, since the corresponding weight distributions at $J_0$ exhibit a lack of overlap, as sketched in (a). Instead, two independent reweight-annealing (RA) processes are performed: the red path corresponds to $\bar{Z}(J)/\bar{Z}(J_0)$, and the blue path corresponds to $Z(J)/Z(J_0)$, as indicated by the colored RA process labels. Along each path, the system is gradually evolved in the annealing direction through small parameter shifts $J’ = J + \delta J$, ensuring sufficient overlap between adjacent distributions, as illustrated in (b) and (c). Once the annealing paths reach an easily solvable reference point, the target ratio $\bar{Z}(J_0)/Z(J_0)$ can be reconstructed.
• Figure 2: Equal-time off-diagonal correlation measurement via the reweight-annealing framework. (a) Ratios of two-point correlations $C_{ij}=\langle S^x_i S^x_j \rangle$ as a function of the Ising coupling strength $\Delta$ for $L=10$ with $\beta=20$. (b) Ratios of four-point correlations $C_{ijkl}=\langle S^x_i S^x_j S^x_k S^x_l \rangle$ in the one-dimensional chain. (c) Ratios of representative two-point ($C_{ij}$) and four-point ($C_{ijkl}$) correlations on a $4\times 2$ lattice with $\beta=8$. (d) Four-point correlation $C_{12,L/2,L/2+1}=\langle S^x_1 S^x_{2} S^x_{L/2} S^x_{L/2+1}\rangle$ on $8\times 8$ and $20\times 20$ square lattices with $\beta=2L$. Panels (a)-(c) include comparisons with exact diagonalization (ED) results. Error bars ($\pm 1\sigma$) from SSE simulation denote the standard error of the mean obtained from the Monte Carlo bins. All calculations are performed on lattices with periodic boundary conditions (PBC).
• Figure 3: Spatial coupling parameters Annealing. The off-diagonal correlation measurement via lattice reweight-annealing method in the 1D XXZ model with $L=48$. (a) The lattice diagram for annealing along the system size $L$. We incrementally tune the coupling $J_2$ from $0^+$ to 1. (b) The lattice diagram for annealing along the distance $r$ between $S^x_1$ and $S^x_{1+r}$. We firstly gradually adjust the coupling $J_2$ from $0^+$ to 1, then we gradually tune the coupling $J_3$ from 1 to $0^+$. (c) Two point off-diagonal correlations for system size $L=48$ obtained from subplots $(a)$ and $(b)$ annealing method. The dashed lines represent the simulation for annealing Ising coupling $\Delta$ with fixed $S^x_1S^x_{1+r}$. Both the hollow symbols and the dashed line represent QMC results, displayed with error bars denoting the mean value and standard error.
• Figure 4: Separability of multipartite off-diagonal observables. (a) When a large system is decomposed into several parts without coupling, the measured observable can also be separated into the product of independent components. (b) The off-diagonal correlations obtained via the annealing from two small part $A$ and $B$. Here $C_4=\langle S^x_1S^x_2S^x_{L/2+1}S^x_{L/2+2}\rangle$. The $S^x_1S^x_2$ is set on the part $A$, and the $S^x_{L/2+1}S^x_{L/2+2}$ is set on the part $B$ ($i=L/2+1$). The colorful dots are QMC results with standard error as error bars. And the dashed lines are the pure ED results.
• Figure 5: The disorder operator $\langle X \rangle$ measurement in the 2D TFIM. Setting $L=16$, $\beta=16$ and $h=1$. The dashed lines are the fitting curves. (a) Scaling behaviors of $\langle X \rangle$ in the paramagnetic phase. (b) Scaling behaviors of $\langle X \rangle$ in the ferromagnetic phase. Each error bar represents the statistical one-standard-error estimates.