Bell sampling in Quantum Monte Carlo simulations

TL;DR

Bell-QMC integrates Bell sampling into quantum Monte Carlo by formulating SSE in the Bell basis and performing two-copy measurements, enabling unbiased estimation of all Pauli operators and the Rényi-2 entanglement entropy for arbitrary subsystems in a single simulation. The method leverages a two-copy Hamiltonian $\mathcal{H}$ and specialized updates (diagonal, cluster, bond/plaquette cluster) to efficiently sample Bell configurations, yielding direct access to $S_2(A)$ and related observables with variance advantages over conventional QMC, even in higher dimensions. Demonstrations on the 1D transverse-field Ising model and 2D $\mathbb{Z}_2$ lattice gauge theory show agreement with exact diagonalization and DMRG, extraction of universal features like the central charge $c\approx 1/2$ and topological entanglement entropy $\gamma$, and the ability to compute Wilson loops and entanglement properties on large lattices. The approach offers a conceptual and practical paradigm shift, enabling measurements of nonlinear and off-diagonal observables in QMC and suggesting extensions to Heisenberg interactions, other QMC flavors, and multi-copy schemes for broader entanglement and spectral investigations.

Abstract

Quantum Monte Carlo (QMC) methods are essential for the numerical study of large-scale quantum many-body systems, yet their utility has been significantly hampered by the difficulty in computing key quantities such as off-diagonal operators and entanglement. This work introduces Bell-QMC, a novel QMC framework leveraging Bell sampling, a two-copy measurement protocol in the transversal Bell basis. We demonstrate that Bell-QMC enables an efficient and unbiased estimation of both challenging classes of observables, offering a significant advantage over previous QMC approaches. Notably, the entanglement across all system partitions can be computed in a single Bell-QMC simulation. We implement this method within the stochastic series expansion (SSE), where we design an efficient update scheme for sampling the configurations in the Bell basis. We demonstrate our algorithm in the one-dimensional transverse-field Ising model and the two-dimensional $\mathbb{Z}_2$ lattice gauge theory, extracting universal quantum features using only simple diagonal measurements. This work establishes Bell-QMC as a powerful framework that significantly expands the accessible quantum properties in QMC simulations, providing a substantial advantage over conventional QMC.

Figures (13)

• Figure 1: (a) Rényi-2 EE $S_2$ in the ground state of the TFIM at the critical point $h=1$ for partitions of sizes $\ell=\{2,4,\dots, 1022\}$ in the middle of the chain, with $L=1024$ and $\beta=3L$. Inset: logarithmic scaling of $S_2$ up to subsystem size $\ell=128$. The solid line denotes a linear fit, from which we extract the central charge $c \approx 0.5030(2)$. (b) The topological entanglement entropy $S^t_\mathrm{topo}$ as a function of the transverse field $h$ around the critical point $h=1$ and $\beta=4L$.
• Figure 2: Rényi-2 EE $S_2$ in the ground state of the $\mathbb{Z}_2$ lattice gauge theory. In (a), we show $S_2$ as a function of the transverse field $h$ around the critical point $h_c \approx 0.33$ for various system sizes with $\beta=4L$. In (b), we set $h=0.3$ and $\beta=L$, extrapolating $S_2$ to obtain the TEE $\gamma=0.71(3)$. In both plots, we consider a square partition of size $L/2 \times L/2$ with boundary length $\ell=2L$.
• Figure 3: The scaling of Wilson loop in the ground state of the $\mathbb{Z}_2$ lattice gauge theory at (a) $h=0.3$ in the deconfined phase and (b) $h=0.35$ in the confined phase. In both plots, we consider a square partition of size $L/2 \times L/2$ with boundary length $\ell=2L$ and temperature $\beta=4L$. The solid line denotes a linear fit.
• Figure S1: Bell sampling from two copies of the state $\rho$. The measurement outcomes are $r_i^z$ and $r_i^x$ which label the Bell states in Eq. \ref{['eq:bell_states']}.
• Figure S2: (a) The state $\ket{r^z, r^x}$ is represented by a circle, with the left half encoding $r^z$ and the right half $r^x$; an empty half denotes 0, and a filled half denotes 1. (b) A site operator is denoted by a square (empty for diagonal, filled for off-diagonal), where the matrix element $\langle r_i'^z ,r_i'^x|\mathcal{H}_{t,i}|r_i^z, r_i^x\rangle$ is represented as shown. From left to right, the three examples correspond to $\langle 0,0|\mathcal{H}_{0,i}|0,0\rangle$, $\langle 0,0|\mathcal{H}_{1,i}|0,1\rangle$, and $\langle 1,0|\mathcal{H}_{0,i}|1,0\rangle$, respectively. (c) A bond operator is denoted by a rectangle (empty for diagonal, filled for off-diagonal), where the matrix element $\langle r_i'^z, r_i'^x| \langle r_j'^z, r_j'^x|\mathcal{H}_{t,\langle ij\rangle}|r_i^z, r_i^x\rangle | r_j^z, r_j^x\rangle$ is represented as shown. From left to right, the three examples correspond to $\langle 0,0| \langle 0,0|\mathcal{H}_{0,\langle ij\rangle}|0,0\rangle |0,0\rangle$, $\langle 0,0|\langle0,0|\mathcal{H}_{1,\langle ij\rangle}|1,0\rangle|1,0\rangle$, and $\langle 0,0|\langle0,1|\mathcal{H}_{0,\langle ij\rangle}|0,0\rangle|0,1\rangle$, respectively.