Concentrated Monte Carlo sampling for local observables in quantum spin chains

TL;DR

The paper introduces Concentrated Monte Carlo Sampling (CMCS) to improve estimation of local observables in quantum spin chains with short-range correlations. CMCS partitions each global configuration into a local region of size $\ell$ and an environment, enumerates all local configurations ($N_\ell=d^{\ell}$), pairs them with all unique environments ($N_e$) to form $\mathcal{U}$ with $N_u=N_\ell N_e$ configurations, and uses renormalized weights $\tilde{P}$ to compute $\langle O_{\rm loc}\rangle=\sum_{\mathbf{u}\in\mathcal{U}}\tilde{P}(\mathbf{u})O_{\rm loc}(\mathbf{u})$. Demonstrations on the ground state of a 1D tilted Ising model and finite-temperature states of a spin-1 bilinear-biquadratic chain show CMCS yields higher accuracy for local observables with fewer effective samples, particularly away from criticality, while remaining practical due to low overhead and parallelizability. The method holds potential to accelerate local observables and gradients in neural-network quantum states and could extend to number-conserving systems with appropriate adaptations, offering meaningful impact for scalable quantum simulations. math notation is used to describe the local-environment reconstruction and renormalization process, such as $N_u=N_\ell N_e$ and $\tilde{P}(\mathbf{u})=P(\mathbf{u})/\sum_{\mathbf{y}\in\mathcal{U}}P(\mathbf{y}).$

Abstract

Monte Carlo methods are widely used to estimate observables in many-body quantum systems. However, conventional sampling schemes often require a large number of samples to achieve sufficient accuracy. In this work we propose the concentrated Monte Carlo sampling approach, which builds on the idea that in systems with only short range correlations, to obtain accurate expectation values for local observables, one would favor detailed information in the surroundings of this observable compared to far away from it. In this approach we consider all possible configurations in the surroundings of a local observable, and unique samples from the remaining of the setup drawn using Markov chain Monte Carlo. We have tested the performance of this approach for ground states of the spin-1/2 tilted Ising model in different phases, and also for thermal states in the a spin-1 bilinear-biquadratic model. Our results demonstrate that CMCS yields higher accuracy for local observables in short-range correlated states while requiring substantially fewer samples, showcasing in which regimes one can obtain acceleration for the evaluation of expectation values.

Figures (5)

• Figure 1: Illustration of CMCS algorithm for a one-dimensional spin-1/2 chain with 6 sites. Red (dark blue) arrows denote spin-up (spin-down) states. Each Monte Carlo sample is partitioned into a local region, in this case the 2 central sites, and the environment, in this case consisting of the remaining 4 sites, depicted with different background colors according to the distinct environment configurations. For the local region, all possible configurations are produced, and they are recombined with the unique environment configurations to form the configurations from the concentrated sampling.
• Figure 2: Performance of CMCS and MCMC for the different ground states of 1D 20 spins tilted Ising model, presented for different local region sizes $\ell$. The panels show the mean absolute error $\epsilon$ of $\langle \sigma^z_i \rangle$ as a function of the site index $i$, where the exact values are obtained by MPS. The system is in (a) the paramagnetic phase with $h_x=10J,\; h_z=J/2$, (b) the critical region with $h_x=0.95J, \;h_z=J/2$, (c) the antiferromagnetic phase with $h_x=J/2, h_z=J/2$ and (d) the ferromagnetic phase with $J<0, h_x=|J|/2,\; h_z=|J|/2$. $\ast$ represents CMCS $\ell = 2$ with sites 10$-$11, $N_l=4$ and $100\le{N_{u}}\le 2000$; $\lozenge$ represents CMCS $\ell = 3$ with sites 9$-$11, $N_l=8$ and $192\le{N_{u}} \le 4000$; $\bigtriangleup$ represents CMCS $\ell = 4$ with sites 9$-$12, $N_l=16$ and $352\le{N_{u}} \le 8000$.
• Figure 3: Comparison between CMCS and MCMC for the ground state of 1D 20 spins tilted Ising model, at different values of $h_z$ and for $J>0$, panels (a) and (c), and $J<0$, panels (b) and (d), while we consider a fixed local region size $\ell=4$ and fix $h_z=|J|/2$. Results are shown for varying sample sizes. The top panels show the mean absolute error $\epsilon$ of the correlation between the two central sites $\langle \sigma^z_{10} \sigma^z_{11}\rangle$, where exact values are obtained by MPS. The bottom panels show the average number of unique configurations, $\overline{N_u}$. Blue markers indicate the different samples in MC: $\circ$ represents $N_s = 1000$; $\times$ represents $N_s = 5000$; $\lozenge$ represents $N_s = 10000$. Red markers indicates the different samples in CMCS: $\triangle$ represents $N_s = 20$; $\square$ represents $N_s = 100$; $\triangleright$ represents $N_s = 200$. (Here, $N_s$ in CMCS denotes the number of raw MCMC samples used to construct the unique environments, not the number of unique configurations.)
• Figure 4: Comparison of the mean absolute error of correlation matrix $\langle \sigma^z_i \sigma^z_j \rangle$ between MCMC and CMCS for the ground state of 1D 20 spins tilted Ising model for the local region size $\ell = 4$ with sites 9$-$12. Results are shown for varying sample sizes $N_s$ and average numbers of unique configurations $\overline{N_u}$. The system is in the antiferromagnetic phase, with parameters $h_x=J/2, h_z=J/2$. The top panels present the mean absolute error $\epsilon$ of the correlation matrix of the local region obtained from MCMC, while the bottom panels show the corresponding results from CMCS.
• Figure 5: Comparison between CMCS and MCMC of 1D spin-1 bilinear-biquadratic model (\ref{['eq:blbq']}) with 12 spins for the local region size $\ell = 4$ with sites 4-6, and with different temperatures. Results are shown for varying sample sizes $N_s$ and numbers of unique configurations $N_u$. The panels show the absolute error $\epsilon$ of the correlation between the two central sites $\langle S^z_5 S^z_6\rangle - \langle S^z_5 \rangle\langle S^z_6 \rangle$. The system is in the (a) antiferromagnetic-Heisenberg(AFH) phase when $\theta=0$; (b) Affleck-Kennedy-Lieb-Tasaki(AKLT) phase when $\theta=\arctan(1/3)$; (c) critical phase when $\theta=\pi/4$; and (d) ferromagnetic phase when $\theta=2\pi/3$. Blue markers indicate the different samples in MC: $\circ$ represents $N_s = 1000$; $\times$ represents $N_s = 5000$; $\lozenge$ represents $N_s = 10000$. Red markers indicate the different samples in CMCS: $\triangle$ represents $N_s = 10$ with $N_u \le 810$; $\square$ represents $N_s = 50$ with $N_u \le 4050$; $\triangleright$ represents $N_s = 100$ with $N_u \le 8100$.