Sampling two-dimensional isometric tensor network states

TL;DR

This work tackles the problem of sampling from probability distributions encoded by 2D quantum states, which is challenging for traditional PEPS-based methods. The authors extend 1D MPS sampling techniques to 2D isometric tensor network states (isoTNS) by introducing an independent-sample algorithm and a greedy top-$K$ algorithm, both with polynomial scaling in system size and bond dimension. They provide detailed cost analyses, address truncation errors from MPO–MPS contractions, and validate the methods on GHZ, W, and random isoTNS test states, including exact recovery for special cases and understanding of truncation effects. The results offer a scalable pathway for sampling and uncertainty quantification in 2D tensor networks, with potential impact on quantum-supremacy verification, METTS-type methods, and tensor-network Monte Carlo approaches.

Abstract

Sampling a quantum systems underlying probability distributions is an important computational task, e.g., for quantum advantage experiments and quantum Monte Carlo algorithms. Tensor networks are an invaluable tool for efficiently representing states of large quantum systems with limited entanglement. Algorithms for sampling one-dimensional (1D) tensor networks are well-established and utilized in several 1D tensor network methods. In this paper we introduce two novel sampling algorithms for two-dimensional (2D) isometric tensor network states (isoTNS) that can be viewed as extensions of algorithms for 1D tensor networks. The first algorithm we propose performs independent sampling and yields a single configuration together with its associated probability. The second algorithm employs a greedy search strategy to identify K high-probability configurations and their corresponding probabilities. Numerical results demonstrate the effectiveness of these algorithms across quantum states with varying entanglement and system size.

Figures (5)

• Figure 1: Median KL divergence between the empirical and exact distributions as a function of $N$, the number of independently drawn samples. Shaded regions indicate the 16th–84th percentile range over independent Monte Carlo trials. The black line shows the expected $N^{-1}$ Monte Carlo convergence rate.
• Figure 2: Left: Empirical distribution of random state from 10000 samples using the proposed isoTNS algorithm compared with the exact distribution. Right: KL divergence of the empirical distribution and the distribution computed from state-vector versus number of samples for a representative set of maximum bond-dimensions $\chi$ in the isoTNS sampling algorithm \ref{['alg:iso_sample']}.
• Figure 3: The KL divergence between the exact and the computed probability distribution of the GHZ-states (left) and W-states (right) obtained from the top-$K$ isoTNS sampling algorithm for $L\times L$ lattices with different $L$'s. For GHZ we set $K=2$ and for W-states, we set $K=L \times L$ which is sufficient to capture the entire probability distribution in both cases.
• Figure 4: top-$K$ isoTNS algorithm applied to random state. With isoTNS bond-dimension 16, 12, 8 the truncation error is machine precision, $5.6\times 10^{-2}$, $1.5\times 10^{-1}$, respectively. Top: Comparison of exact probabilities of the top 20 high probability states and the computed probabilities obtained from the sampling algorithm. Bottom: The absolute difference between the exact and the computed probabilities for 9 high probability states obtained from the sampling algorithm.
• Figure 5: top-$K$ isoTNS algorithm applied to random state. With isoTNS bond-dimension $1,2,\ldots,16$. We plot the sum of the column truncation errors and the sum of the $L1$ errors in the $K$ probabilities found by the isoTNS algorithm.