Re-anchoring Quantum Monte Carlo with Tensor-Train Sketching

TL;DR

The paper tackles the sign and variance challenges in constrained-path AFQMC by re-anchoring with tensor-train (TT) sketching. It introduces a feedback loop that periodically estimates a TT trial wavefunction from CP-AFQMC walkers and uses this TT to guide the next AFQMC episode, reducing bias and variance. Theoretical analysis shows energy estimator variance decreases as the trial approaches the ground state (with a0 = |c_perp/c0| shrinking), while individual walkers remain noisy, motivating explicit TT extraction. Numerical experiments on transverse-field Ising models demonstrate substantial gains in energy accuracy (down to ~10^{-5}) and ground-state overlaps, often outperforming fixed-trial cp-AFQMC and aligning with high-rank DMRG references. Overall, the framework extends TT methods into quantum many-body simulations and points to scalable applications in electronic structure.

Abstract

We propose a novel algorithm for calculating the ground-state energy of quantum many-body systems by combining auxiliary-field quantum Monte Carlo (AFQMC) with tensor-train sketching. In AFQMC, a good trial wavefunction to guide the random walk is crucial for improving the sampling efficiency and controlling the sign problem. Our proposed method iterates between determining a new trial wavefunction in the form of a tensor train, derived from the current walkers, and using this updated trial wavefunction to anchor the next phase of AFQMC. Numerical results demonstrate that the algorithm is highly accurate for large spin systems. The overlap between the estimated trial wavefunction and the ground-state wavefunction also achieves high fidelity. We additionally provide a convergence analysis, highlighting how an effective trial wavefunction can reduce the variance in the AFQMC energy estimation. From a complementary perspective, our algorithm also extends the reach of tensor-train methods for studying quantum many-body systems.

Figures (14)

• Figure 1: Tensor diagrams illustrating the TT-sketching described in Algorithm \ref{['alg:sketching']}. As a demonstration, we present the steps involved constructing the third tensor core $G_3$ for a system with $d=6$. We refer the reader to chen2023committor for a detailed introduction to tensor network diagrams.
• Figure 2: Proposed framework: cp-AFQMC-re-anchoring.
• Figure 3: Results of cp-AFQMC-re-anchoring for $16$ spins in 1D with different transverse-field strength $g$. (a): ground-state energy; (b): transverse-field energy, where the critical point $g=1$ is illustrated with the black dashed line.
• Figure 4: Comparison between the energy convergence of cp-AFQMC and cp-AFQMC-re-anchoring at $g=1$. From (a) to (d): $32$ spins in 1D; $64$ spins in 1D; $96$ spins in 1D; $4\times16$ spins in 2D (cylinder).
• Figure 5: The overlap between the ground-state wavefunction and the sketched wavefunction from cp-AFQMC-re-anchoring, in the form of a rank-$4$ TT. From (a) to (d): $32$ spins in 1D; $64$ spins in 1D; $96$ spins in 1D; $4\times16$ spins in 2D (cylinder).

Theorems & Definitions (12)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• Theorem 5
• Theorem 6
• proof : Proof of Lemma \ref{['lemma:energy error decreasing new']}