Re-anchoring Quantum Monte Carlo with Tensor-Train Sketching

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.

💡 Research Summary

The paper introduces a new hybrid algorithm that integrates constrained‑path auxiliary‑field quantum Monte Carlo (cp‑AFQMC) with tensor‑train (TT) sketching in a self‑consistent loop, aiming to improve the trial wavefunction used to control the sign/phase problem and to reduce the variance of the energy estimator. In standard cp‑AFQMC the many‑body wavefunction is represented by an ensemble of walkers Φₖ, and a trial wavefunction Ψ_tr is required to enforce a non‑negative overlap ⟨Ψ_tr, Φₖ⟩>0. The quality of Ψ_tr directly determines the systematic bias and the statistical variance. The authors propose to periodically replace Ψ_tr by a low‑rank TT approximation constructed from the current ensemble of walkers. This TT approximation is obtained via a randomized sketching procedure that projects each unfolding of the high‑dimensional tensor formed by the walkers onto a set of random Gaussian tensors, then solves for the TT cores sequentially and truncates them to a target rank. The sketching dimension is chosen larger than the desired rank (typically five times) to guarantee with high probability that the range of the sketched matrix captures the dominant subspace.

The algorithm proceeds in outer iterations (M_out). In each outer iteration the current trial wavefunction Ψ_trⁿ is used to run cp‑AFQMC for M_in steps, producing a new ensemble bΨⁿ⁺¹ = ΣₖΦₖ. A TT‑sketch of bΨⁿ⁺¹ yields an updated trial wavefunction Ψ_trⁿ⁺¹, which is then fed back into the next outer iteration. Because the trial wavefunction becomes progressively closer to the exact ground state Ψ₀, the importance‑sampling distribution Qₖ(ξ) becomes more focused on high‑overlap auxiliary fields, thereby reducing the variance of the weight factors. The authors prove two key theoretical results: (1) Lemma 1 shows that the overlap ⟨Ψ_tr, e^{B}_k(ξ)Φₖ⟩ is independent of the sampled auxiliary field ξ, which underlies the variance reduction of the importance‑sampling step; (2) Theorem 1 establishes that the variance of the energy estimator scales as the square of the trial‑wavefunction error, Var