Hybrid Ground-State Quantum Algorithms based on Neural Schrödinger Forging

TL;DR

This work tackles the scalability barrier of entanglement forging in ground-state quantum algorithms by introducing Schrödinger forging guided by autoregressive neural networks (ARNNs). By modeling the distribution of Schmidt coefficients through $|\,\lambda_ au|^2$ and applying a cutoff, the method selects a compact, high‑impact set of bitstrings for a variational quantum eigensolver (VQE), avoiding the exponential cost of summing over all basis states. The authors show that ARNN-based bitstring selection with loss functions like logcosh or maximum mean discrepancy (MMD) yields superior or comparable energy accuracy to standard EF and Heisenberg forging across 1D and 2D spin models and nuclear shell-model configurations, particularly as system size grows. This hybrid classical-quantum approach provides a flexible, permutation-symmetry‑free, resource‑controlled pathway toward scalable ground-state calculations with potential applicability to molecules and nuclei.

Abstract

Entanglement forging based variational algorithms leverage the bi-partition of quantum systems for addressing ground state problems. The primary limitation of these approaches lies in the exponential summation required over the numerous potential basis states, or bitstrings, when performing the Schmidt decomposition of the whole system. To overcome this challenge, we propose a new method for entanglement forging employing generative neural networks to identify the most pertinent bitstrings, eliminating the need for the exponential sum. Through empirical demonstrations on systems of increasing complexity, we show that the proposed algorithm achieves comparable or superior performance compared to the existing standard implementation of entanglement forging. Moreover, by controlling the amount of required resources, this scheme can be applied to larger, as well as non permutation invariant systems, where the latter constraint is associated with the Heisenberg forging procedure. We substantiate our findings through numerical simulations conducted on spins models exhibiting one-dimensional ring, two-dimensional triangular lattice topologies, and nuclear shell model configurations.

Figures (22)

• Figure 1: Schema of the end to end algorithm. The set of bitstrings is first generated by the ARNN, and is then used to perform the Schrödinger forging VQE. This involves iteratively computing the variational energy on the quantum processing unit and classically optimizing the variational parameters until convergence.
• Figure 2: Small models. Training of the generative algorithm for different physical systems. [Top] the number of bitstrings updates between two consecutive iterations. [Bottom] value of the MMD loss at each iteration.
• Figure 3: 1D TFIM 20 spins. Convergence of the variational energy of forged quantum states. The blue curve represents the mean energy over ten sets of $k=8$ random bitstrings, with the shaded area displaying the standard deviation. The purple one is instead showing the training using the set generated by the ARNN. In addition, the simulation with the Heisenberg forging algorithm is shown in pink.
• Figure 4: Correlators in 1D. Correlators $\expval{Z^iZ^j}$ of the Schrödinger and Heisenberg forged states on the TFIM 20 spins in 1D. The pairs $\langle i,j \rangle$ are ordered as follows: [ [$\langle i,j \rangle$ for $i<j$] for $0 \leq i <N$ ]. The neighboring cases, with $j=i+1$, are highlighted with a black vertical line.
• Figure 5: 2D TFIM 12 spins. Convergence of the variational energy of forged quantum states. The colors indicate different boundary conditions, while the shaded curves show the mean energy over ten sets of k = 8 random uniform bitstrings.