Neural-Quantum-States Impurity Solver for Quantum Embedding Problems

Abstract

Neural quantum states (NQS) have emerged as a promising approach to solve second-quantized Hamiltonians, because of their scalability and flexibility. In this work, we design and benchmark an NQS impurity solver for the quantum embedding (QE) methods, focusing on the ghost Gutzwiller Approximation (gGA) framework. We introduce a graph transformer-based NQS framework able to represent arbitrarily connected impurity orbitals of the embedding Hamiltonian (EH) and develop an error control mechanism to stabilize iterative updates throughout the QE loops. We validate the accuracy of our approach with benchmark gGA calculations of the Anderson Lattice Model, yielding results in excellent agreement with the exact diagonalisation impurity solver. Finally, our analysis of the computational budget reveals the method's principal bottleneck to be the high-accuracy sampling of physical observables required by the embedding loop, rather than the NQS variational optimization, directly highlighting the critical need for more efficient inference techniques.

Figures (4)

• Figure 1: The neural quantum states - quantum embedding (NQS-QE) workflow. The many-body Hamiltonian is embedded into impurity models via QE algorithms. It is then solved by the NQS impurity solver. The observables are derived from the graph neural network representation of the many-body wave function, which is then used to update the impurity parameters, thereby forming a self-consistent loop.
• Figure 2: (a) The workflow for the neural quantum states (NQS) impurity solver. The outer loop shows the impurity solver's coupling with the QE method, which generates an impurity model for the NQS to solve. The solving process includes the NQS optimization and properties sampling steps, with error control metrics E-tol and P-tol to guarantee convergence. The E-tol and P-tol criteria are checked against user-defined tolerances $\epsilon_t$ and $\epsilon_p$ for the optimization and sampling steps, respectively. Both steps are driven by the Monte Carlo (MC) sampler to compute expectations. (b) The neural network wave function's architecture. The spin orbital configuration is first encoded with the occupation and positional embedding method, and passes through the main body of the network, which comprises N blocks of graph-transformer-like architecture with a residual connected graph attention (GAT) layer and a feedforward neural network (FFN) layer.
• Figure 3: The comparison of the solution of the Anderson lattice model using ED and NQS, combining with the gGA QE method. The band spectrum and density of states (DOS) are displayed. The convergence criteria are 1e-3 for both methods. The NQS impurity solver reproduces the phase difference correctly.
• Figure 4: Computational time cost and error analysis. (a)--(c) Residual during the self-consistent iterations for three sampling-accuracy settings: (a) P-tol$=10^{-3}$, (b) P-tol$=5\times 10^{-4}$, and (c) P-tol$=10^{-4}$. Within each panel, the three curves correspond to different optimization tolerances: E-tol$=10^{-3}$ (solid), E-tol$=5\times 10^{-4}$ (dashed), and E-tol$=10^{-4}$ (dash-dotted). (d) Average wall time per property-sampling step for the same P-tol values; within each P-tol group, bars correspond to E-tol$=10^{-3}$, $5\times 10^{-4}$, and $10^{-4}$ (as labeled above the bars). Error bars denote the standard deviation over the full set of iterations (400 solves).