Neural-Quantum-States Impurity Solver for Quantum Embedding Problems

Neural quantum states (NQS) have emerged as a promising approach to solve second-quantised Hamiltonians, because of their scalability and flexibility. In this work, we design and benchmark an NQS impurity solver for the quantum embedding methods, focusing on the ghost Gutzwiller Approximation (gGA) framework. We introduce a graph transformer-based NQS framework able to represent arbitrarily connected impurity orbitals and develop an error control mechanism to stabilise iterative updates throughout the quantum embedding 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 optimisation, directly highlighting the critical need for more efficient inference techniques.