Bridging Frustration and Non-Hermiticity via COMPASS: An Adaptive Biorthogonal Neural Quantum State Framework

Research Question and Approach

Non-Hermitian (NH) quantum systems present significant computational challenges because traditional variational methods, designed for Hermitian systems, fail to account for the biorthogonal nature of NH eigenstates. Existing approaches often rely on Markov Chain Monte Carlo (MCMC) sampling, which can suffer from autocorrelation and ergodicity issues, or require complex Hermitian embeddings that limit scalability. The authors address these issues by introducing the Complementary Optimization Method for Progressive and Adaptive State Search (COMPASS).

COMPASS utilizes two independent autoregressive recurrent neural networks (RNNs) to represent the left and right eigenstates of an NH Hamiltonian. By using an autoregressive architecture, the model enables exact, direct sampling of spin configurations, bypassing the need for MCMC. The optimization process is split into two phases: an energy-based phase to identify the ground-state manifold, and a variance-based phase to ensure the state is a true eigenstate. An adaptive curriculum learning strategy is employed, where the network capacity (hidden dimension) is progressively increased during training, allowing for efficient exploration of the variational space.

The study demonstrates that the choice of ansatz is critical for NH simulations. For parity-time (PT)-symmetric Hamiltonians, unconstrained complex ansatze can lead to spontaneous symmetry breaking and spurious imaginary energies, whereas real-valued ansatze maintain physical consistency. Conversely, for generic NH systems, complex ansatze are necessary to capture the complex spectrum.

Applying COMPASS to frustrated spin-1/2 chains, the authors show that gap frustration provides a quantitative shield against NH spectral instability, with the frustration gap setting a threshold for PT-symmetry breaking. Furthermore, by complexifying the frustration coupling, they identify a new topologically nontrivial network of diabolic level crossings—the diabolic ring—which has no analog in Hermitian physics.