Direct/adaptive-mixture phase-gradient learning for neural-network quantum states with complex phase structure

Neural-network quantum states (NQS) are a leading variational tool for quantum many-body physics, yet their optimization is fragile whenever the ground state carries a non-trivial sign or complex phase structure, a situation generic to gauge fields, broken time-reversal symmetry, and fermionic statistics. We trace this fragility to the stochastic estimator of the phase gradient rather than to network expressiveness. The phase sector of the Monte Carlo energy gradient is a noisy score-function estimator; differentiating the local energy instead yields a direct estimator that is unbiased for the same phase force, has far lower variance, and requires only a separated amplitude--phase ansatz. Demonstrated on a 100-site flux ladder, a small network trained this way reaches $0.89\%$ median error, where tuned standard baselines plateau at $1.8\%$ and wider or deeper standard-gradient networks degrade from $8.4\%$ to $24.6\%$. The advantage carries over to chiral XXX chains: the direct estimator again converges to a markedly lower error than the standard one, across $α$ and size; it grows with flux and vanishes in zero-flux controls. An adaptive-mixture of the two estimators is provably never worse in variance than the better endpoint at the optimal mixing coefficient, with seed-resolved diagnostics tracing much of the gain to eliminating failed runs. Estimator design thus emerges as a first-class lever for complex-valued neural quantum states.

A Mathematical Theory of Agency and Intelligence

To operate reliably under changing conditions, complex systems require feedback on how effectively they use resources, not just whether objectives are met. Current AI systems process vast information to produce sophisticated predictions, yet predictions can appear successful while the underlying interaction with the environment degrades. What is missing is a principled measure of how much of the total information a system deploys is actually shared between its observations, actions, and outcomes. We prove this shared fraction, which we term bipredictability, P, is intrinsic to any interaction, derivable from first principles, and strictly bounded: P can reach unity in quantum systems, P equal to, or smaller than 0.5 in classical systems, and lower once agency (action selection) is introduced. We confirm these bounds in a physical system (double pendulum), reinforcement learning agents, and multi turn LLM conversations. These results distinguish agency from intelligence: agency is the capacity to act on predictions, whereas intelligence additionally requires learning from interaction, self-monitoring of its learning effectiveness, and adapting the scope of observations, actions, and outcomes to restore effective learning. By this definition, current AI systems achieve agency but not intelligence. Inspired by thalamocortical regulation in biological systems, we demonstrate a feedback architecture that monitors P in real time, establishing a prerequisite for adaptive, resilient AI.