Time-Conditioned Neural Quantum States with Residual-Based Fidelity Bounds

Learning quantum dynamics by minimizing the Schrödinger residual raises a basic question: what does the residual imply about state accuracy? We introduce Abel-tNQS, a time-conditioned autoregressive neural quantum state with an exact initial-state constraint, and relate its Hilbert residual to time-averaged fidelity. The complex conditional-amplitude construction preserves normalization and direct sampling, provided the local normalization vectors remain nonzero. A linear Abel bound supplies the remaining-time weight $(T-t)$ for a separate training surrogate; a cumulative residual form gives a tighter post-training fidelity bound. Full-basis evaluations on small Ising chains yield numerical lower-bound estimates between $0.9972$ and $0.9999$, without certified numerical error control. In controlled full-gradient $N=10$ ablations with ten seeds per setting, linear weighting reduces final mean magnetization and nearest-neighbor correlation errors by approximately $85$% and $94$% relative to uniform weighting. After $3000$ optimizer updates, the hard-linear setting also has lower mean errors in both observables than all three tested soft-penalty settings. Observable benchmarks cover chains of up to $20$ spins, an open $4\times 4$ lattice, segmented continuation, and time-dependent transverse fields.