Classical Neural Quantum States and the Quest for Quantum Advantage in Many-Body Simulation

Quantum advantage in many-body simulation is meaningful only relative to what the best classical algorithms can do. In this talk I take a deliberately classical perspective, asking how far stochastic variational methods can be pushed for ground states and real-time dynamics of strongly correlated lattice systems before quantum devices become indispensable. I will first discuss "variational benchmarks" of quantum many-body ground states, where a large data set of tensor-network, neural-network and quantum-circuit calculations is compressed into the V-score, a metric built from variational energies and variances (Science 386, 296 (2024)). This allows one to rank Hamiltonians by classical hardness and to identify regimes where robust quantum advantage in ground-state problems remains plausible. I will then highlight recent classical developments based on neural quantum states and time-dependent variational Monte Carlo, including accurate simulations of Rydberg-atom protocols and other far-from-equilibrium dynamics, and of frustrated spin systems on nontrivial geometries (e.g. Nat. Commun. 16, 62098 (2025); Nat. Phys. 21, 2944 (2025)). I will also present new results on the dynamics of 2d interacting fermions and on ground states of triangular-lattice spin Hamiltonians. Throughout, I will use these examples to delineate where state-of-the-art classical stochastic variational methods still track, constrain or challenge near-term quantum simulators, and where current evidence for a decisive quantum advantage remains inconclusive.