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Quadratic Quantum Variational Monte Carlo
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Proposes Q2VMC, a quadratic update for neural-network quantum variational Monte Carlo derived from a discretized imaginary-time Schrödinger evolution projected onto the ansatz via KL divergence, and tests it on Psiformer and LapNet for six small molecules.
Problem
Optimizing neural wavefunction ansatzes in quantum variational Monte Carlo (QVMC) is slow and often needs long convergence times, and continuous imaginary-time evolution is impractical with finite steps.
Approach
Discretize imaginary-time Schrödinger evolution as a multiplicative (1 - tau E_L) update and project the evolved distribution back onto the ansatz by minimizing KL divergence; this yields the standard QVMC gradient plus a quadratic local-energy term at no extra cost.
Key results
- Q2VMC reaches target energies in roughly half the iterations of QVMC (claimed 2x speed-up) on six molecules with Psiformer and LapNet.
- For Psiformer on C4H6 (30 electrons), Q2VMC reaches -155.94665(4) Ha versus -155.94619(8) for the Psiformer reference and -155.94528(4) reproduced by the baseline LapNet reference.
- Tuning QVMC's learning rate on NH3 (best -56.56379(1)) still did not match Q2VMC (-56.56384(1) at eta0=0.2).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add the quadratic Q2VMC update | Psiformer and LapNet · six molecules: Li2, NH3, CO, CH3NH2, C2H6O, C4H6 | ground-state energy: -14.99486(1) -> -14.99490(1); -56.56367(2) -> -56.56374(2); -113.32416(4) -> -113.32442(2); -95.86050(4) -> -95.86073(2) | Helped |
| add the quadratic Q2VMC update | LapNet · six molecules: Li2, NH3, CO, CH3NH2, C2H6O, C4H6 | ground-state energy: -14.99485(1) -> -14.99486(1); -56.56359(2) -> -56.56370(2); -113.32417(4) -> -113.32428(2); -95.86025(3) -> -95.86053(2) | Helped |
| use learning rate η0=0.2 instead of the default η0=0.05 (ablation) | QVMC with Psiformer · NH3 | ground-state energy: -56.56366(2) -> -56.56379(1) | Helped |
| increase learning rate to η0=0.5 (ablation) | QVMC with Psiformer · NH3 | ground-state energy | Hurt |
| use learning rate η0=0.2 instead of η0=0.05 (ablation) | Q2VMC with Psiformer · NH3 | ground-state energy: -56.56374(2) -> -56.56384(1) | Helped |
| increase learning-rate decay time from 1e4 to 3e4 (ablation) | QVMC with Psiformer · NH3 | ground-state energy: -56.56349(3) -> -56.56369(1) with norm constraint 3e-3; -56.56366(2) -> -56.56366(1) with norm constraint 1e-2 | Mixed |
| increase norm constraint to 3e-2 (ablation) | QVMC with Psiformer · NH3 | ground-state energy: Diverge | Hurt |
| reduce learning rate to 1e-2, with decay time 1e4 and norm constraint 1e-2 (ablation) | QVMC with Psiformer · NH3 | ground-state energy: -56.56366(2) | Helped |
| use learning rate 5e-2, decay time 1e4, and norm constraint 1e-2 (ablation) | QVMC with Psiformer · NH3 | ground-state energy: -56.56313(3) | Hurt |
Surprising findings
- Q2VMC with default hyperparameters on NH3 outperforms every tuned QVMC learning rate, and tuned Q2VMC matches or exceeds the Psiformer Large model despite the small model's ~4x smaller size.
Limitations
- Optimal imaginary time step is unresolved and the error from approximate projection is not quantified.
- Experiments limited to systems up to 30 electrons; O(N^4) scaling remains a bottleneck.
- Only ground-state energies tested; relative energies and excited states left for future work.
Experimental setup
Benchmarks: Li2 NH3 CO CH3NH2 C2H6O C4H6 molecular ground-state energies
Metrics: ground state energy convergence iterations energy variance
Baselines: QVMC with Psiformer QVMC with LapNet Psiformer (Large) Stochastic Reconfiguration Wasserstein QMC
Builds on: FermiNet Psiformer LapNet Stochastic Reconfiguration KFAC
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