Parametric Schrödinger operators with conservation at float64 precision
Long-horizon quantum propagation with norm deviation matching machine epsilon. One million time steps. No drift. Verified on both training parameters and interpolated ones.
conservation at the arithmetic floor
What is shown in this dossier is a presentation of measured results. The technology itself is transferred only under an agreed technological cooperation, accepted by all parties involved. What can be made available under such an agreement extends beyond what is presented here.
One million time steps without drift
For a 1D quantum system with parameterized potential, the developed propagation method preserves the state norm at the fundamental limit of float64 precision. Verified through direct diagonalization and comparison against exact solutions.
| Regime | Conventional method | Developed method | Ratio |
|---|---|---|---|
| Rollout 10⁴ steps | 2·10⁻³ | 10⁻¹⁶ | ≈ 10¹³× |
| Rollout 10⁵ steps | 2·10⁻² | 10⁻¹⁶ | ≈ 10¹⁴× |
| Rollout 10⁶ steps | divergent | 10⁻¹⁶ | ─ |
The result holds for interpolated parameters (not seen during training) — not just for training cases. This is a structural property of the method, not simple memorization.
Operator spectrum reproduced exactly
Not only is time propagation at machine precision — the operator spectrum itself is reproduced exactly, to the last digit the arithmetic can hold. The method captures physical structure rather than approximating it.
| Dimension N | Uniform grid | Chebyshev collocation | Developed method |
|---|---|---|---|
| N = 8 | 5.03 | 11.66 | 0.0 |
| N = 16 | 1.73 | 1.30 | 0.0 |
| N = 24 | 0.59 | 0.12 | 0.0 |
| N = 32 | 0.30 | 3·10⁻³ | 0.0 |
| N = 48 | 0.13 | divergent | 0.0 |
maximum error across the first 6 spectral levels · analytically known ground truth
Same results on a second physical system
Verification extends to a second system with two equilibrium points instead of one — a structurally richer class, with quantum tunnelling between them. Same numbers. The method does not depend on a special case.
| Physical system | Operator precision | Stable rollout up to | Interpolation TEST |
|---|---|---|---|
| System with one equilibrium | exact 0 | 10⁶ steps | 10⁻¹⁶ |
| System with two equilibria, with tunnelling | exact 0 | 10⁵ steps | 10⁻¹⁵ |
The second system eliminates the possibility that the results are a coincidence of the first. The method works across a class of physical systems, not on an isolated case.
Computation time for the control loop
A complete rollout of one million time steps completes in under 2 milliseconds. For quantum control with 10,000 optimization iterations, that means a few seconds total — not hours.
Measured on standard hardware (single CPU run), without parallelization, without GPU. This makes the complete design loop for quantum control feasible in execution time measured in seconds, not hours.
Structural parameterization beats generic neural network
For families of operators with known physical structure, a direct parameterization of the operator form achieves precision several orders of magnitude better than a generic neural network trained on the same data — using far fewer parameters.
| Method | Parameters | Interpolation error | Norm at rollout 5·10³ |
|---|---|---|---|
| Generic neural network (MLP) | 297,000 | 3·10⁻¹ | 1.44 (divergent) |
| Structural parameterization | 4,096 | 10⁻⁷ | 1.0000 |
| Final analytic method | 4,096 | 10⁻¹⁶ | 1.0000000000000 |
A generic neural network is a universal approximator, but universality becomes weakness when problem structure is known. Physics-directed parameterization uses the structural constraint as design specification, not as added regularization — and obtains norm conservation as a guaranteed property, not a learned one.
Precision 10⁻⁷ with 15 data points
The best configuration uses only 15 measurements for a representation of 10 dimensions and reaches a precision of 4.6·10⁻⁷. There is a well-defined operating zone in which the data-to-complexity ratio guarantees stable interpolation, and it is identified in advance rather than found by trial.
Half of what standard practice demands. Verified on two completely different parameterizations, with the same result — which indicates that the determining factor is the ratio itself, not the particular choice of representation. The practical consequence: the data-acquisition budget can be sized before the first measurement is taken. Where a single measurement is expensive — a physical experiment, a sensor campaign, a high-fidelity simulation — that is cost avoided rather than cost optimised.
Machine-precision conservation for operator learning
The method combines a parameterization built around the physical constraints of the problem with a representation adapted to the problem itself rather than imposed on it. Result: quantum propagation with conservation at float64 precision, over time horizons ten orders of magnitude longer than grid-based methods. The construction is held as trade secret and is communicated within a cooperation.
Directly demonstrated applicability
- 1D quantum systems with a parameterized potential (Rabi field, Stark field, controlled anharmonic potential)
- Long-time propagation without accumulated drift — critical for quantum calibration simulations
- Stable interpolation between seen parameters, with errors below experimental detection thresholds
- Compatible with optimization loops (pulse design, quantum control)
Known and honestly documented limitations
- Current results are for 1D systems. Structural extension to 2D/3D is direct but requires separate validation.
- The system must have bounded behaviour at the edges of its domain. We check eligibility on your data before any run — the check is part of the evaluation, not a caveat afterwards.
- For systems where the operator depends on the state itself (non-linear), the approach remains an open question.
The presented numbers are measured from experiments with fixed random seeds, reproducible in ~5 minutes on standard hardware without GPU. All errors are reported as maximum deviation, not mean.