netket.errors.MCMixedStateExpectAndGradOnPhysicalOperatorError

exception netket.errors.MCMixedStateExpectAndGradOnPhysicalOperatorError

Error raised when calling netket.vqs.MCMixedState.expect_and_grad() with a physical operator instead of a superoperator.

NetKet only provides built-in mixed-state gradient estimators for superoperators. Calling netket.vqs.MCMixedState.expect_and_grad() with a standard operator such as a Hamiltonian therefore raises this error.

A good estimator for this gradient is, to our knowledeg, not discussed in the literature. Deriving a good one would be an interesting endeavour.

For mixed states and physical operators there is no single obviously canonical Monte Carlo estimator to use. Several estimators may share the same infinite-sample expectation value while exhibiting very different finite-sample variance, signal-to-noise ratio, normalization, bias, and optimization-stability properties.

Implementing a custom estimator

If you want to support this use case, define a custom dispatch rule for netket.vqs.expect_and_grad() matching your variational state and the operator family you care about. In most cases you should dispatch on a suitable operator supertype rather than on one exact concrete operator class.

A minimal registration looks like:

import netket as nk

@nk.vqs.expect_and_grad.dispatch
def expect_and_grad(
    vstate: MyMixedStateType,
    operator: MyOperatorSuperType,
    chunk_size: int | None,
    *args,
    mutable = False,
    **kwargs,
):
    ...
    return expect_value, gradient

Usually MyOperatorSuperType should be a suitable supertype spanning the operator family you want to support, not a single exact concrete operator class.

Choosing among multiple candidate estimators

When several estimators are mathematically possible, selecting the “best” one should not be based only on asymptotic correctness. A useful criterion is to compare their finite-sample behavior, especially the variance and signal-to-noise ratio of the gradient estimator, and the empirical stability they induce during optimization.

A concrete worked example is given in Section 3.2 of Gravina, Savona, and Vicentini, “Neural Projected Quantum Dynamics: a systematic study”. There the authors explicitly separate the analysis of fidelity estimators from the analysis of gradient estimators, note that a good fidelity estimator need not induce a good gradient estimator, and compare candidate estimators by finite-sample signal-to-noise and optimization stability (see in particular Section 3.2.2 and Table 3). Their discussion also gives an example of using Rao-Blackwellization to obtain a lower-variance gradient estimator.

References

Luca Gravina, Vincenzo Savona, and Filippo Vicentini, “Neural Projected Quantum Dynamics: a systematic study”, Quantum 9, 1803 (2025), especially Section 3.2.