QNSPSA¶

class QNSPSA(fidelity, maxiter=100, blocking=True, allowed_increase=None, learning_rate=None, perturbation=None, last_avg=1, resamplings=1, perturbation_dims=None, regularization=None, hessian_delay=0, lse_solver=None, initial_hessian=None, callback=None)[source]¶

Bases: qiskit.algorithms.optimizers.spsa.SPSA

The Quantum Natural SPSA (QN-SPSA) optimizer.

The QN-SPSA optimizer [1] is a stochastic optimizer that belongs to the family of gradient descent methods. This optimizer is based on SPSA but attempts to improve the convergence by sampling the natural gradient instead of the vanilla, first-order gradient. It achieves this by approximating Hessian of the fidelity of the ansatz circuit.

Compared to natural gradients, which require \(\mathcal{O}(d^2)\) expectation value evaluations for a circuit with \(d\) parameters, QN-SPSA only requires \(\mathcal{O}(1)\) and can therefore significantly speed up the natural gradient calculation by sacrificing some accuracy. Compared to SPSA, QN-SPSA requires 4 additional function evaluations of the fidelity.

The stochastic approximation of the natural gradient can be systematically improved by increasing the number of resamplings. This leads to a Monte Carlo-style convergence to the exact, analytic value.

Examples

This short example runs QN-SPSA for the ground state calculation of the Z ^ Z observable where the ansatz is a PauliTwoDesign circuit.

import numpy as np
from qiskit.algorithms.optimizers import QNSPSA
from qiskit.circuit.library import PauliTwoDesign
from qiskit.opflow import Z, StateFn

ansatz = PauliTwoDesign(2, reps=1, seed=2)
observable = Z ^ Z
initial_point = np.random.random(ansatz.num_parameters)

def loss(x):
    bound = ansatz.bind_parameters(x)
    return np.real((StateFn(observable, is_measurement=True) @ StateFn(bound)).eval())

fidelity = QNSPSA.get_fidelity(ansatz)
qnspsa = QNSPSA(fidelity, maxiter=300)
result = qnspsa.optimize(ansatz.num_parameters, loss, initial_point=initial_point)

References

[1] J. Gacon et al, “Simultaneous Perturbation Stochastic Approximation of the Quantum Fisher Information”, arXiv:2103.09232

Parameters

fidelity (Callable[[ndarray, ndarray], float]) – A function to compute the fidelity of the ansatz state with itself for two different sets of parameters.
maxiter (int) – The maximum number of iterations. Note that this is not the maximal number of function evaluations.
blocking (bool) – If True, only accepts updates that improve the loss (up to some allowed increase, see next argument).
allowed_increase (Optional[float]) – If blocking is True, this argument determines by how much the loss can increase with the proposed parameters and still be accepted. If None, the allowed increases is calibrated automatically to be twice the approximated standard deviation of the loss function.
learning_rate (Union[float, Callable[[], Iterator], None]) – The update step is the learning rate is multiplied with the gradient. If the learning rate is a float, it remains constant over the course of the optimization. It can also be a callable returning an iterator which yields the learning rates for each optimization step. If learning_rate is set perturbation must also be provided.
perturbation (Union[float, Callable[[], Iterator], None]) – Specifies the magnitude of the perturbation for the finite difference approximation of the gradients. Can be either a float or a generator yielding the perturbation magnitudes per step. If perturbation is set learning_rate must also be provided.
last_avg (int) – Return the average of the last_avg parameters instead of just the last parameter values.
resamplings (Union[int, Dict[int, int]]) – The number of times the gradient (and Hessian) is sampled using a random direction to construct a gradient estimate. Per default the gradient is estimated using only one random direction. If an integer, all iterations use the same number of resamplings. If a dictionary, this is interpreted as {iteration: number of resamplings per iteration}.
perturbation_dims (Optional[int]) – The number of perturbed dimensions. Per default, all dimensions are perturbed, but a smaller, fixed number can be perturbed. If set, the perturbed dimensions are chosen uniformly at random.
regularization (Optional[float]) – To ensure the preconditioner is symmetric and positive definite, the identity times a small coefficient is added to it. This generator yields that coefficient.
hessian_delay (int) – Start multiplying the gradient with the inverse Hessian only after a certain number of iterations. The Hessian is still evaluated and therefore this argument can be useful to first get a stable average over the last iterations before using it as preconditioner.
lse_solver (Optional[Callable[[ndarray, ndarray], ndarray]]) – The method to solve for the inverse of the Hessian. Per default an exact LSE solver is used, but can e.g. be overwritten by a minimization routine.
initial_hessian (Optional[ndarray]) – The initial guess for the Hessian. By default the identity matrix is used.
callback (Optional[Callable[[ndarray, float, float, int, bool], None]]) – A callback function passed information in each iteration step. The information is, in this order: the parameters, the function value, the number of function evaluations, the stepsize, whether the step was accepted.

Methods

`calibrate`	Calibrate SPSA parameters with a powerseries as learning rate and perturbation coeffs.
`estimate_stddev`	Estimate the standard deviation of the loss function.
`get_fidelity`	Get a function to compute the fidelity of `circuit` with itself.
`get_support_level`	Get the support level dictionary.
`gradient_num_diff`	We compute the gradient with the numeric differentiation in the parallel way, around the point x_center.
`optimize`	Perform optimization.
`print_options`	Print algorithm-specific options.
`set_max_evals_grouped`	Set max evals grouped
`set_options`	Sets or updates values in the options dictionary.
`wrap_function`	Wrap the function to implicitly inject the args at the call of the function.

Attributes

bounds_support_level¶: Returns bounds support level

gradient_support_level¶: Returns gradient support level

initial_point_support_level¶: Returns initial point support level

is_bounds_ignored¶: Returns is bounds ignored

is_bounds_required¶: Returns is bounds required

is_bounds_supported¶: Returns is bounds supported

is_gradient_ignored¶: Returns is gradient ignored

is_gradient_required¶: Returns is gradient required

is_gradient_supported¶: Returns is gradient supported

is_initial_point_ignored¶: Returns is initial point ignored

is_initial_point_required¶: Returns is initial point required

is_initial_point_supported¶: Returns is initial point supported

setting¶: Return setting

settings¶

The optimizer settings in a dictionary format.

Return type: Dict[str, Any]