qiskit.algorithms.optimizers.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]

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.

__init__(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]
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

 __init__(fidelity[, maxiter, blocking, …]) type fidelity Callable[[ndarray, ndarray], float] calibrate(loss, initial_point[, c, …]) Calibrate SPSA parameters with a powerseries as learning rate and perturbation coeffs. estimate_stddev(loss, initial_point[, avg]) Estimate the standard deviation of the loss function. get_fidelity(circuit[, backend, expectation]) Get a function to compute the fidelity of circuit with itself. Get the support level dictionary. gradient_num_diff(x_center, f, epsilon[, …]) We compute the gradient with the numeric differentiation in the parallel way, around the point x_center. optimize(num_vars, objective_function[, …]) Perform optimization. Print algorithm-specific options. Set max evals grouped set_options(**kwargs) Sets or updates values in the options dictionary. wrap_function(function, args) 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.
property bounds_support_level

Returns bounds support level

static calibrate(loss, initial_point, c=0.2, stability_constant=0, target_magnitude=None, alpha=0.602, gamma=0.101, modelspace=False)

Calibrate SPSA parameters with a powerseries as learning rate and perturbation coeffs.

The powerseries are:

$a_k = \frac{a}{(A + k + 1)^\alpha}, c_k = \frac{c}{(k + 1)^\gamma}$
Parameters
• loss (Callable[[ndarray], float]) – The loss function.

• initial_point (ndarray) – The initial guess of the iteration.

• c (float) – The initial perturbation magnitude.

• stability_constant (float) – The value of A.

• target_magnitude (Optional[float]) – The target magnitude for the first update step, defaults to $$2\pi / 10$$.

• alpha (float) – The exponent of the learning rate powerseries.

• gamma (float) – The exponent of the perturbation powerseries.

• modelspace (bool) – Whether the target magnitude is the difference of parameter values or function values (= model space).

Returns

A tuple of powerseries generators, the first one for the

learning rate and the second one for the perturbation.

Return type

tuple(generator, generator)

static estimate_stddev(loss, initial_point, avg=25)

Estimate the standard deviation of the loss function.

Return type

float

static get_fidelity(circuit, backend=None, expectation=None)[source]

Get a function to compute the fidelity of circuit with itself.

Let circuit be a parameterized quantum circuit performing the operation $$U(\theta)$$ given a set of parameters $$\theta$$. Then this method returns a function to evaluate

$F(\theta, \phi) = \big|\langle 0 | U^\dagger(\theta) U(\phi) |0\rangle \big|^2.$

The output of this function can be used as input for the fidelity to the :class:~qiskit.algorithms.optimizers.QNSPSA optimizer.

Parameters
• circuit (QuantumCircuit) – The circuit preparing the parameterized ansatz.

• backend (Union[Backend, QuantumInstance, None]) – A backend of quantum instance to evaluate the circuits. If None, plain matrix multiplication will be used.

• expectation (Optional[ExpectationBase]) – An expectation converter to specify how the expected value is computed. If a shot-based readout is used this should be set to PauliExpectation.

Return type

Callable[[ndarray, ndarray], float]

Returns

A handle to the function $$F$$.

get_support_level()

Get the support level dictionary.

static gradient_num_diff(x_center, f, epsilon, max_evals_grouped=1)

We compute the gradient with the numeric differentiation in the parallel way, around the point x_center.

Parameters
• x_center (ndarray) – point around which we compute the gradient

• f (func) – the function of which the gradient is to be computed.

• epsilon (float) – the epsilon used in the numeric differentiation.

• max_evals_grouped (int) – max evals grouped

Returns

Return type

property gradient_support_level

property initial_point_support_level

Returns initial point support level

property is_bounds_ignored

Returns is bounds ignored

property is_bounds_required

Returns is bounds required

property is_bounds_supported

Returns is bounds supported

property is_gradient_ignored

property is_gradient_required

property is_gradient_supported

property is_initial_point_ignored

Returns is initial point ignored

property is_initial_point_required

Returns is initial point required

property is_initial_point_supported

Returns is initial point supported

optimize(num_vars, objective_function, gradient_function=None, variable_bounds=None, initial_point=None)

Perform optimization.

Parameters
• num_vars (int) – Number of parameters to be optimized.

• objective_function (callable) – A function that computes the objective function.

• gradient_function (callable) – A function that computes the gradient of the objective function, or None if not available.

• variable_bounds (list[(float, float)]) – List of variable bounds, given as pairs (lower, upper). None means unbounded.

• initial_point (numpy.ndarray[float]) – Initial point.

Returns

point, value, nfev

point: is a 1D numpy.ndarray[float] containing the solution value: is a float with the objective function value nfev: number of objective function calls made if available or None

Raises

ValueError – invalid input

print_options()

Print algorithm-specific options.

set_max_evals_grouped(limit)

Set max evals grouped

set_options(**kwargs)

Sets or updates values in the options dictionary.

The options dictionary may be used internally by a given optimizer to pass additional optional values for the underlying optimizer/optimization function used. The options dictionary may be initially populated with a set of key/values when the given optimizer is constructed.

Parameters

kwargs (dict) – options, given as name=value.

property setting

Return setting

property settings

The optimizer settings in a dictionary format.

Note

The fidelity property cannot be serialized and will not be contained in the dictionary. To construct a QNSPSA object from a dictionary you have to add it manually with the key "fidelity".

Return type

Dict[str, Any]

static wrap_function(function, args)

Wrap the function to implicitly inject the args at the call of the function.

Parameters
• function (func) – the target function

• args (tuple) – the args to be injected

Returns

wrapper

Return type

function_wrapper