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 (QNSPSA) optimizer.
The QNSPSA 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, firstorder 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, QNSPSA only requires \(\mathcal{O}(1)\) and can therefore significantly speed up the natural gradient calculation by sacrificing some accuracy. Compared to SPSA, QNSPSA 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 Carlostyle convergence to the exact, analytic value.Examples
This short example runs QNSPSA for the ground state calculation of the
Z ^ Z
observable where the ansatz is aPauliTwoDesign
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
]) – Ifblocking
isTrue
, this argument determines by how much the loss can increase with the proposed parameters and still be accepted. IfNone
, 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. Iflearning_rate
is setperturbation
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. Ifperturbation
is setlearning_rate
must also be provided.last_avg (
int
) – Return the average of thelast_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
]) – Ifblocking
isTrue
, this argument determines by how much the loss can increase with the proposed parameters and still be accepted. IfNone
, 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. Iflearning_rate
is setperturbation
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. Ifperturbation
is setlearning_rate
must also be provided.last_avg (
int
) – Return the average of thelast_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 algorithmspecific options.
set_max_evals_grouped
(limit)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
Returns bounds support level
Returns gradient support level
Returns initial point support level
Returns is bounds ignored
Returns is bounds required
Returns is bounds supported
Returns is gradient ignored
Returns is gradient required
Returns is gradient supported
Returns is initial point ignored
Returns is initial point required
Returns is initial point supported
Return setting
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 shotbased readout is used this should be set toPauliExpectation
.
 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
the gradient computed
 Return type
grad

property
gradient_support_level
¶ Returns 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
¶ Returns is gradient ignored

property
is_gradient_required
¶ Returns is gradient required

property
is_gradient_supported
¶ Returns 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 algorithmspecific 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 aQNSPSA
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