# This code is part of a Qiskit project.
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# (C) Copyright IBM 2021, 2023.
#
# This code is licensed under the Apache License, Version 2.0. You may
# obtain a copy of this license in the LICENSE.txt file in the root directory
# of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.
#
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# copyright notice, and modified files need to carry a notice indicating
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"""The QN-SPSA optimizer."""
from __future__ import annotations
from collections.abc import Iterator
from typing import Any, Callable
import numpy as np
from qiskit.circuit import QuantumCircuit
from qiskit.primitives import BaseSampler
from qiskit_algorithms.state_fidelities import ComputeUncompute
from .spsa import SPSA, CALLBACK, TERMINATIONCHECKER, _batch_evaluate
# the function to compute the fidelity
FIDELITY = Callable[[np.ndarray, np.ndarray], float]
[docs]class QNSPSA(SPSA):
r"""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 :math:`\mathcal{O}(d^2)` expectation value
evaluations for a circuit with :math:`d` parameters, QN-SPSA only requires
:math:`\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.
.. note::
This component has some function that is normally random. If you want to reproduce behavior
then you should set the random number generator seed in the algorithm_globals
(``qiskit_algorithms.utils.algorithm_globals.random_seed = seed``).
Examples:
This short example runs QN-SPSA for the ground state calculation of the ``Z ^ Z``
observable where the ansatz is a ``PauliTwoDesign`` circuit.
.. code-block:: python
import numpy as np
from qiskit_algorithms.optimizers import QNSPSA
from qiskit.circuit.library import PauliTwoDesign
from qiskit.primitives import Estimator, Sampler
from qiskit.quantum_info import Pauli
# problem setup
ansatz = PauliTwoDesign(2, reps=1, seed=2)
observable = Pauli("ZZ")
initial_point = np.random.random(ansatz.num_parameters)
# loss function
estimator = Estimator()
def loss(x):
result = estimator.run([ansatz], [observable], [x]).result()
return np.real(result.values[0])
# fidelity for estimation of the geometric tensor
sampler = Sampler()
fidelity = QNSPSA.get_fidelity(ansatz, sampler)
# run QN-SPSA
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 <https://arxiv.org/abs/2103.09232>`_
"""
def __init__(
self,
fidelity: FIDELITY,
maxiter: int = 100,
blocking: bool = True,
allowed_increase: float | None = None,
learning_rate: float | Callable[[], Iterator] | None = None,
perturbation: float | Callable[[], Iterator] | None = None,
resamplings: int | dict[int, int] = 1,
perturbation_dims: int | None = None,
regularization: float | None = None,
hessian_delay: int = 0,
lse_solver: Callable[[np.ndarray, np.ndarray], np.ndarray] | None = None,
initial_hessian: np.ndarray | None = None,
callback: CALLBACK | None = None,
termination_checker: TERMINATIONCHECKER | None = None,
) -> None:
r"""
Args:
fidelity: A function to compute the fidelity of the ansatz state with itself for
two different sets of parameters.
maxiter: The maximum number of iterations. Note that this is not the maximal number
of function evaluations.
blocking: If True, only accepts updates that improve the loss (up to some allowed
increase, see next argument).
allowed_increase: 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: 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: 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.
resamplings: 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: 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: 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: 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: 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: The initial guess for the Hessian. By default the identity matrix
is used.
callback: 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.
termination_checker: A callback function executed at the end of each iteration step. The
arguments are, in this order: the parameters, the function value, the number
of function evaluations, the stepsize, whether the step was accepted. If the callback
returns True, the optimization is terminated.
To prevent additional evaluations of the objective method, if the objective has not yet
been evaluated, the objective is estimated by taking the mean of the objective
evaluations used in the estimate of the gradient.
"""
super().__init__(
maxiter,
blocking,
allowed_increase,
# trust region *must* be false for natural gradients to work
trust_region=False,
learning_rate=learning_rate,
perturbation=perturbation,
resamplings=resamplings,
callback=callback,
second_order=True,
hessian_delay=hessian_delay,
lse_solver=lse_solver,
regularization=regularization,
perturbation_dims=perturbation_dims,
initial_hessian=initial_hessian,
termination_checker=termination_checker,
)
self.fidelity = fidelity
def _point_sample(self, loss, x, eps, delta1, delta2):
loss_points = [x + eps * delta1, x - eps * delta1]
fidelity_points = [
(x, x + eps * delta1),
(x, x - eps * delta1),
(x, x + eps * (delta1 + delta2)),
(x, x + eps * (-delta1 + delta2)),
]
self._nfev += 6
loss_values = _batch_evaluate(loss, loss_points, self._max_evals_grouped)
fidelity_values = _batch_evaluate(
self.fidelity, fidelity_points, self._max_evals_grouped, unpack_points=True
)
# compute the gradient approximation and additionally return the loss function evaluations
gradient_estimate = (loss_values[0] - loss_values[1]) / (2 * eps) * delta1
# compute the preconditioner point estimate
fidelity_values = np.asarray(fidelity_values, dtype=float)
diff = fidelity_values[2] - fidelity_values[0]
diff = diff - (fidelity_values[3] - fidelity_values[1])
diff = diff / (2 * eps**2)
rank_one = np.outer(delta1, delta2)
# -0.5 factor comes from the fact that we need -0.5 * fidelity
hessian_estimate = -0.5 * diff * (rank_one + rank_one.T) / 2
return np.mean(loss_values), gradient_estimate, hessian_estimate
@property
def settings(self) -> dict[str, Any]:
"""The optimizer settings in a dictionary format."""
# re-use serialization from SPSA
settings = super().settings
settings.update({"fidelity": self.fidelity})
# remove SPSA-specific arguments not in QNSPSA
settings.pop("trust_region")
settings.pop("second_order")
return settings
[docs] @staticmethod
def get_fidelity(
circuit: QuantumCircuit,
*,
sampler: BaseSampler | None = None,
) -> Callable[[np.ndarray, np.ndarray], float]:
r"""Get a function to compute the fidelity of ``circuit`` with itself.
Let ``circuit`` be a parameterized quantum circuit performing the operation
:math:`U(\theta)` given a set of parameters :math:`\theta`. Then this method returns
a function to evaluate
.. math::
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:`~.QNSPSA` optimizer.
Args:
circuit: The circuit preparing the parameterized ansatz.
sampler: A sampler primitive to sample from a quantum state.
Returns:
A handle to the function :math:`F`.
"""
fid = ComputeUncompute(sampler)
num_parameters = circuit.num_parameters
def fidelity(values_x, values_y):
values_x = np.reshape(values_x, (-1, num_parameters)).tolist()
batch_size_x = len(values_x)
values_y = np.reshape(values_y, (-1, num_parameters)).tolist()
batch_size_y = len(values_y)
result = fid.run(
batch_size_x * [circuit], batch_size_y * [circuit], values_x, values_y
).result()
return np.asarray(result.fidelities)
return fidelity