# This code is part of a Qiskit project.
#
# (C) Copyright IBM 2018, 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.
#
# Any modifications or derivative works of this code must retain this
# copyright notice, and modified files need to carry a notice indicating
# that they have been altered from the originals.
"""Univariate Marginal Distribution Algorithm (Estimation-of-Distribution-Algorithm)."""
from __future__ import annotations
from collections.abc import Callable
from typing import Any
import numpy as np
from scipy.stats import norm
from qiskit_algorithms.utils import algorithm_globals
from .optimizer import OptimizerResult, POINT
from .scipy_optimizer import Optimizer, OptimizerSupportLevel
[docs]class UMDA(Optimizer):
"""Continuous Univariate Marginal Distribution Algorithm (UMDA).
UMDA [1] is a specific type of Estimation of Distribution Algorithm (EDA) where new individuals
are sampled from univariate normal distributions and are updated in each iteration of the
algorithm by the best individuals found in the previous iteration.
.. seealso::
This original implementation of the UDMA optimizer for Qiskit was inspired by my
(Vicente P. Soloviev) work on the EDAspy Python package [2].
EDAs are stochastic search algorithms and belong to the family of the evolutionary algorithms.
The main difference is that EDAs have a probabilistic model which is updated in each iteration
from the best individuals of previous generations (elite selection). Depending on the complexity
of the probabilistic model, EDAs can be classified in different ways. In this case, UMDA is a
univariate EDA as the embedded probabilistic model is univariate.
UMDA has been compared to some of the already implemented algorithms in Qiskit library to
optimize the parameters of variational algorithms such as QAOA or VQE and competitive results
have been obtained [1]. UMDA seems to provide very good solutions for those circuits in which
the number of layers is not big.
The optimization process can be personalized depending on the parameters chosen in the
initialization. The main parameter is the population size. The bigger it is, the final result
will be better. However, this increases the complexity of the algorithm and the runtime will
be much heavier. In the work [1] different experiments have been performed where population
size has been set to 20 - 30.
.. note::
The UMDA implementation has more parameters but these have default values for the
initialization for better understanding of the user. For example, ``\alpha`` parameter has
been set to 0.5 and is the percentage of the population which is selected in each iteration
to update the probabilistic model.
Example:
This short example runs UMDA to optimize the parameters of a variational algorithm. Here we
will use the same operator as used in the algorithms introduction, which was originally
computed by Qiskit Nature for an H2 molecule. The minimum energy of the H2 Hamiltonian can
be found quite easily so we are able to set maxiters to a small value.
.. code-block:: python
from qiskit_algorithms.optimizers import UMDA
from qiskit_algorithms import QAOA
from qiskit.quantum_info import Pauli
from qiskit.primitives import Sampler
X = Pauli("X")
I = Pauli("I")
Z = Pauli("Z")
H2_op = (-1.052373245772859 * I ^ I) + \
(0.39793742484318045 * I ^ Z) + \
(-0.39793742484318045 * Z ^ I) + \
(-0.01128010425623538 * Z ^ Z) + \
(0.18093119978423156 * X ^ X)
p = 2 # Toy example: 2 layers with 2 parameters in each layer: 4 variables
opt = UMDA(maxiter=100, size_gen=20)
qaoa = QAOA(Sampler(), opt,reps=p)
result = qaoa.compute_minimum_eigenvalue(operator=H2_op)
If it is desired to modify the percentage of individuals considered to update the
probabilistic model, then this code can be used. Here for example we set the 60% instead
of the 50% predefined.
.. code-block:: python
opt = UMDA(maxiter=100, size_gen=20, alpha = 0.6)
qaoa = QAOA(Sampler(), opt,reps=p)
result = qaoa.compute_minimum_eigenvalue(operator=H2_op)
.. 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``).
References:
[1]: Vicente P. Soloviev, Pedro Larrañaga and Concha Bielza (2022, July). Quantum Parametric
Circuit Optimization with Estimation of Distribution Algorithms. In 2022 The Genetic and
Evolutionary Computation Conference (GECCO). DOI: https://doi.org/10.1145/3520304.3533963
[2]: Vicente P. Soloviev. Python package EDAspy.
https://github.com/VicentePerezSoloviev/EDAspy.
"""
ELITE_FACTOR = 0.4
STD_BOUND = 0.3
def __init__(
self,
maxiter: int = 100,
size_gen: int = 20,
alpha: float = 0.5,
callback: Callable[[int, np.ndarray, float], None] | None = None,
) -> None:
r"""
Args:
maxiter: Maximum number of iterations.
size_gen: Population size of each generation.
alpha: Percentage (0, 1] of the population to be selected as elite selection.
callback: A callback function passed information in each iteration step. The
information is, in this order: the number of function evaluations, the parameters,
the best function value in this iteration.
"""
self.size_gen = size_gen
self.maxiter = maxiter
self.alpha = alpha
self._vector: np.ndarray | None = None
# initialization of generation
self._generation: np.ndarray | None = None
self._dead_iter = int(self._maxiter / 5)
self._truncation_length = int(size_gen * alpha)
super().__init__()
self._best_cost_global: float | None = None
self._best_ind_global: np.ndarray | None = None
self._evaluations: np.ndarray | None = None
self._n_variables: int | None = None
self.callback = callback
def _initialization(self) -> np.ndarray:
vector = np.zeros((4, self._n_variables))
vector[0, :] = np.pi # mu
vector[1, :] = 0.5 # std
return vector
# build a generation of size SIZE_GEN from prob vector
def _new_generation(self):
"""Build a new generation sampled from the vector of probabilities.
Updates the generation pandas dataframe
"""
gen = algorithm_globals.random.normal(
self._vector[0, :], self._vector[1, :], [self._size_gen, self._n_variables]
)
self._generation = self._generation[: int(self.ELITE_FACTOR * len(self._generation))]
self._generation = np.vstack((self._generation, gen))
# truncate the generation at alpha percent
def _truncation(self):
"""Selection of the best individuals of the actual generation.
Updates the generation by selecting the best individuals.
"""
best_indices = self._evaluations.argsort()[: self._truncation_length]
self._generation = self._generation[best_indices, :]
self._evaluations = np.take(self._evaluations, best_indices)
# check each individual of the generation
def _check_generation(self, objective_function):
"""Check the cost of each individual in the cost function implemented by the user."""
self._evaluations = np.apply_along_axis(objective_function, 1, self._generation)
# update the probability vector
def _update_vector(self):
"""From the best individuals update the vector of normal distributions in order to the next
generation can sample from it. Update the vector of normal distributions
"""
for i in range(self._n_variables):
self._vector[0, i], self._vector[1, i] = norm.fit(self._generation[:, i])
if self._vector[1, i] < self.STD_BOUND:
self._vector[1, i] = self.STD_BOUND
[docs] def minimize(
self,
fun: Callable[[POINT], float],
x0: POINT,
jac: Callable[[POINT], POINT] | None = None,
bounds: list[tuple[float, float]] | None = None,
) -> OptimizerResult:
not_better_count = 0
result = OptimizerResult()
if isinstance(x0, float):
x0 = np.asarray([x0])
self._n_variables = len(x0)
self._best_cost_global = 999999999999
self._best_ind_global = None
history = []
self._evaluations = np.array(0)
self._vector = self._initialization()
# initialization of generation
self._generation = algorithm_globals.random.normal(
self._vector[0, :], self._vector[1, :], [self._size_gen, self._n_variables]
)
for _ in range(self._maxiter):
self._check_generation(fun)
self._truncation()
self._update_vector()
best_mae_local: float = min(self._evaluations)
history.append(best_mae_local)
best_ind_local = np.where(self._evaluations == best_mae_local)[0][0]
best_ind_local = self._generation[best_ind_local]
# update the best values ever
if best_mae_local < self._best_cost_global:
self._best_cost_global = best_mae_local
self._best_ind_global = best_ind_local
not_better_count = 0
else:
not_better_count += 1
if not_better_count >= self._dead_iter:
break
if self.callback is not None:
self.callback(
len(history) * self._size_gen, self._best_ind_global, self._best_cost_global
)
self._new_generation()
result.x = self._best_ind_global
result.fun = self._best_cost_global
result.nfev = len(history) * self._size_gen
return result
@property
def size_gen(self) -> int:
"""Returns the size of the generations (number of individuals per generation)"""
return self._size_gen
@size_gen.setter
def size_gen(self, value: int):
"""
Sets the size of the generations of the algorithm.
Args:
value: Size of the generations (number of individuals per generation).
Raises:
ValueError: If `value` is lower than 1.
"""
if value <= 0:
raise ValueError("The size of the generation should be greater than 0.")
self._size_gen = value
@property
def maxiter(self) -> int:
"""Returns the maximum number of iterations"""
return self._maxiter
@maxiter.setter
def maxiter(self, value: int):
"""
Sets the maximum number of iterations of the algorithm.
Args:
value: Maximum number of iterations of the algorithm.
Raises:
ValueError: If `value` is lower than 1.
"""
if value <= 0:
raise ValueError("The maximum number of iterations should be greater than 0.")
self._maxiter = value
@property
def alpha(self) -> float:
"""Returns the alpha parameter value (percentage of population selected to update
probabilistic model)"""
return self._alpha
@alpha.setter
def alpha(self, value: float):
"""
Sets the alpha parameter (percentage of individuals selected to update the probabilistic
model)
Args:
value: Percentage (0,1] of generation selected to update the probabilistic model.
Raises:
ValueError: If `value` is lower than 0 or greater than 1.
"""
if (value <= 0) or (value > 1):
raise ValueError(f"alpha must be in the range (0, 1], value given was {value}")
self._alpha = value
@property
def settings(self) -> dict[str, Any]:
return {
"maxiter": self.maxiter,
"alpha": self.alpha,
"size_gen": self.size_gen,
"callback": self.callback,
}
[docs] def get_support_level(self):
"""Get the support level dictionary."""
return {
"gradient": OptimizerSupportLevel.ignored,
"bounds": OptimizerSupportLevel.ignored,
"initial_point": OptimizerSupportLevel.required,
}