UMDA
class UMDA(maxiter=100, size_gen=20, alpha=0.5)
Bases: qiskit.algorithms.optimizers.optimizer.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.
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 paremeters 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.
The UMDA implementation has more parameters but these have default values for the initialization for better understanding of the user. For example, lpha 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.
from qiskit.opflow import X, Z, I
from qiskit import Aer
from qiskit.algorithms.optimizers import UMDA
from qiskit.algorithms import QAOA
from qiskit.utils import QuantumInstance
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)
backend = Aer.get_backend('statevector_simulator')
vqe = QAOA(opt,
quantum_instance=QuantumInstance(backend=backend),
reps=p)
result = vqe.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.
opt = UMDA(maxiter=100, size_gen=20, alpha = 0.6)
backend = Aer.get_backend('statevector_simulator')
vqe = QAOA(opt,
quantum_instance=QuantumInstance(backend=backend),
reps=p)
result = vqe.compute_minimum_eigenvalue(operator=qubit_op)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.
Parameters
- maxiter (
int) – Maximum number of iterations. - size_gen (
int) – Population size of each generation. - alpha (
float) – Percentage (0, 1] of the population to be selected as elite selection.
Methods
get_support_level
UMDA.get_support_level()
Get the support level dictionary.
gradient_num_diff
static UMDA.gradient_num_diff(x_center, f, epsilon, max_evals_grouped=None)
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, defaults to 1 (i.e. no batching).
Returns
the gradient computed
Return type
grad
minimize
UMDA.minimize(fun, x0, jac=None, bounds=None)
Minimize the scalar function.
Parameters
- fun (
Callable[[Union[float,ndarray]],float]) – The scalar function to minimize. - x0 (
Union[float,ndarray]) – The initial point for the minimization. - jac (
Optional[Callable[[Union[float,ndarray]],Union[float,ndarray]]]) – The gradient of the scalar functionfun. - bounds (
Optional[List[Tuple[float,float]]]) – Bounds for the variables offun. This argument might be ignored if the optimizer does not support bounds.
Return type
Returns
The result of the optimization, containing e.g. the result as attribute x.
print_options
UMDA.print_options()
Print algorithm-specific options.
set_max_evals_grouped
UMDA.set_max_evals_grouped(limit)
Set max evals grouped
set_options
UMDA.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.
wrap_function
static UMDA.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
Attributes
ELITE_FACTOR
Default value: 0.4
STD_BOUND
Default value: 0.3
alpha
Returns the alpha parameter value (percentage of population selected to update probabilistic model)
Return type
float
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
maxiter
Returns the maximum number of iterations
Return type
int
setting
Return setting
settings
Return type
Dict[str, Any]
size_gen
Returns the size of the generations (number of individuals per generation)
Return type
int