# This code is part of a Qiskit project.
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# (C) Copyright IBM 2017, 2024.
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"""The log-normal probability distribution circuit."""
from typing import Tuple, List, Union, Optional
import numpy as np
from qiskit.circuit import QuantumCircuit
from qiskit.circuit.library import Initialize, Isometry
from .normal import _check_bounds_valid, _check_dimensions_match
[docs]class LogNormalDistribution(QuantumCircuit):
r"""A circuit to encode a discretized log-normal distribution in qubit amplitudes.
A random variable :math:`X` is log-normal distributed if
.. math::
\log(X) \sim \mathcal{N}(\mu, \sigma^2)
for a normal distribution :math:`\mathcal{N}(\mu, \sigma^2)`.
The probability density function of the log-normal distribution is defined as
.. math::
\mathbb{P}(X = x) = \frac{1}{x\sqrt{2\pi\sigma^2}} e^{-\frac{(\log(x) - \mu)^2}{\sigma^2}}
.. note::
The parameter ``sigma`` in this class equals the **variance**, :math:`\sigma^2` and not the
standard deviation. This is for consistency with multivariate distributions, where the
uppercase sigma, :math:`\Sigma`, is associated with the covariance.
This circuit considers the discretized version of :math:`X` on ``2 ** num_qubits`` equidistant
points, :math:`x_i`, truncated to ``bounds``. The action of this circuit can be written as
.. math::
\mathcal{P}_X |0\rangle^n = \sum_{i=0}^{2^n - 1} \sqrt{\mathbb{P}(x_i)} |i\rangle
where :math:`n` is `num_qubits`.
.. note::
The circuit loads the **square root** of the probabilities into the qubit amplitudes such
that the sampling probability, which is the square of the amplitude, equals the
probability of the distribution.
This circuit is for example used in amplitude estimation applications, such as finance [1, 2],
where customer demand or the return of a portfolio could be modeled using a log-normal
distribution.
Examples:
This class can be used for both univariate and multivariate distributions.
>>> from qiskit_finance.circuit.library.probability_distributions import LogNormalDistribution
>>> mu = [1, 0.9, 0.2]
>>> sigma = [[1, -0.2, 0.2], [-0.2, 1, 0.4], [0.2, 0.4, 1]]
>>> circuit = LogNormalDistribution([2, 2, 2], mu, sigma)
>>> circuit.num_qubits
6
References:
[1]: Gacon, J., Zoufal, C., & Woerner, S. (2020).
Quantum-Enhanced Simulation-Based Optimization.
`arXiv:2005.10780 <http://arxiv.org/abs/2005.10780>`_
[2]: Woerner, S., & Egger, D. J. (2018).
Quantum Risk Analysis.
`arXiv:1806.06893 <http://arxiv.org/abs/1806.06893>`_
"""
def __init__(
self,
num_qubits: Union[int, List[int]],
mu: Optional[Union[float, List[float]]] = None,
sigma: Optional[Union[float, List[float]]] = None,
bounds: Optional[Union[Tuple[float, float], List[Tuple[float, float]]]] = None,
upto_diag: bool = False,
name: str = "P(X)",
) -> None:
r"""
Args:
num_qubits: The number of qubits used to discretize the random variable. For a 1d
random variable, ``num_qubits`` is an integer, for multiple dimensions a list
of integers indicating the number of qubits to use in each dimension.
mu: The parameter :math:`\mu` of the distribution.
Can be either a float for a 1d random variable or a list of floats for a higher
dimensional random variable.
sigma: The parameter :math:`\sigma^2` or :math:`\Sigma`, which is the variance or
covariance matrix.
bounds: The truncation bounds of the distribution as tuples. For multiple dimensions,
``bounds`` is a list of tuples ``[(low0, high0), (low1, high1), ...]``.
If ``None``, the bounds are set to ``(0, 1)`` for each dimension.
upto_diag: If True, load the square root of the probabilities up to multiplication
with a diagonal for a more efficient circuit.
name: The name of the circuit.
"""
_check_dimensions_match(num_qubits, mu, sigma, bounds)
_check_bounds_valid(bounds)
# set default arguments
dim = 1 if isinstance(num_qubits, int) else len(num_qubits)
if mu is None:
mu = 0 if dim == 1 else [0] * dim
if sigma is None:
sigma = 1 if dim == 1 else np.eye(dim) # type: ignore[assignment]
if bounds is None:
bounds = (0, 1) if dim == 1 else [(0, 1)] * dim
if isinstance(num_qubits, int): # univariate case
inner = QuantumCircuit(num_qubits, name=name)
x = np.linspace(bounds[0], bounds[1], num=2**num_qubits)
else: # multivariate case
inner = QuantumCircuit(sum(num_qubits), name=name)
# compute the evaluation points using meshgrid of numpy
# indexing 'ij' yields the "column-based" indexing
meshgrid = np.meshgrid(
*[
np.linspace(bound[0], bound[1], num=2 ** num_qubits[i]) # type: ignore
for i, bound in enumerate(bounds)
],
indexing="ij",
)
# flatten into a list of points
x = list(zip(*[grid.flatten() for grid in meshgrid])) # type: ignore
# compute the normalized, truncated probabilities
probabilities = []
from scipy.stats import multivariate_normal
for x_i in x:
# map probabilities from normal to log-normal reference:
# https://stats.stackexchange.com/questions/214997/multivariate-log-normal-probabiltiy-density-function-pdf
if np.min(x_i) > 0:
det = 1 / np.prod(x_i)
probability = multivariate_normal.pdf(np.log(x_i), mu, sigma) * det
else:
probability = 0
probabilities += [probability]
normalized_probabilities = probabilities / np.sum(probabilities)
# store as properties
self._values = x
self._probabilities = normalized_probabilities
self._bounds = bounds
super().__init__(*inner.qregs, name=name)
# use default the isometry (or initialize w/o resets) algorithm to construct the circuit
if upto_diag:
inner.append(Isometry(np.sqrt(normalized_probabilities), 0, 0), inner.qubits)
self.append(inner.to_instruction(), inner.qubits) # Isometry is not a Gate
else:
initialize = Initialize(np.sqrt(normalized_probabilities))
circuit = initialize.gates_to_uncompute().inverse()
inner.compose(circuit, inplace=True)
self.append(inner.to_gate(), inner.qubits)
@property
def values(self) -> np.ndarray:
"""Return the discretized points of the random variable."""
return self._values
@property
def probabilities(self) -> np.ndarray:
"""Return the sampling probabilities for the values."""
return self._probabilities
@property
def bounds(self) -> Union[Tuple[float, float], List[Tuple[float, float]]]:
"""Return the bounds of the probability distribution."""
return self._bounds