# LogNormalDistribution¶

class LogNormalDistribution(num_qubits, mu=None, sigma=None, bounds=None, upto_diag=False, name='P(X)')[ソース]

A circuit to encode a discretized log-normal distribution in qubit amplitudes.

A random variable $$X$$ is log-normal distributed if

$\log(X) \sim \mathcal{N}(\mu, \sigma^2)$

for a normal distribution $$\mathcal{N}(\mu, \sigma^2)$$. The probability density function of the log-normal distribution is defined as

$\mathbb{P}(X = x) = \frac{1}{x\sqrt{2\pi\sigma^2}} e^{-\frac{(\log(x) - \mu)^2}{\sigma^2}}$

The parameter sigma in this class equals the variance, $$\sigma^2$$ and not the standard deviation. This is for consistency with multivariate distributions, where the uppercase sigma, $$\Sigma$$, is associated with the covariance.

This circuit considers the discretized version of $$X$$ on 2 ** num_qubits equidistant points, $$x_i$$, truncated to bounds. The action of this circuit can be written as

$\mathcal{P}_X |0\rangle^n = \sum_{i=0}^{2^n - 1} \sqrt{\mathbb{P}(x_i)} |i\rangle$

where $$n$$ is num_qubits.

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 modelled using a log-normal distribution.

サンプル

This class can be used for both univariate and multivariate distributions. >>> 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

[1]: Gacon, J., Zoufal, C., & Woerner, S. (2020).

Quantum-Enhanced Simulation-Based Optimization. arXiv:2005.10780

[2]: Woerner, S., & Egger, D. J. (2018).

Quantum Risk Analysis. arXiv:1806.06893

パラメータ
• num_qubits (Union[int, List[int]]) – 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 (Union[float, List[float], None]) – The parameter $$\mu$$ 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 (Union[float, List[float], None]) – The parameter $$\sigma^2$$ or $$\Sigma$$, which is the variance or covariance matrix.

• bounds (Union[Tuple[float, float], List[Tuple[float, float]], None]) – 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 (bool) – If True, load the square root of the probabilities up to multiplication with a diagonal for a more efficient circuit.

• name (str) – The name of the circuit.

Attributes

ancillas

Returns a list of ancilla bits in the order that the registers were added.

List[AncillaQubit]

bounds

Return the bounds of the probability distribution.

Union[Tuple[float, float], List[Tuple[float, float]]]

calibrations

Return calibration dictionary.

The custom pulse definition of a given gate is of the form

{『gate_name』: {(qubits, params): schedule}}

dict

clbits

Returns a list of classical bits in the order that the registers were added.

List[Clbit]

data

Return the circuit data (instructions and context).

a list-like object containing the tuples for the circuit’s data.

Each tuple is in the format (instruction, qargs, cargs), where instruction is an Instruction (or subclass) object, qargs is a list of Qubit objects, and cargs is a list of Clbit objects.

QuantumCircuitData

extension_lib = 'include "qelib1.inc";'
global_phase

Return the global phase of the circuit in radians.

Union[ParameterExpression, float]

instances = 9

The user provided metadata associated with the circuit

The metadata for the circuit is a user provided dict of metadata for the circuit. It will not be used to influence the execution or operation of the circuit, but it is expected to be passed between all transforms of the circuit (ie transpilation) and that providers will associate any circuit metadata with the results it returns from execution of that circuit.

dict

num_ancillas

Return the number of ancilla qubits.

int

num_clbits

Return number of classical bits.

int

num_parameters

Convenience function to get the number of parameter objects in the circuit.

int

num_qubits

Return number of qubits.

int

parameters

Convenience function to get the parameters defined in the parameter table.

ParameterView

prefix = 'circuit'
probabilities

Return the sampling probabilities for the values.

ndarray

qubits

Returns a list of quantum bits in the order that the registers were added.

List[Qubit]

values

Return the discretized points of the random variable.

ndarray