NormalDistribution

class NormalDistribution(num_qubits, mu=None, sigma=None, bounds=None, upto_diag=False, name='P(X)')

Bases: qiskit.circuit.quantumcircuit.QuantumCircuit

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

The probability density function of the normal distribution is defined as

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

Note

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 the normal distribution on 2 ** num_qubits equidistant points, \(x_i\), truncated to bounds. For a one-dimensional random variable, meaning num_qubits is a single integer, it applies the operation

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

where \(n\) 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.

In the multi-dimensional case, the distribution is defined as

\[\mathbb{P}(X = x) = \frac{\Sigma^{-1}}{\sqrt{2\pi}} e^{-\frac{(x - \mu)^2}{\Sigma}}\]

where \(\Sigma\) is the covariance. To specify a multivariate normal distribution, num_qubits is a list of integers, each specifying how many qubits are used to discretize the respective dimension. The arguments mu and sigma in this case are a vector and square matrix. If for instance, num_qubits = [2, 3] then mu is a 2d vector and sigma is the \(2 \times 2\) covariance matrix. The first dimension is discretized using 2 qubits, hence on 4 points, and the second dimension on 3 qubits, hence 8 points. Therefore the random variable is discretized on \(4 \times 8 = 32\) points.

Since, in general, it is not yet known how to efficiently prepare the qubit amplitudes to represent a normal distribution, this class computes the expected amplitudes and then uses the QuantumCircuit.initialize method to construct the corresponding circuit.

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 normal distribution.

Examples

>>> circuit = NormalDistribution(3, mu=1, sigma=1, bounds=(0, 2))
>>> circuit.draw()
     ┌────────────────────────────────────────────────────────────────────────────┐
q_0: ┤0                                                                           ├
     │                                                                            │
q_1: ┤1 initialize(0.30391,0.3435,0.37271,0.38824,0.38824,0.37271,0.3435,0.30391) ├
     │                                                                            │
q_2: ┤2                                                                           ├
     └────────────────────────────────────────────────────────────────────────────┘

>>> mu = [1, 0.9]
>>> sigma = [[1, -0.2], [-0.2, 1]]
>>> circuit = NormalDistribution([2, 3], mu, sigma)
>>> circuit.num_qubits
5

>>> from qiskit import QuantumCircuit
>>> mu = [1, 0.9]
>>> sigma = [[1, -0.2], [-0.2, 1]]
>>> bounds = [(0, 1), (-1, 1)]
>>> p_x = NormalDistribution([2, 3], mu, sigma, bounds)
>>> circuit = QuantumCircuit(6)
>>> circuit.append(p_x, list(range(5)))
>>> for i in range(5):
...    circuit.cry(2 ** i, i, 5)
>>> circuit.draw()
     ┌───────┐
q_0: ┤0      ├────■─────────────────────────────────────────
     │       │    │
q_1: ┤1      ├────┼────────■────────────────────────────────
     │       │    │        │
q_2: ┤2 P(X) ├────┼────────┼────────■───────────────────────
     │       │    │        │        │
q_3: ┤3      ├────┼────────┼────────┼────────■──────────────
     │       │    │        │        │        │
q_4: ┤4      ├────┼────────┼────────┼────────┼────────■─────
     └───────┘┌───┴───┐┌───┴───┐┌───┴───┐┌───┴───┐┌───┴────┐
q_5: ─────────┤ RY(1) ├┤ RY(2) ├┤ RY(4) ├┤ RY(8) ├┤ RY(16) ├
              └───────┘└───────┘└───────┘└───────┘└────────┘

References

[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

Parameters

• 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 \(\mu\), which is the expected value of the distribution. Can be either a float for a 1d random variable or a list of floats for a higher dimensional random variable. Defaults to 0.

• sigma (Union[float, List[float], None]) – The parameter \(\sigma^2\) or \(\Sigma\), which is the variance or covariance matrix. Default to the identity matrix of appropriate size.

• 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 (-1, 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.

bounds

Return the bounds of the probability distribution.

Return type

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}}

clbits

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

data

Return the circuit data (instructions and context).

Returns

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.

Return type

QuantumCircuitData

extension_lib = 'include "qelib1.inc";'

global_phase

Return the global phase of the circuit in radians.

header = 'OPENQASM 2.0;'

instances = 16

metadata

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.

num_ancillas

Return the number of ancilla qubits.

num_clbits

Return number of classical bits.

num_parameters

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

num_qubits

Return number of qubits.

parameters

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

prefix = 'circuit'

probabilities

Return the sampling probabilities for the values.

Return type

ndarray

qubits

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

values

Return the discretized points of the random variable.

Return type

ndarray