NormalDistribution#

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

Bases: 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 modeled using a normal distribution.

Examples

>>> from qiskit_finance.circuit.library.probability_distributions import NormalDistribution
>>> circuit = NormalDistribution(3, mu=1, sigma=1, bounds=(0, 2))
>>> circuit.decompose().draw()
                                                        »
q_0: ───────────────────────────────────────────────────»
                              ┌────────────────────────┐»
q_1: ─────────────────────────┤0                       ├»
     ┌───────────────────────┐│  multiplex2_reverse_dg │»
q_2: ┤ multiplex1_reverse_dg ├┤1                       ├»
     └───────────────────────┘└────────────────────────┘»
«     ┌────────────────────────┐
«q_0: ┤0                       ├
«     │                        │
«q_1: ┤1 multiplex3_reverse_dg ├
«     │                        │
«q_2: ┤2                       ├
«     └────────────────────────┘

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

>>> import os
>>> 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)
>>> # strip trailing white spaces to match the expected output
>>> print(os.linesep.join([s.rstrip() for s in circuit.draw().lines()]))
     ┌───────┐
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 (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 (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 (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 (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

bounds#: Return the bounds of the probability distribution.

probabilities#: Return the sampling probabilities for the values.

values#: Return the discretized points of the random variable.

Methods