NormalDistribution
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
The parameter sigma
in this class equals the variance, and not the standard deviation. This is for consistency with multivariate distributions, where the uppercase sigma, , is associated with the covariance.
This circuit considers the discretized version of the normal distribution on 2 ** num_qubits
equidistant points, , truncated to bounds
. For a one-dimensional random variable, meaning num_qubits is a single integer, it applies the operation
where 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.
In the multi-dimensional case, the distribution is defined as
where 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 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 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(opens in a new tab)
[2]: Woerner, S., & Egger, D. J. (2018).
Quantum Risk Analysis. arXiv:1806.06893(opens in a new tab)
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 , 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 or , 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), ...]
. IfNone
, 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.
Return type
List
[AncillaQubit
]
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}}
Return type
dict
clbits
Returns a list of classical bits in the order that the registers were added.
Return type
List
[Clbit
]
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.
Return type
Union
[ParameterExpression
, float
]
header
= 'OPENQASM 2.0;'
instances
= 9
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.
Return type
dict
num_ancillas
Return the number of ancilla qubits.
Return type
int
num_clbits
Return number of classical bits.
Return type
int
num_parameters
Convenience function to get the number of parameter objects in the circuit.
Return type
int
num_qubits
Return number of qubits.
Return type
int
parameters
Convenience function to get the parameters defined in the parameter table.
Return type
ParameterView
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.
Return type
List
[Qubit
]
values
Return the discretized points of the random variable.
Return type
ndarray