LinearAmplitudeFunction

class LinearAmplitudeFunction(num_state_qubits, slope, offset, domain, image, rescaling_factor=1, breakpoints=None, name='F')

Bases: qiskit.circuit.quantumcircuit.QuantumCircuit

A circuit implementing a (piecewise) linear function on qubit amplitudes.

An amplitude function \(F\) of a function \(f\) is a mapping

\[F|x\rangle|0\rangle = \sqrt{1 - \hat{f}(x)} |x\rangle|0\rangle + \sqrt{\hat{f}(x)} |x\rangle|1\rangle.\]

for a function \(\hat{f}: \{ 0, ..., 2^n - 1 \} \rightarrow [0, 1]\), where \(|x\rangle\) is a \(n\) qubit state.

This circuit implements \(F\) for piecewise linear functions \(\hat{f}\). In this case, the mapping \(F\) can be approximately implemented using a Taylor expansion and linearly controlled Pauli-Y rotations, see [1, 2] for more detail. This approximation uses a rescaling_factor to determine the accuracy of the Taylor expansion.

In general, the function of interest \(f\) is defined from some interval \([a,b]\), the domain to \([c,d]\), the image, instead of \(\{ 1, ..., N \}\) to \([0, 1]\). Using an affine transformation we can rescale \(f\) to \(\hat{f}\):

\[\hat{f}(x) = \frac{f(\phi(x)) - c}{d - c}\]

with

\[\phi(x) = a + \frac{b - a}{2^n - 1} x.\]

If \(f\) is a piecewise linear function on \(m\) intervals \([p_{i-1}, p_i], i \in \{1, ..., m\}\) with slopes \(\alpha_i\) and offsets \(\beta_i\) it can be written as

\[f(x) = \sum_{i=1}^m 1_{[p_{i-1}, p_i]}(x) (\alpha_i x + \beta_i)\]

where \(1_{[a, b]}\) is an indication function that is 1 if the argument is in the interval \([a, b]\) and otherwise 0. The breakpoints \(p_i\) can be specified by the breakpoints argument.

References

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

Quantum Risk Analysis. arXiv:1806.06893

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

Quantum-Enhanced Simulation-Based Optimization. arXiv:2005.10780

Parameters

• num_state_qubits (int) – The number of qubits used to encode the variable \(x\).

• slope (Union[float, List[float]]) – The slope of the linear function. Can be a list of slopes if it is a piecewise linear function.

• offset (Union[float, List[float]]) – The offset of the linear function. Can be a list of offsets if it is a piecewise linear function.

• domain (Tuple[float, float]) – The domain of the function as tuple \((x_\min{}, x_\max{})\).

• image (Tuple[float, float]) – The image of the function as tuple \((f_\min{}, f_\max{})\).

• rescaling_factor (float) – The rescaling factor to adjust the accuracy in the Taylor approximation.

• breakpoints (Optional[List[float]]) – The breakpoints if the function is piecewise linear. If None, the function is not piecewise.

• name (str) – Name of the circuit.

Methods Defined Here

post_processing

Map the function value of the approximated \(\hat{f}\) to \(f\).

Attributes

ancillas

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

Return type

List[AncillaQubit]

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'

qubits

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

Return type

List[Qubit]