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LinearAmplitudeFunction

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

GitHub

Bases: qiskit.circuit.quantumcircuit.QuantumCircuit

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

An amplitude function FF of a function ff is a mapping

F∣x⟩∣0⟩=1−f^(x)∣x⟩∣0⟩+f^(x)∣x⟩∣1⟩.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 f^:{0,...,2n−1}→[0,1]\hat{f}: \{ 0, ..., 2^n - 1 \} \rightarrow [0, 1], where ∣x⟩|x\rangle is a nn qubit state.

This circuit implements FF for piecewise linear functions f^\hat{f}. In this case, the mapping FF 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 ff is defined from some interval [a,b][a,b], the domain to [c,d][c,d], the image, instead of {1,...,N}\{ 1, ..., N \} to [0,1][0, 1]. Using an affine transformation we can rescale ff to f^\hat{f}:

f^(x)=f(ϕ(x))−cd−c\hat{f}(x) = \frac{f(\phi(x)) - c}{d - c}

with

ϕ(x)=a+b−a2n−1x.\phi(x) = a + \frac{b - a}{2^n - 1} x.

If ff is a piecewise linear function on mm intervals [pi−1,pi],i∈{1,...,m}[p_{i-1}, p_i], i \in \{1, ..., m\} with slopes αi\alpha_i and offsets βi\beta_i it can be written as

f(x)=∑i=1m1[pi−1,pi](x)(αix+βi)f(x) = \sum_{i=1}^m 1_{[p_{i-1}, p_i]}(x) (\alpha_i x + \beta_i)

where 1[a,b]1_{[a, b]} is an indication function that is 1 if the argument is in the interval [a,b][a, b] and otherwise 0. The breakpoints pip_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 xx.
  • 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 (xmin⁡,xmax⁡)(x_{\min}, x_{\max}).
  • image (Tuple[float, float]) – The image of the function as tuple (fmin⁡,fmax⁡)(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

LinearAmplitudeFunction.post_processing(scaled_value)

Map the function value of the approximated f^\hat{f} to ff.

Parameters

scaled_value (float) – A function value from the Taylor expansion of f^(x)\hat{f}(x).

Return type

float

Returns

The scaled_value mapped back to the domain of ff, by first inverting the transformation used for the Taylor approximation and then mapping back from [0,1][0, 1] to the original domain.

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 CircuitInstructions for each instruction.

Return type

QuantumCircuitData

extension_lib

Default value: 'include "qelib1.inc";'

global_phase

Return the global phase of the circuit in radians.

Return type

Union[ParameterExpression, float]

header

Default value: 'OPENQASM 2.0;'

instances

Default value: 2368

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

The number of parameter objects in the circuit.

Return type

int

num_qubits

Return number of qubits.

Return type

int

op_start_times

Return a list of operation start times.

This attribute is enabled once one of scheduling analysis passes runs on the quantum circuit.

Return type

List[int]

Returns

List of integers representing instruction start times. The index corresponds to the index of instruction in QuantumCircuit.data.

Raises

AttributeError – When circuit is not scheduled.

parameters

The parameters defined in the circuit.

This attribute returns the Parameter objects in the circuit sorted alphabetically. Note that parameters instantiated with a ParameterVector are still sorted numerically.

Examples

The snippet below shows that insertion order of parameters does not matter.

>>> from qiskit.circuit import QuantumCircuit, Parameter
>>> a, b, elephant = Parameter("a"), Parameter("b"), Parameter("elephant")
>>> circuit = QuantumCircuit(1)
>>> circuit.rx(b, 0)
>>> circuit.rz(elephant, 0)
>>> circuit.ry(a, 0)
>>> circuit.parameters  # sorted alphabetically!
ParameterView([Parameter(a), Parameter(b), Parameter(elephant)])

Bear in mind that alphabetical sorting might be unituitive when it comes to numbers. The literal “10” comes before “2” in strict alphabetical sorting.

>>> from qiskit.circuit import QuantumCircuit, Parameter
>>> angles = [Parameter("angle_1"), Parameter("angle_2"), Parameter("angle_10")]
>>> circuit = QuantumCircuit(1)
>>> circuit.u(*angles, 0)
>>> circuit.draw()
   ┌─────────────────────────────┐
q: ┤ U(angle_1,angle_2,angle_10) ├
   └─────────────────────────────┘
>>> circuit.parameters
ParameterView([Parameter(angle_1), Parameter(angle_10), Parameter(angle_2)])

To respect numerical sorting, a ParameterVector can be used.

 
>>> from qiskit.circuit import QuantumCircuit, Parameter, ParameterVector
>>> x = ParameterVector("x", 12)
>>> circuit = QuantumCircuit(1)
>>> for x_i in x:
...     circuit.rx(x_i, 0)
>>> circuit.parameters
ParameterView([
    ParameterVectorElement(x[0]), ParameterVectorElement(x[1]),
    ParameterVectorElement(x[2]), ParameterVectorElement(x[3]),
    ..., ParameterVectorElement(x[11])
])

Return type

ParameterView

Returns

The sorted Parameter objects in the circuit.

prefix

Default value: 'circuit'

qubits

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

Return type

List[Qubit]

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