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# LinearAmplitudeFunction¶

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

Bases: 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

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

Quantum Risk Analysis. arXiv:1806.06893

: 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 (float | list[float]) – The slope of the linear function. Can be a list of slopes if it is a piecewise linear function.

• offset (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 (list[float] | None) – 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.

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 CircuitInstructions for each instruction.

Return type

QuantumCircuitData

extension_lib = 'include "qelib1.inc";'
global_phase

Return the global phase of the circuit in radians.

instances = 181
layout

Return any associated layout information anout the circuit

This attribute contains an optional TranspileLayout object. This is typically set on the output from transpile() or PassManager.run() to retain information about the permutations caused on the input circuit by transpilation.

There are two types of permutations caused by the transpile() function, an initial layout which permutes the qubits based on the selected physical qubits on the Target, and a final layout which is an output permutation caused by SwapGates inserted during routing.

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

The number of parameter objects in the circuit.

num_qubits

Return number of qubits.

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.

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), ParameterVectorElement(x),
ParameterVectorElement(x), ParameterVectorElement(x),
..., ParameterVectorElement(x)
])

Returns

The sorted Parameter objects in the circuit.

prefix = 'circuit'
qubits

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