LinearAmplitudeFunction

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

GitHub

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

[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 (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.

Attributes

ancillas

A list of AncillaQubits in the order that they were added. You should not mutate this.

calibrations

Return calibration dictionary.

The custom pulse definition of a given gate is of the form {'gate_name': {(qubits, params): schedule}}

Deprecated since version 1.3

The property qiskit.circuit.quantumcircuit.QuantumCircuit.calibrations is deprecated as of Qiskit 1.3. It will be removed in Qiskit 2.0. The entire Qiskit Pulse package is being deprecated and will be moved to the Qiskit Dynamics repository: https://github.com/qiskit-community/qiskit-dynamics. Note that once removed, qiskit.circuit.quantumcircuit.QuantumCircuit.calibrations will have no alternative in Qiskit.

clbits

A list of Clbits in the order that they were added. You should not mutate this.

data

The circuit data (instructions and context).

Returns

a list-like object containing the CircuitInstructions for each instruction.

Return type

QuantumCircuitData

duration

The total duration of the circuit, set by a scheduling transpiler pass. Its unit is specified by unit.

Deprecated since version 1.3.0

The property qiskit.circuit.quantumcircuit.QuantumCircuit.duration is deprecated as of qiskit 1.3.0. It will be removed in Qiskit 2.0.0.

global_phase

The global phase of the current circuit scope in radians.

instances

Default value: 178

layout

Return any associated layout information about 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.

metadata

Arbitrary user-defined metadata for the circuit.

Qiskit will not examine the content of this mapping, but it will pass it through the transpiler and reattach it to the output, so you can track your own metadata.

num_ancillas

Return the number of ancilla qubits.

num_captured_vars

The number of real-time classical variables in the circuit marked as captured from an enclosing scope.

This is the length of the iter_captured_vars() iterable. If this is non-zero, num_input_vars must be zero.

num_clbits

Return number of classical bits.

num_declared_vars

The number of real-time classical variables in the circuit that are declared by this circuit scope, excluding inputs or captures.

This is the length of the iter_declared_vars() iterable.

num_input_vars

The number of real-time classical variables in the circuit marked as circuit inputs.

This is the length of the iter_input_vars() iterable. If this is non-zero, num_captured_vars must be zero.

num_parameters

The number of parameter objects in the circuit.

num_qubits

Return number of qubits.

num_vars

The number of real-time classical variables in the circuit.

This is the length of the iter_vars() iterable.

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 unintuitive 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])
])

Returns

The sorted Parameter objects in the circuit.

prefix

Default value: 'circuit'

qubits

A list of Qubits in the order that they were added. You should not mutate this.

unit

The unit that duration is specified in.

Deprecated since version 1.3.0

The property qiskit.circuit.quantumcircuit.QuantumCircuit.unit is deprecated as of qiskit 1.3.0. It will be removed in Qiskit 2.0.0.

name

Type: str

A human-readable name for the circuit.

qregs

Type: list[QuantumRegister]

A list of the QuantumRegisters in this circuit. You should not mutate this.

cregs

Type: list[ClassicalRegister]

A list of the ClassicalRegisters in this circuit. You should not mutate this.

Methods

post_processing

post_processing(scaled_value)

GitHub

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

Parameters

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

Returns

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

Return type

float

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