LinearAmplitudeFunction

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

Bases: QuantumCircuit

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

An amplitude function $F$ of a function $f$ is a mapping

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 $\hat{f}$:

with

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

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

name

from qiskit import QuantumCircuit
 
qc = QuantumCircuit(2, 2, name="my_circuit")
print(qc.name)

my_circuit

Methods

post_processing