PiecewiseChebyshev

class qiskit.circuit.library.PiecewiseChebyshev(f_x, degree=None, breakpoints=None, num_state_qubits=None, name='pw_cheb')

Bases: BlueprintCircuit

Piecewise Chebyshev approximation to an input function.

For a given function $f(x)$ and degree $d$, this class implements a piecewise polynomial Chebyshev approximation on $n$ qubits to $f(x)$ on the given intervals. All the polynomials in the approximation are of degree $d$.

The values of the parameters are calculated according to [1] and see [2] for a more detailed explanation of the circuit construction and how it acts on the qubits.

Examples:

import numpy as np
from qiskit import QuantumCircuit
from qiskit.circuit.library.arithmetic.piecewise_chebyshev import PiecewiseChebyshev
f_x, degree, breakpoints, num_state_qubits = lambda x: np.arcsin(1 / x), 2, [2, 4], 2
pw_approximation = PiecewiseChebyshev(f_x, degree, breakpoints, num_state_qubits)
pw_approximation._build()
qc = QuantumCircuit(pw_approximation.num_qubits)
qc.h(list(range(num_state_qubits)))
qc.append(pw_approximation.to_instruction(), qc.qubits)
qc.draw(output='mpl')

References:

[1] Haener, T., Roetteler, M., & Svore, K. M. (2018). Optimizing Quantum Circuits for Arithmetic. arXiv:1805.12445

[2] Carrera Vazquez, A., Hiptmair, H., & Woerner, S. (2022). Enhancing the Quantum Linear Systems Algorithm Using Richardson Extrapolation. ACM Transactions on Quantum Computing 3, 1, Article 2

Parameters

• f_x (float | Callable[[int], float]) – the function to be approximated. Constant functions should be specified as f_x = constant.
• degree (int | None) – the degree of the polynomials. Defaults to 1.
• breakpoints (list[int] | None) – the breakpoints to define the piecewise-linear function. Defaults to the full interval.
• num_state_qubits (int | None) – number of qubits representing the state.
• name (str) – The name of the circuit object.

Attributes

breakpoints

degree

f_x

num_state_qubits

polynomials

name

from qiskit import QuantumCircuit

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

my_circuit