# PiecewiseChebyshev

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 (opens in a new tab)

[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 (opens in a new tab)

Parameters

## Attributes

### ancillas

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

### breakpoints

The breakpoints for the piecewise approximation.

Returns

The breakpoints for the piecewise approximation.

### 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.

### degree

The degree of the polynomials.

Returns

The degree of the polynomials.

### extension_lib

= 'include "qelib1.inc";'

### f_x

The function to be approximated.

Returns

The function to be approximated.

### global_phase

Return the global phase of the current circuit scope in radians.

= 'OPENQASM 2.0;'

### instances

= 306

### 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.

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_qubits

Return number of qubits.

### num_state_qubits

The number of state qubits representing the state $|x\rangle$.

Returns

The number of state 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 (opens in a new tab) – When circuit is not scheduled.

### polynomials

The polynomials for the piecewise approximation.

Returns

The polynomials for the piecewise approximation.

Raises

TypeError (opens in a new tab) – If the input function is not in the correct format.

### prefix

= 'circuit'

### qregs

list[QuantumRegister]

A list of the quantum registers associated with the circuit.

### qubits

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