# 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

### 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.

### data

### degree

The degree of the polynomials.

**Returns**

The degree of the polynomials.

### extension_lib

Default value: `'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.

### header

Default value: `'OPENQASM 2.0;'`

### instances

Default value: `250`

### 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 `SwapGate`

s inserted during routing.

### metadata

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

### 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** – When circuit is not scheduled.

### parameters

### polynomials

The polynomials for the piecewise approximation.

**Returns**

The polynomials for the piecewise approximation.

**Raises**

**TypeError** – If the input function is not in the correct format.

### prefix

Default value: `'circuit'`

### qregs

Type: `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.