# 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(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**

**f_x**(*float*(opens in a new tab)*| Callable[[**int*(opens in a new tab)*],**float*(opens in a new tab)*]*) – the function to be approximated. Constant functions should be specified as f_x = constant.**degree**(*int*(opens in a new tab)*| None*) – the degree of the polynomials. Defaults to`1`

.**breakpoints**(*list*(opens in a new tab)*[**int*(opens in a new tab)*] | None*) – the breakpoints to define the piecewise-linear function. Defaults to the full interval.**num_state_qubits**(*int*(opens in a new tab)*| None*) – number of qubits representing the state.**name**(*str*(opens in a new tab)) – The name of the circuit object.

## Attributes

### ancillas

A list of `AncillaQubit`

s in the order that they were added. You should not mutate this.

### 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

A list of `Clbit`

s in the order that they were added. You should not mutate this.

### data

### degree

The degree of the polynomials.

**Returns**

The degree of the polynomials.

### f_x

The function to be approximated.

**Returns**

The function to be approximated.

### global_phase

The global phase of the current circuit scope in radians.

### instances

Default value: `256`

### 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

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

### 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.

### 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**(opens in a new tab) – When circuit is not scheduled.

### parameters

### 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

Default value: `'circuit'`

### qregs

Type: `list[QuantumRegister]`

A list of the `QuantumRegister`

s in this circuit. You should not mutate this.

### qubits

A list of `Qubit`

s in the order that they were added. You should not mutate this.

### name

Type: `str`

A human-readable name for the circuit.

### cregs

Type: `list[ClassicalRegister]`

A list of the `ClassicalRegister`

s in this circuit. You should not mutate this.

### duration

Type: `int | float | None`

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

.

### unit

The unit that `duration`

is specified in.