# PiecewisePolynomialPauliRotations

*class *`qiskit.circuit.library.PiecewisePolynomialPauliRotations(num_state_qubits=None, breakpoints=None, coeffs=None, basis='Y', name='pw_poly')`

Bases: `FunctionalPauliRotations`

Piecewise-polynomially-controlled Pauli rotations.

This class implements a piecewise polynomial (not necessarily continuous) function, $f(x)$, on qubit amplitudes, which is defined through breakpoints and coefficients as follows. Suppose the breakpoints $(x_0, ..., x_J)$ are a subset of $[0, 2^n-1]$, where $n$ is the number of state qubits. Further on, denote the corresponding coefficients by $[a_{j,1},...,a_{j,d}]$, where $d$ is the highest degree among all polynomials.

Then $f(x)$ is defined as:

$f(x) = \begin{cases} 0, x < x_0 \\ \sum_{i=0}^{i=d}a_{j,i}/2 x^i, x_j \leq x < x_{j+1} \end{cases}$where if given the same number of breakpoints as polynomials, we implicitly assume $x_{J+1} = 2^n$.

Note the $1/2$ factor in the coefficients of $f(x)$, this is consistent with Qiskit’s Pauli rotations.

**Examples**

```
>>> from qiskit import QuantumCircuit
>>> from qiskit.circuit.library.arithmetic.piecewise_polynomial_pauli_rotations import\
... PiecewisePolynomialPauliRotations
>>> qubits, breakpoints, coeffs = (2, [0, 2], [[0, -1.2],[-1, 1, 3]])
>>> poly_r = PiecewisePolynomialPauliRotations(num_state_qubits=qubits,
...breakpoints=breakpoints, coeffs=coeffs)
>>>
>>> qc = QuantumCircuit(poly_r.num_qubits)
>>> qc.h(list(range(qubits)));
>>> qc.append(poly_r.to_instruction(), list(range(qc.num_qubits)));
>>> qc.draw()
┌───┐┌──────────┐
q_0: ┤ H ├┤0 ├
├───┤│ │
q_1: ┤ H ├┤1 ├
└───┘│ │
q_2: ─────┤2 ├
│ pw_poly │
q_3: ─────┤3 ├
│ │
q_4: ─────┤4 ├
│ │
q_5: ─────┤5 ├
└──────────┘
```

**References**

**[1]: Haener, T., Roetteler, M., & Svore, K. M. (2018).**

Optimizing Quantum Circuits for Arithmetic. arXiv:1805.12445

**[2]: Carrera Vazquez, A., Hiptmair, R., & Woerner, S. (2022).**

Enhancing the Quantum Linear Systems Algorithm using Richardson Extrapolation. ACM Transactions on Quantum Computing 3, 1, Article 2

**Parameters**

**num_state_qubits**(*Optional[**int**]*) – The number of qubits representing the state.**breakpoints**(*Optional[List[**int**]]*) – The breakpoints to define the piecewise-linear function. Defaults to`[0]`

.**coeffs**(*Optional[List[List[**float**]]]*) – The coefficients of the polynomials for different segments of the**x**(*piecewise-linear function. coeffs[j][i] is the coefficient of the i-th power of*) –**polynomial.**(*for the j-th*) – Defaults to linear:`[[1]]`

.**basis**(*str*) – The type of Pauli rotation (`'X'`

,`'Y'`

,`'Z'`

).**name**(*str*) – The name of the circuit.

## Attributes

### ancillas

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

### basis

The kind of Pauli rotation to be used.

Set the basis to ‘X’, ‘Y’ or ‘Z’ for controlled-X, -Y, or -Z rotations respectively.

**Returns**

The kind of Pauli rotation used in controlled rotation.

### breakpoints

The breakpoints of the piecewise polynomial function.

The function is polynomial in the intervals `[point_i, point_{i+1}]`

where the last point implicitly is `2**(num_state_qubits + 1)`

.

**Returns**

The list of breakpoints.

### 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.

### coeffs

The coefficients of the polynomials.

**Returns**

The polynomial coefficients per interval as nested lists.

### contains_zero_breakpoint

Whether 0 is the first breakpoint.

**Returns**

True, if 0 is the first breakpoint, otherwise False.

### data

### extension_lib

Default value: `'include "qelib1.inc";'`

### 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.

### mapped_coeffs

The coefficients mapped to the internal representation, since we only compare x>=breakpoint.

**Returns**

The mapped coefficients.

### 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_ancilla_qubits

The minimum number of ancilla qubits in the circuit.

**Returns**

The minimal number of ancillas required.

### 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

### 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.

## Methods

### evaluate

`evaluate(x)`

Classically evaluate the piecewise polynomial rotation.

**Parameters**

**x** (*float*) – Value to be evaluated at.

**Returns**

Value of piecewise polynomial function at x.

**Return type**