# 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(opens in a new tab)

**[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(opens in a new tab)

**Parameters**

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

.**coeffs**(*Optional[List[List[**float*(opens in a new tab)*]]]*) – 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*(opens in a new tab)) – The type of Pauli rotation (`'X'`

,`'Y'`

,`'Z'`

).**name**(*str*(opens in a new tab)) – The name of the circuit.

## Attributes

### ancillas

A list of `AncillaQubit`

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

### 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

A list of `Clbit`

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

### 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

### 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.

### mapped_coeffs

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

**Returns**

The mapped coefficients.

### 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_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_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

### 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.

## Methods

### evaluate

`evaluate(x)`

Classically evaluate the piecewise polynomial rotation.

**Parameters**

**x** (*float*(opens in a new tab)) – Value to be evaluated at.

**Returns**

Value of piecewise polynomial function at x.

**Return type**