---
title: PolynomialPauliRotations (v1.2)
description: API reference for qiskit.circuit.library.PolynomialPauliRotations in qiskit v1.2
source: https://eu-de.quantum.cloud.ibm.com/docs/en/api/qiskit/1.2/qiskit.circuit.library.PolynomialPauliRotations
---

# PolynomialPauliRotations

*class* `qiskit.circuit.library.PolynomialPauliRotations(num_state_qubits=None, coeffs=None, basis='Y', name='poly')`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.2/qiskit/circuit/library/arithmetic/polynomial_pauli_rotations.py#L134-L335)

Bases: [`FunctionalPauliRotations`](/docs/api/qiskit/1.2/qiskit.circuit.library.FunctionalPauliRotations "qiskit.circuit.library.arithmetic.functional_pauli_rotations.FunctionalPauliRotations")

A circuit implementing polynomial Pauli rotations.

For a polynomial $p(x)$, a basis state $|i\rangle$ and a target qubit $|0\rangle$ this operator acts as:

$$
|i\rangle |0\rangle \mapsto \cos\left(\frac{p(i)}{2}\right) |i\rangle |0\rangle
+ \sin\left(\frac{p(i)}{2}\right) |i\rangle |1\rangle
$$

Let n be the number of qubits representing the state, d the degree of p(x) and q\_i the qubits, where q\_0 is the least significant qubit. Then for

$$
x = \sum_{i=0}^{n-1} 2^i q_i,
$$

we can write

$$
p(x) = \sum_{j=0}^{j=d} c_j x^j
$$

where $c$ are the input coefficients, `coeffs`.

Prepare an approximation to a state with amplitudes specified by a polynomial.

**Parameters**

- **num\_state\_qubits** ([*int*](https://docs.python.org/3/library/functions.html#int) *| None*) – The number of qubits representing the state.
- **coeffs** ([*list*](https://docs.python.org/3/library/stdtypes.html#list)*\[*[*float*](https://docs.python.org/3/library/functions.html#float)*] | None*) – The coefficients of the polynomial. `coeffs[i]` is the coefficient of the i-th power of x. Defaults to linear: \[0, 1].
- **basis** ([*str*](https://docs.python.org/3/library/stdtypes.html#str)) – The type of Pauli rotation (‘X’, ‘Y’, ‘Z’).
- **name** ([*str*](https://docs.python.org/3/library/stdtypes.html#str)) – 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.

### 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 polynomial.

`coeffs[i]` is the coefficient of the i-th power of the function input $x$, that means that the rotation angles are based on the coefficients value, following the formula

$$
c_j x^j ,  j=0, ..., d
$$

where $d$ is the degree of the polynomial $p(x)$ and $c$ are the coefficients `coeffs`.

**Returns**

The coefficients of the polynomial.

### data

The circuit data (instructions and context).

**Returns**

a list-like object containing the [`CircuitInstruction`](/docs/api/qiskit/1.2/qiskit.circuit.CircuitInstruction "qiskit.circuit.CircuitInstruction")s for each instruction.

**Return type**

QuantumCircuitData

### degree

Return the degree of the polynomial, equals to the number of coefficients minus 1.

**Returns**

The degree of the polynomial. If the coefficients have not been set, return 0.

### global\_phase

The global phase of the current circuit scope in radians.

### instances

Default value: `259`

### layout

Return any associated layout information about the circuit

This attribute contains an optional [`TranspileLayout`](/docs/api/qiskit/1.2/qiskit.transpiler.TranspileLayout "qiskit.transpiler.TranspileLayout") object. This is typically set on the output from [`transpile()`](/docs/api/qiskit/1.2/compiler#qiskit.compiler.transpile "qiskit.compiler.transpile") or [`PassManager.run()`](/docs/api/qiskit/1.2/qiskit.transpiler.PassManager#run "qiskit.transpiler.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()`](/docs/api/qiskit/1.2/compiler#qiskit.compiler.transpile "qiskit.compiler.transpile") function, an initial layout which permutes the qubits based on the selected physical qubits on the [`Target`](/docs/api/qiskit/1.2/qiskit.transpiler.Target "qiskit.transpiler.Target"), and a final layout which is an output permutation caused by [`SwapGate`](/docs/api/qiskit/1.2/qiskit.circuit.library.SwapGate "qiskit.circuit.library.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\_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`](#qiskit.circuit.library.PolynomialPauliRotations.num_input_vars "qiskit.circuit.library.PolynomialPauliRotations.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`](#qiskit.circuit.library.PolynomialPauliRotations.num_captured_vars "qiskit.circuit.library.PolynomialPauliRotations.num_captured_vars") must be zero.

### num\_parameters

The number of parameter objects in the circuit.

### 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**](https://docs.python.org/3/library/exceptions.html#AttributeError) – When circuit is not scheduled.

### parameters

The parameters defined in the circuit.

This attribute returns the [`Parameter`](/docs/api/qiskit/1.2/qiskit.circuit.Parameter "qiskit.circuit.Parameter") objects in the circuit sorted alphabetically. Note that parameters instantiated with a [`ParameterVector`](/docs/api/qiskit/1.2/qiskit.circuit.ParameterVector "qiskit.circuit.ParameterVector") are still sorted numerically.

**Examples**

The snippet below shows that insertion order of parameters does not matter.

```python
>>> from qiskit.circuit import QuantumCircuit, Parameter
>>> a, b, elephant = Parameter("a"), Parameter("b"), Parameter("elephant")
>>> circuit = QuantumCircuit(1)
>>> circuit.rx(b, 0)
>>> circuit.rz(elephant, 0)
>>> circuit.ry(a, 0)
>>> circuit.parameters  # sorted alphabetically!
ParameterView([Parameter(a), Parameter(b), Parameter(elephant)])
```

Bear in mind that alphabetical sorting might be unintuitive when it comes to numbers. The literal “10” comes before “2” in strict alphabetical sorting.

```python
>>> from qiskit.circuit import QuantumCircuit, Parameter
>>> angles = [Parameter("angle_1"), Parameter("angle_2"), Parameter("angle_10")]
>>> circuit = QuantumCircuit(1)
>>> circuit.u(*angles, 0)
>>> circuit.draw()
   ┌─────────────────────────────┐
q: ┤ U(angle_1,angle_2,angle_10) ├
   └─────────────────────────────┘
>>> circuit.parameters
ParameterView([Parameter(angle_1), Parameter(angle_10), Parameter(angle_2)])
```

To respect numerical sorting, a [`ParameterVector`](/docs/api/qiskit/1.2/qiskit.circuit.ParameterVector "qiskit.circuit.ParameterVector") can be used.

```python
>>> from qiskit.circuit import QuantumCircuit, Parameter, ParameterVector
>>> x = ParameterVector("x", 12)
>>> circuit = QuantumCircuit(1)
>>> for x_i in x:
...     circuit.rx(x_i, 0)
>>> circuit.parameters
ParameterView([
    ParameterVectorElement(x[0]), ParameterVectorElement(x[1]),
    ParameterVectorElement(x[2]), ParameterVectorElement(x[3]),
    ..., ParameterVectorElement(x[11])
])
```

**Returns**

The sorted [`Parameter`](/docs/api/qiskit/1.2/qiskit.circuit.Parameter "qiskit.circuit.Parameter") objects in the circuit.

### 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`](#qiskit.circuit.library.PolynomialPauliRotations.unit "qiskit.circuit.library.PolynomialPauliRotations.unit").

### unit

The unit that [`duration`](#qiskit.circuit.library.PolynomialPauliRotations.duration "qiskit.circuit.library.PolynomialPauliRotations.duration") is specified in.