# PolynomialPauliRotations

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

Bases: `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*(opens in a new tab)*| None*) – The number of qubits representing the state.**coeffs**(*list*(opens in a new tab)*[**float*(opens in a new tab)*] | 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*(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

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.

### 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 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

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

.

**Returns**

The coefficients of the polynomial.

### data

### 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.

### 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: `271`

### 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_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**(opens in a new tab) – 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.