---
title: PauliFeatureMap (latest version)
description: API reference for qiskit.circuit.library.PauliFeatureMap in the latest version of qiskit
source: https://eu-de.quantum.cloud.ibm.com/docs/en/api/qiskit/qiskit.circuit.library.PauliFeatureMap
---

# PauliFeatureMap

*class* `qiskit.circuit.library.PauliFeatureMap(feature_dimension=None, reps=2, entanglement='full', alpha=2.0, paulis=None, data_map_func=None, parameter_prefix='x', insert_barriers=False, name='PauliFeatureMap')`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/2.5/qiskit/circuit/library/data_preparation/pauli_feature_map.py#L387-L703)

Bases: [`NLocal`](/docs/api/qiskit/qiskit.circuit.library.NLocal "qiskit.circuit.library.n_local.n_local.NLocal")

The Pauli Expansion circuit.

The Pauli Expansion circuit is a data encoding circuit that transforms input data $\vec{x} \in \mathbb{R}^n$, where n is the `feature_dimension`, as

$$
U_{\Phi(\vec{x})}=\exp\left(i\sum_{S \in \mathcal{I}}
\phi_S(\vec{x})\prod_{i\in S} P_i\right).
$$

Here, $S$ is a set of qubit indices that describes the connections in the feature map, $\mathcal{I}$ is a set containing all these index sets, and $P_i \in \{I, X, Y, Z\}$. Per default the data-mapping $\phi_S$ is

$$
\phi_S(\vec{x}) = \begin{cases}
x_i \text{ if } S = \{i\} \\
\prod_{j \in S} (\pi - x_j) \text{ if } |S| > 1
\end{cases}.
$$

The possible connections can be set using the `entanglement` and `paulis` arguments. For example, for single-qubit $Z$ rotations and two-qubit $YY$ interactions between all qubit pairs, we can set:

```python
feature_map = PauliFeatureMap(..., paulis=["Z", "YY"], entanglement="full")
```

which will produce blocks of the form

```text
┌───┐┌─────────────┐┌──────────┐                                            ┌───────────┐
┤ H ├┤ P(2.0*x[0]) ├┤ RX(pi/2) ├──■──────────────────────────────────────■──┤ RX(-pi/2) ├
├───┤├─────────────┤├──────────┤┌─┴─┐┌────────────────────────────────┐┌─┴─┐├───────────┤
┤ H ├┤ P(2.0*x[1]) ├┤ RX(pi/2) ├┤ X ├┤ P(2.0*(pi - x[0])*(pi - x[1])) ├┤ X ├┤ RX(-pi/2) ├
└───┘└─────────────┘└──────────┘└───┘└────────────────────────────────┘└───┘└───────────┘
```

The circuit contains `reps` repetitions of this transformation.

Please refer to [`ZFeatureMap`](/docs/api/qiskit/qiskit.circuit.library.ZFeatureMap "qiskit.circuit.library.ZFeatureMap") for the case of single-qubit Pauli-$Z$ rotations and to [`ZZFeatureMap`](/docs/api/qiskit/qiskit.circuit.library.ZZFeatureMap "qiskit.circuit.library.ZZFeatureMap") for the single- and two-qubit Pauli-$Z$ rotations.

**Examples**

```python
>>> prep = PauliFeatureMap(2, reps=1, paulis=['ZZ'])
>>> print(prep.decompose())
     ┌───┐
q_0: ┤ H ├──■──────────────────────────────────────■──
     ├───┤┌─┴─┐┌────────────────────────────────┐┌─┴─┐
q_1: ┤ H ├┤ X ├┤ P(2.0*(pi - x[0])*(pi - x[1])) ├┤ X ├
     └───┘└───┘└────────────────────────────────┘└───┘
```

```python
>>> prep = PauliFeatureMap(2, reps=1, paulis=['Z', 'XX'])
>>> print(prep.decompose())
     ┌───┐┌─────────────┐┌───┐                                            ┌───┐
q_0: ┤ H ├┤ P(2.0*x[0]) ├┤ H ├──■──────────────────────────────────────■──┤ H ├
     ├───┤├─────────────┤├───┤┌─┴─┐┌────────────────────────────────┐┌─┴─┐├───┤
q_1: ┤ H ├┤ P(2.0*x[1]) ├┤ H ├┤ X ├┤ P(2.0*(pi - x[0])*(pi - x[1])) ├┤ X ├┤ H ├
     └───┘└─────────────┘└───┘└───┘└────────────────────────────────┘└───┘└───┘
```

```python
>>> prep = PauliFeatureMap(2, reps=1, paulis=['ZY'])
>>> print(prep.decompose())
     ┌───┐┌──────────┐                                            ┌───────────┐
q_0: ┤ H ├┤ RX(pi/2) ├──■──────────────────────────────────────■──┤ RX(-pi/2) ├
     ├───┤└──────────┘┌─┴─┐┌────────────────────────────────┐┌─┴─┐└───────────┘
q_1: ┤ H ├────────────┤ X ├┤ P(2.0*(pi - x[0])*(pi - x[1])) ├┤ X ├─────────────
     └───┘            └───┘└────────────────────────────────┘└───┘
```

```python
>>> from qiskit.circuit.library import EfficientSU2
>>> prep = PauliFeatureMap(3, reps=3, paulis=['Z', 'YY', 'ZXZ'])
>>> wavefunction = EfficientSU2(3)
>>> classifier = prep.compose(wavefunction)
>>> classifier.num_parameters
27
>>> classifier.count_ops()
OrderedDict([('cx', 39), ('rx', 36), ('u1', 21), ('h', 15), ('ry', 12), ('rz', 12)])
```

References:

\[1] Havlicek et al. Supervised learning with quantum enhanced feature spaces, [Nature 567, 209-212 (2019)](https://www.nature.com/articles/s41586-019-0980-2).

> **Deprecated since version 2.1**
>
> The class `qiskit.circuit.library.data_preparation.pauli_feature_map.PauliFeatureMap` is deprecated as of Qiskit 2.1. It will be removed in Qiskit 3.0. Use the pauli\_feature\_map function as a replacement. Note that this will no longer return a BlueprintCircuit, but just a plain QuantumCircuit.

**Parameters**

- **feature\_dimension** ([*int*](https://docs.python.org/3/library/functions.html#int) *| None*) – Number of qubits in the circuit.
- **reps** ([*int*](https://docs.python.org/3/library/functions.html#int)) – The number of times the evolution layers are repeated.
- **entanglement** ([*str*](https://docs.python.org/3/library/stdtypes.html#str)  *|*[*dict*](https://docs.python.org/3/library/stdtypes.html#dict)*\[*[*int*](https://docs.python.org/3/library/functions.html#int)*,* [*list*](https://docs.python.org/3/library/stdtypes.html#list)*\[*[*tuple*](https://docs.python.org/3/library/stdtypes.html#tuple)*\[*[*int*](https://docs.python.org/3/library/functions.html#int)*]]] | Callable\[\[*[*int*](https://docs.python.org/3/library/functions.html#int)*],* [*str*](https://docs.python.org/3/library/stdtypes.html#str)  *|*[*dict*](https://docs.python.org/3/library/stdtypes.html#dict)*\[*[*int*](https://docs.python.org/3/library/functions.html#int)*,* [*list*](https://docs.python.org/3/library/stdtypes.html#list)*\[*[*tuple*](https://docs.python.org/3/library/stdtypes.html#tuple)*\[*[*int*](https://docs.python.org/3/library/functions.html#int)*]]]]*) – Specifies the entanglement structure. Can be a string (`'full'`, `'linear'`, `'reverse_linear'`, `'circular'` or `'sca'`) or can be a dictionary where the keys represent the number of qubits and the values are list of integer-pairs specifying the indices of qubits that are entangled with one another, for example: `{1: [(0,), (2,)], 2: [(0,1), (2,0)]}` or can be a `Callable[[int], Union[str | Dict[...]]]` to return an entanglement specific for a repetition
- **alpha** ([*float*](https://docs.python.org/3/library/functions.html#float)) – The Pauli rotation factor, multiplicative to the pauli rotations
- **paulis** ([*list*](https://docs.python.org/3/library/stdtypes.html#list)*\[*[*str*](https://docs.python.org/3/library/stdtypes.html#str)*] | None*) – A list of strings for to-be-used paulis. If None are provided, `['Z', 'ZZ']` will be used.
- **data\_map\_func** (*Callable\[\[np.ndarray],* [*float*](https://docs.python.org/3/library/functions.html#float)*] | None*) – A mapping function for the data `x` which can be supplied to override the default mapping.
- **parameter\_prefix** ([*str*](https://docs.python.org/3/library/stdtypes.html#str)) – The prefix used if default parameters are generated.
- **insert\_barriers** ([*bool*](https://docs.python.org/3/library/functions.html#bool)) – If `True`, barriers are inserted in between the evolution instructions and Hadamard layers.
- **name** ([*str*](https://docs.python.org/3/library/stdtypes.html#str)) – Name of the circuit.

## Attributes

### alpha

The Pauli rotation factor (alpha).

**Returns**

The Pauli rotation factor.

### entanglement\_blocks

The blocks in the entanglement layers.

**Returns**

The blocks in the entanglement layers.

### feature\_dimension

Returns the feature dimension (which is equal to the number of qubits).

**Returns**

The feature dimension of this feature map.

### num\_parameters\_settable

The number of distinct parameters.

### paulis

The Pauli strings used in the entanglement of the qubits.

**Returns**

The Pauli strings as list.

### name

Type: `str`

A human-readable name for the circuit.

**Example**

```python
from qiskit import QuantumCircuit

qc = QuantumCircuit(2, 2, name="my_circuit")
print(qc.name)
```

```text
my_circuit
```

## Methods

### get\_entangler\_map

`get_entangler_map(rep_num, block_num, num_block_qubits)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/2.5/qiskit/circuit/library/data_preparation/pauli_feature_map.py#L666-L703)

Get the entangler map for in the repetition `rep_num` and the block `block_num`.

The entangler map for the current block is derived from the value of `self.entanglement`. Below the different cases are listed, where `i` and `j` denote the repetition number and the block number, respectively, and `n` the number of qubits in the block.

| entanglement type                | entangler map                                      |
| -------------------------------- | -------------------------------------------------- |
| `None`                           | `[[0, ..., n - 1]]`                                |
| `str` (e.g `'full'`)             | the specified connectivity on `n` qubits           |
| `List[int]`                      | \[`entanglement`]                                  |
| `List[List[int]]`                | `entanglement`                                     |
| `List[List[List[int]]]`          | `entanglement[i]`                                  |
| `List[List[List[List[int]]]]`    | `entanglement[i][j]`                               |
| `List[str]`                      | the connectivity specified in `entanglement[i]`    |
| `List[List[str]]`                | the connectivity specified in `entanglement[i][j]` |
| `Callable[int, str]`             | same as `List[str]`                                |
| `Callable[int, List[List[int]]]` | same as `List[List[List[int]]]`                    |

Note that all indices are to be taken modulo the length of the array they act on, i.e. no out-of-bounds index error will be raised but we re-iterate from the beginning of the list.

**Parameters**

- **rep\_num** ([*int*](https://docs.python.org/3/library/functions.html#int)) – The current repetition we are in.
- **block\_num** ([*int*](https://docs.python.org/3/library/functions.html#int)) – The block number within the entanglement layers.
- **num\_block\_qubits** ([*int*](https://docs.python.org/3/library/functions.html#int)) – The number of qubits in the block.

**Returns**

The entangler map for the current block in the current repetition.

**Raises**

[**ValueError**](https://docs.python.org/3/library/exceptions.html#ValueError) – If the value of `entanglement` could not be cast to a corresponding entangler map.

**Return type**

[*Sequence*](https://docs.python.org/3/library/collections.abc.html#collections.abc.Sequence)\[[*Sequence*](https://docs.python.org/3/library/collections.abc.html#collections.abc.Sequence)\[[int](https://docs.python.org/3/library/functions.html#int)]]

### pauli\_block

`pauli_block(pauli_string)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/2.5/qiskit/circuit/library/data_preparation/pauli_feature_map.py#L620-L624)

Get the Pauli block for the feature map circuit.

### pauli\_evolution

`pauli_evolution(pauli_string, time)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/2.5/qiskit/circuit/library/data_preparation/pauli_feature_map.py#L626-L664)

Get the evolution block for the given pauli string.