---
title: PauliFeatureMap (v1.2)
description: API reference for qiskit.circuit.library.PauliFeatureMap in qiskit v1.2
source: https://eu-de.quantum.cloud.ibm.com/docs/en/api/qiskit/1.2/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/1.2/qiskit/circuit/library/data_preparation/pauli_feature_map.py#L26-L287)

Bases: [`NLocal`](/docs/api/qiskit/1.2/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

```python
┌───┐┌─────────────┐┌──────────┐                                            ┌───────────┐
┤ 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/1.2/qiskit.circuit.library.ZFeatureMap "qiskit.circuit.library.ZFeatureMap") for the case of single-qubit Pauli-$Z$ rotations and to [`ZZFeatureMap`](/docs/api/qiskit/1.2/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).decompose()
>>> classifier.num_parameters
27
>>> classifier.count_ops()
OrderedDict([('cx', 39), ('rx', 36), ('p', 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).

Create a new Pauli expansion circuit.

**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 repeated circuits.
- **entanglement** ([*str*](https://docs.python.org/3/library/stdtypes.html#str)  *|*[*List*](https://docs.python.org/3/library/typing.html#typing.List)*\[*[*List*](https://docs.python.org/3/library/typing.html#typing.List)*\[*[*int*](https://docs.python.org/3/library/functions.html#int)*]] |* [*Callable*](https://docs.python.org/3/library/typing.html#typing.Callable)*\[\[*[*int*](https://docs.python.org/3/library/functions.html#int)*],* [*List*](https://docs.python.org/3/library/typing.html#typing.List)*\[*[*int*](https://docs.python.org/3/library/functions.html#int)*]]*) – Specifies the entanglement structure. Refer to [`NLocal`](/docs/api/qiskit/1.2/qiskit.circuit.library.NLocal "qiskit.circuit.library.NLocal") for detail.
- **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/typing.html#typing.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*](https://docs.python.org/3/library/typing.html#typing.Callable)*\[\[*[*ndarray*](https://numpy.org/doc/stable/reference/generated/numpy.ndarray.html#numpy.ndarray)*],* [*float*](https://docs.python.org/3/library/functions.html#float)*] | None*) – A mapping function for data x which can be supplied to override the default mapping from `self_product()`.
- **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.

## Attributes

### alpha

The Pauli rotation factor (alpha).

**Returns**

The Pauli rotation factor.

### ancillas

A list of `AncillaQubit`s in the order that they were added. You should not mutate this.

### 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.

### 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

### entanglement

Get the entanglement strategy.

**Returns**

The entanglement strategy, see `get_entangler_map()` for more detail on how the format is interpreted.

### 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.

### flatten

Returns whether the circuit is wrapped in nested gates/instructions or flattened.

### global\_phase

The global phase of the current circuit scope in radians.

### initial\_state

Return the initial state that is added in front of the n-local circuit.

**Returns**

The initial state.

### insert\_barriers

If barriers are inserted in between the layers or not.

**Returns**

`True`, if barriers are inserted in between the layers, `False` if not.

### instances

Default value: `188`

### 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\_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.PauliFeatureMap.num_input_vars "qiskit.circuit.library.PauliFeatureMap.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.PauliFeatureMap.num_captured_vars "qiskit.circuit.library.PauliFeatureMap.num_captured_vars") must be zero.

### num\_layers

Return the number of layers in the n-local circuit.

**Returns**

The number of layers in the circuit.

### num\_parameters

The number of parameter objects in the circuit.

### num\_parameters\_settable

The number of distinct parameters.

### num\_qubits

Returns the number of qubits in this circuit.

**Returns**

The number of 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.

### ordered\_parameters

The parameters used in the underlying circuit.

This includes float values and duplicates.

**Examples**

```python
>>> # prepare circuit ...
>>> print(nlocal)
     ┌───────┐┌──────────┐┌──────────┐┌──────────┐
q_0: ┤ Ry(1) ├┤ Ry(θ[1]) ├┤ Ry(θ[1]) ├┤ Ry(θ[3]) ├
     └───────┘└──────────┘└──────────┘└──────────┘
>>> nlocal.parameters
{Parameter(θ[1]), Parameter(θ[3])}
>>> nlocal.ordered_parameters
[1, Parameter(θ[1]), Parameter(θ[1]), Parameter(θ[3])]
```

**Returns**

The parameters objects used in the circuit.

### parameter\_bounds

The parameter bounds for the unbound parameters in the circuit.

**Returns**

A list of pairs indicating the bounds, as (lower, upper). None indicates an unbounded parameter in the corresponding direction. If `None` is returned, problem is fully unbounded.

### 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.

### paulis

The Pauli strings used in the entanglement of the qubits.

**Returns**

The Pauli strings as list.

### preferred\_init\_points

The initial points for the parameters. Can be stored as initial guess in optimization.

**Returns**

The initial values for the parameters, or None, if none have been set.

### 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.

### reps

The number of times rotation and entanglement block are repeated.

**Returns**

The number of repetitions.

### rotation\_blocks

The blocks in the rotation layers.

**Returns**

The blocks in the rotation layers.

### 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.PauliFeatureMap.unit "qiskit.circuit.library.PauliFeatureMap.unit").

### unit

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

## Methods

### pauli\_block

`pauli_block(pauli_string)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.2/qiskit/circuit/library/data_preparation/pauli_feature_map.py#L246-L250)

Get the Pauli block for the feature map circuit.

### pauli\_evolution

`pauli_evolution(pauli_string, time)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.2/qiskit/circuit/library/data_preparation/pauli_feature_map.py#L252-L287)

Get the evolution block for the given pauli string.