# 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')[source]#

Bases: 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

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

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:

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


which will produce blocks of the form

┌───┐┌──────────────┐┌──────────┐                                             ┌───────────┐
┤ H ├┤ U1(2.0*x[0]) ├┤ RX(pi/2) ├──■───────────────────────────────────────■──┤ RX(-pi/2) ├
├───┤├──────────────┤├──────────┤┌─┴─┐┌─────────────────────────────────┐┌─┴─┐├───────────┤
┤ H ├┤ U1(2.0*x[1]) ├┤ RX(pi/2) ├┤ X ├┤ U1(2.0*(pi - x[0])*(pi - x[1])) ├┤ X ├┤ RX(-pi/2) ├
└───┘└──────────────┘└──────────┘└───┘└─────────────────────────────────┘└───┘└───────────┘


The circuit contains reps repetitions of this transformation.

Please refer to ZFeatureMap for the case of single-qubit Pauli-$$Z$$ rotations and to ZZFeatureMap for the single- and two-qubit Pauli-$$Z$$ rotations.

Examples

>>> prep = PauliFeatureMap(2, reps=1, paulis=['ZZ'])
>>> print(prep)
┌───┐
q_0: ┤ H ├──■───────────────────────────────────────■──
├───┤┌─┴─┐┌─────────────────────────────────┐┌─┴─┐
q_1: ┤ H ├┤ X ├┤ U1(2.0*(pi - x[0])*(pi - x[1])) ├┤ X ├
└───┘└───┘└─────────────────────────────────┘└───┘

>>> prep = PauliFeatureMap(2, reps=1, paulis=['Z', 'XX'])
>>> print(prep)
┌───┐┌──────────────┐┌───┐                                             ┌───┐
q_0: ┤ H ├┤ U1(2.0*x[0]) ├┤ H ├──■───────────────────────────────────────■──┤ H ├
├───┤├──────────────┤├───┤┌─┴─┐┌─────────────────────────────────┐┌─┴─┐├───┤
q_1: ┤ H ├┤ U1(2.0*x[1]) ├┤ H ├┤ X ├┤ U1(2.0*(pi - x[0])*(pi - x[1])) ├┤ X ├┤ H ├
└───┘└──────────────┘└───┘└───┘└─────────────────────────────────┘└───┘└───┘

>>> prep = PauliFeatureMap(2, reps=1, paulis=['ZY'])
>>> print(prep)
┌───┐┌──────────┐                                             ┌───────────┐
q_0: ┤ H ├┤ RX(pi/2) ├──■───────────────────────────────────────■──┤ RX(-pi/2) ├
├───┤└──────────┘┌─┴─┐┌─────────────────────────────────┐┌─┴─┐└───────────┘
q_1: ┤ H ├────────────┤ X ├┤ U1(2.0*(pi - x[0])*(pi - x[1])) ├┤ X ├─────────────
└───┘            └───┘└─────────────────────────────────┘└───┘

>>> 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).

Create a new Pauli expansion circuit.

Parameters:
• feature_dimension (int | None) – Number of qubits in the circuit.

• reps (int) – The number of repeated circuits.

• entanglement (str | List[List[int]] | Callable[[int], List[int]]) – Specifies the entanglement structure. Refer to NLocal for detail.

• alpha (float) – The Pauli rotation factor, multiplicative to the pauli rotations

• paulis (List[str] | None) – A list of strings for to-be-used paulis. If None are provided, ['Z', 'ZZ'] will be used.

• data_map_func (Callable[[ndarray], float] | None) – A mapping function for data x which can be supplied to override the default mapping from self_product().

• parameter_prefix (str) – The prefix used if default parameters are generated.

• insert_barriers (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#

Returns a list of ancilla bits in the order that the registers were added.

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.

data#
entanglement#

Get the entanglement strategy.

Returns:

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

entanglement_blocks#
extension_lib = 'include "qelib1.inc";'#
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#

Return the global phase of the circuit 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 = 223#
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 SwapGates inserted during routing.

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_ancillas#

Return the number of ancilla qubits.

num_clbits#

Return number of classical bits.

num_layers#

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

Returns:

The number of layers in the circuit.

num_parameters#
num_parameters_settable#

The number of distinct parameters.

num_qubits#

Returns the number of qubits in this circuit.

Returns:

The number of 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 – When circuit is not scheduled.

ordered_parameters#

The parameters used in the underlying circuit.

This includes float values and duplicates.

Examples

>>> # 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#
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 = 'circuit'#
qregs: 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.

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.

Methods

pauli_block(pauli_string)[source]#

Get the Pauli block for the feature map circuit.

pauli_evolution(pauli_string, time)[source]#

Get the evolution block for the given pauli string.