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# PauliFeatureMap¶

class 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 (Optional[int]) -- Number of qubits in the circuit.

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

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

• data_map_func (Optional[Callable[[ndarray], float]]) -- 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.

Methods Defined Here

 pauli_block Get the Pauli block for the feature map circuit. pauli_evolution Get the evolution block for the given pauli string.

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.

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 = 217
layout

Return any associated layout information anout 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.