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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')

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Bases: 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 x⃗∈Rn\vec{x} \in \mathbb{R}^n as

UΦ(x⃗)=exp⁡(i∑S⊆[n]ϕS(x⃗)∏i∈SPi)U_{\Phi(\vec{x})}=\exp\left(i\sum_{S\subseteq [n]} \phi_S(\vec{x})\prod_{i\in S} P_i\right)

The circuit contains reps repetitions of this transformation. The variable Pi∈{I,X,Y,Z}P_i \in \{ I, X, Y, Z \} denotes the Pauli matrices. The index SS describes connectivities between different qubits or datapoints: S∈{(nk) combinations, k=1,...n}S \in \{\binom{n}{k}\ combinations,\ k = 1,... n \}. Per default the data-mapping ϕS\phi_S is

ϕS(x⃗)={x0 if k=1∏j∈S(π−xj) otherwise \begin{split}\phi_S(\vec{x}) = \begin{cases} x_0 \text{ if } k = 1 \\ \prod_{j \in S} (\pi - x_j) \text{ otherwise } \end{cases}\end{split}

For example, if the Pauli strings are chosen to be P0=ZP_0 = Z and P0,1=YYP_{0,1} = YY on 2 qubits and with 1 repetition using the default data-mapping, the Pauli evolution feature map is represented by:

┌───┐┌──────────────┐┌──────────┐                                             ┌───────────┐
┤ 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) ├
└───┘└──────────────┘└──────────┘└───┘└─────────────────────────────────┘└───┘└───────────┘

Please refer to ZFeatureMap for the case k=1k = 1, P0=ZP_0 = Z and to ZZFeatureMap for the case k=2k = 2, P0=ZP_0 = Z and P0,1=ZZP_{0,1} = ZZ.

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. (2018), Supervised learning with quantum enhanced feature spaces.

arXiv:1804.11326

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

PauliFeatureMap.pauli_block(pauli_string)

Get the Pauli block for the feature map circuit.

pauli_evolution

PauliFeatureMap.pauli_evolution(pauli_string, time)

Get the evolution block for the given pauli string.

Attributes

alpha

The Pauli rotation factor (alpha).

Return type

float

Returns

The Pauli rotation factor.

ancillas

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

Return type

List[AncillaQubit]

calibrations

Return calibration dictionary.

The custom pulse definition of a given gate is of the form {'gate_name': {(qubits, params): schedule}}

Return type

dict

clbits

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

Return type

List[Clbit]

data

entanglement

Get the entanglement strategy.

Return type

Union[str, List[str], List[List[str]], List[int], List[List[int]], List[List[List[int]]], List[List[List[List[int]]]], Callable[[int], str], Callable[[int], List[List[int]]]]

Returns

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

entanglement_blocks

extension_lib

Default value: 'include "qelib1.inc";'

feature_dimension

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

Return type

int

Returns

The feature dimension of this feature map.

global_phase

Return the global phase of the circuit in radians.

Return type

Union[ParameterExpression, float]

header

Default value: 'OPENQASM 2.0;'

initial_state

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

Return type

QuantumCircuit

Returns

The initial state.

insert_barriers

If barriers are inserted in between the layers or not.

Return type

bool

Returns

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

instances

Default value: 2404

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.

Return type

dict

num_ancillas

Return the number of ancilla qubits.

Return type

int

num_clbits

Return number of classical bits.

Return type

int

num_layers

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

Return type

int

Returns

The number of layers in the circuit.

num_parameters

Return type

int

num_parameters_settable

The number of distinct parameters.

num_qubits

Returns the number of qubits in this circuit.

Return type

int

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.

Return type

List[int]

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])]

Return type

List[Parameter]

Returns

The parameters objects used in the circuit.

parameter_bounds

The parameter bounds for the unbound parameters in the circuit.

Return type

Optional[List[Tuple[float, float]]]

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

Return type

ParameterView

paulis

The Pauli strings used in the entanglement of the qubits.

Return type

List[str]

Returns

The Pauli strings as list.

preferred_init_points

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

Return type

Optional[List[float]]

Returns

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

prefix

Default value: 'circuit'

qregs

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.

Return type

List[Qubit]

reps

The number of times rotation and entanglement block are repeated.

Return type

int

Returns

The number of repetitions.

rotation_blocks

The blocks in the rotation layers.

Return type

List[Instruction]

Returns

The blocks in the rotation layers.

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