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: 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 \(\vec{x} \in \mathbb{R}^n\) as

\[U_{\Phi(\vec{x})}=\exp\left(i\sum_{S\subseteq [n]} \phi_S(\vec{x})\prod_{i\in S} P_i\right)\]

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

\[\begin{split}\phi_S(\vec{x}) = \begin{cases} x_0 \text{ if } k = 1 \\ \prod_{j \in S} (\pi - x_j) \text{ otherwise } \end{cases}\end{split}\]

For example, if the Pauli strings are chosen to be \(P_0 = Z\) and \(P_{0,1} = YY\) 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 \(k = 1\), \(P_0 = Z\) and to ZZFeatureMap for the case \(k = 2\), \(P_0 = Z\) and \(P_{0,1} = ZZ\).

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`	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).

Return type: float
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.

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 = '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.

header = 'OPENQASM 2.0;'¶

initial_state¶

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

Return type: Any
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 = 16¶

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.

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.

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.

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 = '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.

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.