# This code is part of Qiskit.
#
# (C) Copyright IBM 2017, 2020.
#
# This code is licensed under the Apache License, Version 2.0. You may
# obtain a copy of this license in the LICENSE.txt file in the root directory
# of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.
#
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"""The Pauli expansion circuit module."""
from typing import Optional, Callable, List, Union
from functools import reduce
import numpy as np
from qiskit.circuit import QuantumCircuit
from qiskit.circuit import Parameter, ParameterVector
from qiskit.circuit.library.standard_gates import HGate
from ..n_local.n_local import NLocal
[docs]class PauliFeatureMap(NLocal):
r"""The Pauli Expansion circuit.
The Pauli Expansion circuit is a data encoding circuit that transforms input data
:math:`\vec{x} \in \mathbb{R}^n`, where `n` is the ``feature_dimension``, as
.. math::
U_{\Phi(\vec{x})}=\exp\left(i\sum_{S \in \mathcal{I}}
\phi_S(\vec{x})\prod_{i\in S} P_i\right).
Here, :math:`S` is a set of qubit indices that describes the connections in the feature map,
:math:`\mathcal{I}` is a set containing all these index sets, and
:math:`P_i \in \{I, X, Y, Z\}`. Per default the data-mapping
:math:`\phi_S` is
.. math::
\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 :math:`Z` rotations and two-qubit :math:`YY` interactions
between all qubit pairs, we can set::
feature_map = PauliFeatureMap(..., paulis=["Z", "YY"], entanglement="full")
which will produce blocks of the form
.. parsed-literal::
┌───┐┌──────────────┐┌──────────┐ ┌───────────┐
┤ 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 :class:`.ZFeatureMap` for the case of single-qubit Pauli-:math:`Z` rotations
and to :class:`.ZZFeatureMap` for the single- and two-qubit Pauli-:math:`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) <https://www.nature.com/articles/s41586-019-0980-2>`__.
"""
def __init__(
self,
feature_dimension: Optional[int] = None,
reps: int = 2,
entanglement: Union[str, List[List[int]], Callable[[int], List[int]]] = "full",
alpha: float = 2.0,
paulis: Optional[List[str]] = None,
data_map_func: Optional[Callable[[np.ndarray], float]] = None,
parameter_prefix: str = "x",
insert_barriers: bool = False,
name: str = "PauliFeatureMap",
) -> None:
"""Create a new Pauli expansion circuit.
Args:
feature_dimension: Number of qubits in the circuit.
reps: The number of repeated circuits.
entanglement: Specifies the entanglement structure. Refer to
:class:`~qiskit.circuit.library.NLocal` for detail.
alpha: The Pauli rotation factor, multiplicative to the pauli rotations
paulis: A list of strings for to-be-used paulis. If None are provided, ``['Z', 'ZZ']``
will be used.
data_map_func: A mapping function for data x which can be supplied to override the
default mapping from :meth:`self_product`.
parameter_prefix: The prefix used if default parameters are generated.
insert_barriers: If True, barriers are inserted in between the evolution instructions
and hadamard layers.
"""
super().__init__(
num_qubits=feature_dimension,
reps=reps,
rotation_blocks=HGate(),
entanglement=entanglement,
parameter_prefix=parameter_prefix,
insert_barriers=insert_barriers,
skip_final_rotation_layer=True,
name=name,
)
self._data_map_func = data_map_func or self_product
self._paulis = paulis or ["Z", "ZZ"]
self._alpha = alpha
def _parameter_generator(
self, rep: int, block: int, indices: List[int]
) -> Optional[List[Parameter]]:
"""If certain blocks should use certain parameters this method can be overridden."""
params = [self.ordered_parameters[i] for i in indices]
return params
@property
def num_parameters_settable(self):
"""The number of distinct parameters."""
return self.feature_dimension
@property
def paulis(self) -> List[str]:
"""The Pauli strings used in the entanglement of the qubits.
Returns:
The Pauli strings as list.
"""
return self._paulis
@paulis.setter
def paulis(self, paulis: List[str]) -> None:
"""Set the pauli strings.
Args:
paulis: The new pauli strings.
"""
self._invalidate()
self._paulis = paulis
@property
def alpha(self) -> float:
"""The Pauli rotation factor (alpha).
Returns:
The Pauli rotation factor.
"""
return self._alpha
@alpha.setter
def alpha(self, alpha: float) -> None:
"""Set the Pauli rotation factor (alpha).
Args:
alpha: Pauli rotation factor
"""
self._invalidate()
self._alpha = alpha
@property
def entanglement_blocks(self):
return [self.pauli_block(pauli) for pauli in self._paulis]
@entanglement_blocks.setter
def entanglement_blocks(self, entanglement_blocks):
self._entanglement_blocks = entanglement_blocks
@property
def feature_dimension(self) -> int:
"""Returns the feature dimension (which is equal to the number of qubits).
Returns:
The feature dimension of this feature map.
"""
return self.num_qubits
@feature_dimension.setter
def feature_dimension(self, feature_dimension: int) -> None:
"""Set the feature dimension.
Args:
feature_dimension: The new feature dimension.
"""
self.num_qubits = feature_dimension
def _extract_data_for_rotation(self, pauli, x):
where_non_i = np.where(np.asarray(list(pauli[::-1])) != "I")[0]
x = np.asarray(x)
return x[where_non_i]
[docs] def pauli_block(self, pauli_string):
"""Get the Pauli block for the feature map circuit."""
params = ParameterVector("_", length=len(pauli_string))
time = self._data_map_func(np.asarray(params))
return self.pauli_evolution(pauli_string, time)
[docs] def pauli_evolution(self, pauli_string, time):
"""Get the evolution block for the given pauli string."""
# for some reason this is in reversed order
pauli_string = pauli_string[::-1]
# trim the pauli string if identities are included
trimmed = []
indices = []
for i, pauli in enumerate(pauli_string):
if pauli != "I":
trimmed += [pauli]
indices += [i]
evo = QuantumCircuit(len(pauli_string))
if len(trimmed) == 0:
return evo
def basis_change(circuit, inverse=False):
for i, pauli in enumerate(pauli_string):
if pauli == "X":
circuit.h(i)
elif pauli == "Y":
circuit.rx(-np.pi / 2 if inverse else np.pi / 2, i)
def cx_chain(circuit, inverse=False):
num_cx = len(indices) - 1
for i in reversed(range(num_cx)) if inverse else range(num_cx):
circuit.cx(indices[i], indices[i + 1])
basis_change(evo)
cx_chain(evo)
evo.p(self.alpha * time, indices[-1])
cx_chain(evo, inverse=True)
basis_change(evo, inverse=True)
return evo
def self_product(x: np.ndarray) -> float:
"""
Define a function map from R^n to R.
Args:
x: data
Returns:
float: the mapped value
"""
coeff = x[0] if len(x) == 1 else reduce(lambda m, n: m * n, np.pi - x)
return coeff