# Copyright 2018-2021 Xanadu Quantum Technologies Inc.
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
# http://www.apache.org/licenses/LICENSE-2.0
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
r"""
The default.qubit device is PennyLane's standard qubit-based device.
It implements the necessary :class:`~pennylane._device.Device` methods as well as some built-in
:mod:`qubit operations <pennylane.ops.qubit>`, and provides a very simple pure state
simulation of a qubit-based quantum circuit architecture.
"""
import itertools
import functools
from string import ascii_letters as ABC
import numpy as np
from scipy.sparse import coo_matrix
from pennylane import QubitDevice, DeviceError, QubitStateVector, BasisState
from pennylane.operation import DiagonalOperation
from pennylane.wires import WireError
from .._version import __version__
ABC_ARRAY = np.array(list(ABC))
# tolerance for numerical errors
tolerance = 1e-10
SQRT2INV = 1 / np.sqrt(2)
TPHASE = np.exp(1j * np.pi / 4)
def _get_slice(index, axis, num_axes):
"""Allows slicing along an arbitrary axis of an array or tensor.
Args:
index (int): the index to access
axis (int): the axis to slice into
num_axes (int): total number of axes
Returns:
tuple[slice or int]: a tuple that can be used to slice into an array or tensor
**Example:**
Accessing the 2 index along axis 1 of a 3-axis array:
>>> sl = _get_slice(2, 1, 3)
>>> sl
(slice(None, None, None), 2, slice(None, None, None))
>>> a = np.arange(27).reshape((3, 3, 3))
>>> a[sl]
array([[ 6, 7, 8],
[15, 16, 17],
[24, 25, 26]])
"""
idx = [slice(None)] * num_axes
idx[axis] = index
return tuple(idx)
# pylint: disable=unused-argument
[docs]class DefaultQubit(QubitDevice):
"""Default qubit device for PennyLane.
Args:
wires (int, Iterable[Number, str]): Number of subsystems represented by the device,
or iterable that contains unique labels for the subsystems as numbers (i.e., ``[-1, 0, 2]``)
or strings (``['ancilla', 'q1', 'q2']``). Default 1 if not specified.
shots (None, int): How many times the circuit should be evaluated (or sampled) to estimate
the expectation values. Defaults to ``None`` if not specified, which means that the device
returns analytical results.
cache (int): Number of device executions to store in a cache to speed up subsequent
executions. A value of ``0`` indicates that no caching will take place. Once filled,
older elements of the cache are removed and replaced with the most recent device
executions to keep the cache up to date.
"""
name = "Default qubit PennyLane plugin"
short_name = "default.qubit"
pennylane_requires = __version__
version = __version__
author = "Xanadu Inc."
operations = {
"BasisState",
"QubitStateVector",
"QubitUnitary",
"ControlledQubitUnitary",
"MultiControlledX",
"DiagonalQubitUnitary",
"PauliX",
"PauliY",
"PauliZ",
"MultiRZ",
"Hadamard",
"S",
"T",
"SX",
"CNOT",
"SWAP",
"ISWAP",
"CSWAP",
"Toffoli",
"CY",
"CZ",
"PhaseShift",
"ControlledPhaseShift",
"CPhase",
"RX",
"RY",
"RZ",
"Rot",
"CRX",
"CRY",
"CRZ",
"CRot",
"QFT",
"IsingXX",
"IsingYY",
"IsingZZ",
"SingleExcitation",
"SingleExcitationPlus",
"SingleExcitationMinus",
"DoubleExcitation",
"DoubleExcitationPlus",
"DoubleExcitationMinus",
"QubitCarry",
"QubitSum",
}
observables = {
"PauliX",
"PauliY",
"PauliZ",
"Hadamard",
"Hermitian",
"Identity",
"Projector",
"SparseHamiltonian",
}
def __init__(self, wires, *, shots=None, cache=0, analytic=None):
super().__init__(wires, shots, cache=cache, analytic=analytic)
# Create the initial state. Internally, we store the
# state as an array of dimension [2]*wires.
self._state = self._create_basis_state(0)
self._pre_rotated_state = self._state
self._apply_ops = {
"PauliX": self._apply_x,
"PauliY": self._apply_y,
"PauliZ": self._apply_z,
"Hadamard": self._apply_hadamard,
"S": self._apply_s,
"T": self._apply_t,
"SX": self._apply_sx,
"CNOT": self._apply_cnot,
"SWAP": self._apply_swap,
"CZ": self._apply_cz,
"Toffoli": self._apply_toffoli,
}
[docs] @functools.lru_cache()
def map_wires(self, wires):
# temporarily overwrite this method to bypass
# wire map that produces Wires objects
try:
mapped_wires = [self.wire_map[w] for w in wires]
except KeyError as e:
raise WireError(
"Did not find some of the wires {} on device with wires {}.".format(
wires.labels, self.wires.labels
)
) from e
return mapped_wires
[docs] def define_wire_map(self, wires):
# temporarily overwrite this method to bypass
# wire map that produces Wires objects
consecutive_wires = range(self.num_wires)
wire_map = zip(wires, consecutive_wires)
return dict(wire_map)
[docs] def apply(self, operations, rotations=None, **kwargs):
rotations = rotations or []
# apply the circuit operations
for i, operation in enumerate(operations):
if i > 0 and isinstance(operation, (QubitStateVector, BasisState)):
raise DeviceError(
"Operation {} cannot be used after other Operations have already been applied "
"on a {} device.".format(operation.name, self.short_name)
)
if isinstance(operation, QubitStateVector):
self._apply_state_vector(operation.parameters[0], operation.wires)
elif isinstance(operation, BasisState):
self._apply_basis_state(operation.parameters[0], operation.wires)
else:
self._state = self._apply_operation(self._state, operation)
# store the pre-rotated state
self._pre_rotated_state = self._state
# apply the circuit rotations
for operation in rotations:
self._state = self._apply_operation(self._state, operation)
def _apply_operation(self, state, operation):
"""Applies operations to the input state.
Args:
state (array[complex]): input state
operation (~.Operation): operation to apply on the device
Returns:
array[complex]: output state
"""
wires = operation.wires
if operation.base_name in self._apply_ops:
axes = self.wires.indices(wires)
return self._apply_ops[operation.base_name](state, axes, inverse=operation.inverse)
matrix = self._get_unitary_matrix(operation)
if isinstance(operation, DiagonalOperation):
return self._apply_diagonal_unitary(state, matrix, wires)
if len(wires) <= 2:
# Einsum is faster for small gates
return self._apply_unitary_einsum(state, matrix, wires)
return self._apply_unitary(state, matrix, wires)
def _apply_x(self, state, axes, **kwargs):
"""Applies a PauliX gate by rolling 1 unit along the axis specified in ``axes``.
Rolling by 1 unit along the axis means that the :math:`|0 \rangle` state with index ``0`` is
shifted to the :math:`|1 \rangle` state with index ``1``. Likewise, since rolling beyond
the last index loops back to the first, :math:`|1 \rangle` is transformed to
:math:`|0\rangle`.
Args:
state (array[complex]): input state
axes (List[int]): target axes to apply transformation
Returns:
array[complex]: output state
"""
return self._roll(state, 1, axes[0])
def _apply_y(self, state, axes, **kwargs):
"""Applies a PauliY gate by adding a negative sign to the 1 index along the axis specified
in ``axes``, rolling one unit along the same axis, and multiplying the result by 1j.
Args:
state (array[complex]): input state
axes (List[int]): target axes to apply transformation
Returns:
array[complex]: output state
"""
return 1j * self._apply_x(self._apply_z(state, axes), axes)
def _apply_z(self, state, axes, **kwargs):
"""Applies a PauliZ gate by adding a negative sign to the 1 index along the axis specified
in ``axes``.
Args:
state (array[complex]): input state
axes (List[int]): target axes to apply transformation
Returns:
array[complex]: output state
"""
return self._apply_phase(state, axes, -1)
def _apply_hadamard(self, state, axes, **kwargs):
"""Apply the Hadamard gate by combining the results of applying the PauliX and PauliZ gates.
Args:
state (array[complex]): input state
axes (List[int]): target axes to apply transformation
Returns:
array[complex]: output state
"""
state_x = self._apply_x(state, axes)
state_z = self._apply_z(state, axes)
return SQRT2INV * (state_x + state_z)
def _apply_s(self, state, axes, inverse=False):
return self._apply_phase(state, axes, 1j, inverse)
def _apply_t(self, state, axes, inverse=False):
return self._apply_phase(state, axes, TPHASE, inverse)
def _apply_sx(self, state, axes, inverse=False):
"""Apply the Square Root X gate.
Args:
state (array[complex]): input state
axes (List[int]): target axes to apply transformation
Returns:
array[complex]: output state
"""
if inverse:
return 0.5 * ((1 - 1j) * state + (1 + 1j) * self._apply_x(state, axes))
return 0.5 * ((1 + 1j) * state + (1 - 1j) * self._apply_x(state, axes))
def _apply_cnot(self, state, axes, **kwargs):
"""Applies a CNOT gate by slicing along the first axis specified in ``axes`` and then
applying an X transformation along the second axis.
By slicing along the first axis, we are able to select all of the amplitudes with a
corresponding :math:`|1\rangle` for the control qubit. This means we then just need to apply
a :class:`~.PauliX` (NOT) gate to the result.
Args:
state (array[complex]): input state
axes (List[int]): target axes to apply transformation
Returns:
array[complex]: output state
"""
sl_0 = _get_slice(0, axes[0], self.num_wires)
sl_1 = _get_slice(1, axes[0], self.num_wires)
# We will be slicing into the state according to state[sl_1], giving us all of the
# amplitudes with a |1> for the control qubit. The resulting array has lost an axis
# relative to state and we need to be careful about the axis we apply the PauliX rotation
# to. If axes[1] is larger than axes[0], then we need to shift the target axis down by
# one, otherwise we can leave as-is. For example: a state has [0, 1, 2, 3], control=1,
# target=3. Then, state[sl_1] has 3 axes and target=3 now corresponds to the second axis.
if axes[1] > axes[0]:
target_axes = [axes[1] - 1]
else:
target_axes = [axes[1]]
state_x = self._apply_x(state[sl_1], axes=target_axes)
return self._stack([state[sl_0], state_x], axis=axes[0])
def _apply_toffoli(self, state, axes, **kwargs):
"""Applies a Toffoli gate by slicing along the axis of the greater control qubit, slicing
each of the resulting sub-arrays along the axis of the smaller control qubit, and then applying
an X transformation along the axis of the target qubit of the fourth sub-sub-array.
By performing two consecutive slices in this way, we are able to select all of the amplitudes with
a corresponding :math:`|11\rangle` for the two control qubits. This means we then just need to apply
a :class:`~.PauliX` (NOT) gate to the result.
Args:
state (array[complex]): input state
axes (List[int]): target axes to apply transformation
Returns:
array[complex]: output state
"""
cntrl_max = np.argmax(axes[:2])
cntrl_min = cntrl_max ^ 1
sl_a0 = _get_slice(0, axes[cntrl_max], self.num_wires)
sl_a1 = _get_slice(1, axes[cntrl_max], self.num_wires)
sl_b0 = _get_slice(0, axes[cntrl_min], self.num_wires - 1)
sl_b1 = _get_slice(1, axes[cntrl_min], self.num_wires - 1)
# If both controls are smaller than the target, shift the target axis down by two. If one
# control is greater and one control is smaller than the target, shift the target axis
# down by one. If both controls are greater than the target, leave the target axis as-is.
if axes[cntrl_min] > axes[2]:
target_axes = [axes[2]]
elif axes[cntrl_max] > axes[2]:
target_axes = [axes[2] - 1]
else:
target_axes = [axes[2] - 2]
# state[sl_a1][sl_b1] gives us all of the amplitudes with a |11> for the two control qubits.
state_x = self._apply_x(state[sl_a1][sl_b1], axes=target_axes)
state_stacked_a1 = self._stack([state[sl_a1][sl_b0], state_x], axis=axes[cntrl_min])
return self._stack([state[sl_a0], state_stacked_a1], axis=axes[cntrl_max])
def _apply_swap(self, state, axes, **kwargs):
"""Applies a SWAP gate by performing a partial transposition along the specified axes.
Args:
state (array[complex]): input state
axes (List[int]): target axes to apply transformation
Returns:
array[complex]: output state
"""
all_axes = list(range(len(state.shape)))
all_axes[axes[0]] = axes[1]
all_axes[axes[1]] = axes[0]
return self._transpose(state, all_axes)
def _apply_cz(self, state, axes, **kwargs):
"""Applies a CZ gate by slicing along the first axis specified in ``axes`` and then
applying a Z transformation along the second axis.
Args:
state (array[complex]): input state
axes (List[int]): target axes to apply transformation
Returns:
array[complex]: output state
"""
sl_0 = _get_slice(0, axes[0], self.num_wires)
sl_1 = _get_slice(1, axes[0], self.num_wires)
if axes[1] > axes[0]:
target_axes = [axes[1] - 1]
else:
target_axes = [axes[1]]
state_z = self._apply_z(state[sl_1], axes=target_axes)
return self._stack([state[sl_0], state_z], axis=axes[0])
def _apply_phase(self, state, axes, parameters, inverse=False):
"""Applies a phase onto the 1 index along the axis specified in ``axes``.
Args:
state (array[complex]): input state
axes (List[int]): target axes to apply transformation
parameters (float): phase to apply
inverse (bool): whether to apply the inverse phase
Returns:
array[complex]: output state
"""
num_wires = len(state.shape)
sl_0 = _get_slice(0, axes[0], num_wires)
sl_1 = _get_slice(1, axes[0], num_wires)
phase = self._conj(parameters) if inverse else parameters
return self._stack([state[sl_0], phase * state[sl_1]], axis=axes[0])
[docs] def expval(self, observable, shot_range=None, bin_size=None):
"""Returns the expectation value of a Hamiltonian observable. When the observable is a
``SparseHamiltonian`` object, the expectation value is computed directly for the full
Hamiltonian, which leads to faster execution.
Args:
observable (~.Observable): a PennyLane observable
shot_range (tuple[int]): 2-tuple of integers specifying the range of samples
to use. If not specified, all samples are used.
bin_size (int): Divides the shot range into bins of size ``bin_size``, and
returns the measurement statistic separately over each bin. If not
provided, the entire shot range is treated as a single bin.
Returns:
float: returns the expectation value of the observable
"""
if observable.name == "SparseHamiltonian":
if self.shots is not None:
raise DeviceError("SparseHamiltonian must be used with shots=None")
if observable.name == "SparseHamiltonian" and self.shots is None:
ev = coo_matrix.dot(
coo_matrix(self._conj(self.state)),
coo_matrix.dot(
observable.matrix, coo_matrix(self.state.reshape(len(self.state), 1))
),
)
return np.real(ev.toarray()[0])
return super().expval(observable, shot_range=shot_range, bin_size=bin_size)
def _get_unitary_matrix(self, unitary): # pylint: disable=no-self-use
"""Return the matrix representing a unitary operation.
Args:
unitary (~.Operation): a PennyLane unitary operation
Returns:
array[complex]: Returns a 2D matrix representation of
the unitary in the computational basis, or, in the case of a diagonal unitary,
a 1D array representing the matrix diagonal.
"""
if isinstance(unitary, DiagonalOperation):
return unitary.eigvals
return unitary.matrix
[docs] @classmethod
def capabilities(cls):
capabilities = super().capabilities().copy()
capabilities.update(
model="qubit",
supports_reversible_diff=True,
supports_inverse_operations=True,
supports_analytic_computation=True,
returns_state=True,
passthru_devices={
"tf": "default.qubit.tf",
"autograd": "default.qubit.autograd",
"jax": "default.qubit.jax",
},
)
return capabilities
def _create_basis_state(self, index):
"""Return a computational basis state over all wires.
Args:
index (int): integer representing the computational basis state
Returns:
array[complex]: complex array of shape ``[2]*self.num_wires``
representing the statevector of the basis state
"""
state = np.zeros(2 ** self.num_wires, dtype=np.complex128)
state[index] = 1
state = self._asarray(state, dtype=self.C_DTYPE)
return self._reshape(state, [2] * self.num_wires)
@property
def state(self):
return self._flatten(self._pre_rotated_state)
[docs] def density_matrix(self, wires):
"""Returns the reduced density matrix of a given set of wires.
Args:
wires (Wires): wires of the reduced system.
Returns:
array[complex]: complex tensor of shape ``(2 ** len(wires), 2 ** len(wires))``
representing the reduced density matrix.
"""
dim = self.num_wires
state = self._pre_rotated_state
# Return the full density matrix by using numpy tensor product
if wires == self.wires:
density_matrix = self._tensordot(state, self._conj(state), 0)
density_matrix = self._reshape(density_matrix, (2 ** len(wires), 2 ** len(wires)))
return density_matrix
complete_system = list(range(0, dim))
traced_system = [x for x in complete_system if x not in wires.labels]
# Return the reduced density matrix by using numpy tensor product
density_matrix = self._tensordot(
state, self._conj(state), axes=(traced_system, traced_system)
)
density_matrix = self._reshape(density_matrix, (2 ** len(wires), 2 ** len(wires)))
return density_matrix
def _apply_state_vector(self, state, device_wires):
"""Initialize the internal state vector in a specified state.
Args:
state (array[complex]): normalized input state of length
``2**len(wires)``
device_wires (Wires): wires that get initialized in the state
"""
# translate to wire labels used by device
device_wires = self.map_wires(device_wires)
state = self._asarray(state, dtype=self.C_DTYPE)
n_state_vector = state.shape[0]
if state.ndim != 1 or n_state_vector != 2 ** len(device_wires):
raise ValueError("State vector must be of length 2**wires.")
if not np.allclose(np.linalg.norm(state, ord=2), 1.0, atol=tolerance):
raise ValueError("Sum of amplitudes-squared does not equal one.")
if len(device_wires) == self.num_wires and sorted(device_wires) == device_wires:
# Initialize the entire wires with the state
self._state = self._reshape(state, [2] * self.num_wires)
return
# generate basis states on subset of qubits via the cartesian product
basis_states = np.array(list(itertools.product([0, 1], repeat=len(device_wires))))
# get basis states to alter on full set of qubits
unravelled_indices = np.zeros((2 ** len(device_wires), self.num_wires), dtype=int)
unravelled_indices[:, device_wires] = basis_states
# get indices for which the state is changed to input state vector elements
ravelled_indices = np.ravel_multi_index(unravelled_indices.T, [2] * self.num_wires)
state = self._scatter(ravelled_indices, state, [2 ** self.num_wires])
state = self._reshape(state, [2] * self.num_wires)
self._state = self._asarray(state, dtype=self.C_DTYPE)
def _apply_basis_state(self, state, wires):
"""Initialize the state vector in a specified computational basis state.
Args:
state (array[int]): computational basis state of shape ``(wires,)``
consisting of 0s and 1s.
wires (Wires): wires that the provided computational state should be initialized on
"""
# translate to wire labels used by device
device_wires = self.map_wires(wires)
# length of basis state parameter
n_basis_state = len(state)
if not set(state.tolist()).issubset({0, 1}):
raise ValueError("BasisState parameter must consist of 0 or 1 integers.")
if n_basis_state != len(device_wires):
raise ValueError("BasisState parameter and wires must be of equal length.")
# get computational basis state number
basis_states = 2 ** (self.num_wires - 1 - np.array(device_wires))
num = int(np.dot(state, basis_states))
self._state = self._create_basis_state(num)
def _apply_unitary(self, state, mat, wires):
r"""Apply multiplication of a matrix to subsystems of the quantum state.
Args:
state (array[complex]): input state
mat (array): matrix to multiply
wires (Wires): target wires
Returns:
array[complex]: output state
"""
# translate to wire labels used by device
device_wires = self.map_wires(wires)
mat = self._cast(self._reshape(mat, [2] * len(device_wires) * 2), dtype=self.C_DTYPE)
axes = (np.arange(len(device_wires), 2 * len(device_wires)), device_wires)
tdot = self._tensordot(mat, state, axes=axes)
# tensordot causes the axes given in `wires` to end up in the first positions
# of the resulting tensor. This corresponds to a (partial) transpose of
# the correct output state
# We'll need to invert this permutation to put the indices in the correct place
unused_idxs = [idx for idx in range(self.num_wires) if idx not in device_wires]
perm = list(device_wires) + unused_idxs
inv_perm = np.argsort(perm) # argsort gives inverse permutation
return self._transpose(tdot, inv_perm)
def _apply_unitary_einsum(self, state, mat, wires):
r"""Apply multiplication of a matrix to subsystems of the quantum state.
This function uses einsum instead of tensordot. This approach is only
faster for single- and two-qubit gates.
Args:
state (array[complex]): input state
mat (array): matrix to multiply
wires (Wires): target wires
Returns:
array[complex]: output state
"""
# translate to wire labels used by device
device_wires = self.map_wires(wires)
mat = self._cast(self._reshape(mat, [2] * len(device_wires) * 2), dtype=self.C_DTYPE)
# Tensor indices of the quantum state
state_indices = ABC[: self.num_wires]
# Indices of the quantum state affected by this operation
affected_indices = "".join(ABC_ARRAY[list(device_wires)].tolist())
# All affected indices will be summed over, so we need the same number of new indices
new_indices = ABC[self.num_wires : self.num_wires + len(device_wires)]
# The new indices of the state are given by the old ones with the affected indices
# replaced by the new_indices
new_state_indices = functools.reduce(
lambda old_string, idx_pair: old_string.replace(idx_pair[0], idx_pair[1]),
zip(affected_indices, new_indices),
state_indices,
)
# We now put together the indices in the notation numpy's einsum requires
einsum_indices = (
"{new_indices}{affected_indices},{state_indices}->{new_state_indices}".format(
affected_indices=affected_indices,
state_indices=state_indices,
new_indices=new_indices,
new_state_indices=new_state_indices,
)
)
return self._einsum(einsum_indices, mat, state)
def _apply_diagonal_unitary(self, state, phases, wires):
r"""Apply multiplication of a phase vector to subsystems of the quantum state.
This represents the multiplication with diagonal gates in a more efficient manner.
Args:
state (array[complex]): input state
phases (array): vector to multiply
wires (Wires): target wires
Returns:
array[complex]: output state
"""
# translate to wire labels used by device
device_wires = self.map_wires(wires)
# reshape vectors
phases = self._cast(self._reshape(phases, [2] * len(device_wires)), dtype=self.C_DTYPE)
state_indices = ABC[: self.num_wires]
affected_indices = "".join(ABC_ARRAY[list(device_wires)].tolist())
einsum_indices = "{affected_indices},{state_indices}->{state_indices}".format(
affected_indices=affected_indices, state_indices=state_indices
)
return self._einsum(einsum_indices, phases, state)
[docs] def reset(self):
"""Reset the device"""
super().reset()
# init the state vector to |00..0>
self._state = self._create_basis_state(0)
self._pre_rotated_state = self._state
[docs] def analytic_probability(self, wires=None):
if self._state is None:
return None
prob = self.marginal_prob(self._abs(self._flatten(self._state)) ** 2, wires)
return prob
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