# Source code for pennylane.utils

# Copyright 2018-2021 Xanadu Quantum Technologies Inc.

# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at

# Unless required by applicable law or agreed to in writing, software
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
"""
This module contains utilities and auxiliary functions which are shared
across the PennyLane submodules.
"""
# pylint: disable=protected-access,too-many-branches
from collections.abc import Iterable
import functools
import inspect
import itertools
import numbers
import warnings
from operator import matmul

import numpy as np
import scipy

import pennylane as qml

def __getattr__(name):
# for more information on overwriting __getattr__, see https://peps.python.org/pep-0562/
if name == "expand":
warning_string = (
"qml.utils.expand is deprecated; using qml.operation.expand_matrix instead."
)
warnings.warn(warning_string, UserWarning)
return qml.operation.expand_matrix
try:
return globals()[name]
except KeyError as e:
raise AttributeError from e

[docs]def decompose_hamiltonian(H, hide_identity=False, wire_order=None):
r"""Decomposes a Hermitian matrix into a linear combination of Pauli operators.

Args:
H (array[complex]): a Hermitian matrix of dimension :math:2^n\times 2^n
hide_identity (bool): does not include the :class:~.Identity observable within
the tensor products of the decomposition if True

Returns:
tuple[list[float], list[~.Observable]]: a list of coefficients and a list
of corresponding tensor products of Pauli observables that decompose the Hamiltonian.

**Example:**

We can use this function to compute the Pauli operator decomposition of an arbitrary Hermitian
matrix:

>>> A = np.array(
... [[-2, -2+1j, -2, -2], [-2-1j,  0,  0, -1], [-2,  0, -2, -1], [-2, -1, -1,  0]])
>>> coeffs, obs_list = decompose_hamiltonian(A)
>>> coeffs
[-1.0, -1.5, -0.5, -1.0, -1.5, -1.0, -0.5, 1.0, -0.5, -0.5]

We can use the output coefficients and tensor Pauli terms to construct a :class:~.Hamiltonian:

>>> H = qml.Hamiltonian(coeffs, obs_list)
>>> print(H)
(-1.0) [I0 I1]
+ (-1.5) [X1]
+ (-0.5) [Y1]
+ (-1.0) [Z1]
+ (-1.5) [X0]
+ (-1.0) [X0 X1]
+ (-0.5) [X0 Z1]
+ (1.0) [Y0 Y1]
+ (-0.5) [Z0 X1]
+ (-0.5) [Z0 Y1]

This Hamiltonian can then be used in defining VQE problems using :class:~.ExpvalCost.
"""
n = int(np.log2(len(H)))
N = 2**n

if wire_order is None:
wire_order = range(n)

if H.shape != (N, N):
raise ValueError(
"The Hamiltonian should have shape (2**n, 2**n), for any qubit number n>=1"
)

if not np.allclose(H, H.conj().T):
raise ValueError("The Hamiltonian is not Hermitian")

paulis = [qml.Identity, qml.PauliX, qml.PauliY, qml.PauliZ]
obs = []
coeffs = []

for term in itertools.product(paulis, repeat=n):
matrices = [i.compute_matrix() for i in term]
coeff = np.trace(functools.reduce(np.kron, matrices) @ H) / N
coeff = np.real_if_close(coeff).item()

if not np.allclose(coeff, 0):
coeffs.append(coeff)

if not all(t is qml.Identity for t in term) and hide_identity:
obs.append(
functools.reduce(
matmul,
[t(i) for i, t in zip(wire_order, term) if t is not qml.Identity],
)
)
else:
obs.append(functools.reduce(matmul, [t(i) for i, t in enumerate(term)]))

return coeffs, obs

[docs]def sparse_hamiltonian(H, wires=None):
r"""Computes the sparse matrix representation a Hamiltonian in the computational basis.

Args:
H (~.Hamiltonian): Hamiltonian operator for which the matrix representation should be
computed
wires (Iterable): Wire labels that indicate the order of wires according to which the matrix
is constructed. If not profided, H.wires is used.

Returns:
csr_matrix: a sparse matrix in scipy Compressed Sparse Row (CSR) format with dimension
:math:(2^n, 2^n), where :math:n is the number of wires

**Example:**

This function can be used by passing a qml.Hamiltonian object as:

>>> coeffs = [1, -0.45]
>>> obs = [qml.PauliZ(0) @ qml.PauliZ(1), qml.PauliY(0) @ qml.PauliZ(1)]
>>> H = qml.Hamiltonian(coeffs, obs)
>>> H_sparse = sparse_hamiltonian(H)
>>> H_sparse
<4x4 sparse matrix of type '<class 'numpy.complex128'>'
with 2 stored elements in COOrdinate format>

The resulting sparse matrix can be either used directly or transformed into a numpy array:

>>> H_sparse.toarray()
array([[ 1.+0.j  ,  0.+0.j  ,  0.+0.45j,  0.+0.j  ],
[ 0.+0.j  , -1.+0.j  ,  0.+0.j  ,  0.-0.45j],
[ 0.-0.45j,  0.+0.j  , -1.+0.j  ,  0.+0.j  ],
[ 0.+0.j  ,  0.+0.45j,  0.+0.j  ,  1.+0.j  ]])
"""
if not isinstance(H, qml.Hamiltonian):
raise TypeError("Passed Hamiltonian must be of type qml.Hamiltonian")

if wires is None:
wires = H.wires
else:
wires = qml.wires.Wires(wires)

n = len(wires)
matrix = scipy.sparse.csr_matrix((2**n, 2**n), dtype="complex128")

coeffs = qml.math.toarray(H.data)

temp_mats = []
for coeff, op in zip(coeffs, H.ops):
obs = []
for o in qml.operation.Tensor(op).obs:
if len(o.wires) > 1:
# todo: deal with operations created from multi-qubit operations such as Hermitian
raise ValueError(
f"Can only sparsify Hamiltonians whose constituent observables consist of "
f"(tensor products of) single-qubit operators; got {op}."
)
obs.append(o.matrix())

# Array to store the single-wire observables which will be Kronecker producted together
mat = []
# i_count tracks the number of consecutive single-wire identity matrices encountered
# in order to avoid unnecessary Kronecker products, since I_n x I_m = I_{n+m}
i_count = 0
for wire_lab in wires:
if wire_lab in op.wires:
if i_count > 0:
mat.append(scipy.sparse.eye(2**i_count, format="coo"))
i_count = 0
idx = op.wires.index(wire_lab)
# obs is an array storing the single-wire observables which
# make up the full Hamiltonian term
sp_obs = scipy.sparse.coo_matrix(obs[idx])
mat.append(sp_obs)
else:
i_count += 1

if i_count > 0:
mat.append(scipy.sparse.eye(2**i_count, format="coo"))

red_mat = functools.reduce(lambda i, j: scipy.sparse.kron(i, j, format="coo"), mat) * coeff

temp_mats.append(red_mat.tocsr())
# Value of 100 arrived at empirically to balance time savings vs memory use. At this point
# the temp_mats are summed into the final result and the temporary storage array is
# cleared.
if (len(temp_mats) % 100) == 0:
matrix += sum(temp_mats)
temp_mats = []

matrix += sum(temp_mats)
return matrix

def _flatten(x):
"""Iterate recursively through an arbitrarily nested structure in depth-first order.

See also :func:_unflatten.

Args:
x (array, Iterable, Any): each element of an array or an Iterable may itself be any of these types

Yields:
Any: elements of x in depth-first order
"""
if isinstance(x, np.ndarray):
yield from _flatten(x.flat)  # should we allow object arrays? or just "yield from x.flat"?
elif isinstance(x, qml.wires.Wires):
# Reursive calls to flatten Wires will cause infinite recursion (Wires atoms are Wires).
# Since Wires are always flat, just yield.
for item in x:
yield item
elif isinstance(x, Iterable) and not isinstance(x, (str, bytes)):
for item in x:
yield from _flatten(item)
else:
yield x

def _unflatten(flat, model):
"""Restores an arbitrary nested structure to a flattened iterable.

See also :func:_flatten.

Args:
flat (array): 1D array of items
model (array, Iterable, Number): model nested structure

Raises:
TypeError: if model contains an object of unsupported type

Returns:
Union[array, list, Any], array: first elements of flat arranged into the nested
structure of model, unused elements of flat
"""
if isinstance(model, (numbers.Number, str)):
return flat[0], flat[1:]

if isinstance(model, np.ndarray):
idx = model.size
res = np.array(flat)[:idx].reshape(model.shape)
return res, flat[idx:]

if isinstance(model, Iterable):
res = []
for x in model:
val, flat = _unflatten(flat, x)
res.append(val)
return res, flat

raise TypeError(f"Unsupported type in the model: {type(model)}")

[docs]def unflatten(flat, model):
"""Wrapper for :func:_unflatten.

Args:
flat (array): 1D array of items
model (array, Iterable, Number): model nested structure

Raises:
ValueError: if flat has more elements than model
"""
# pylint:disable=len-as-condition
res, tail = _unflatten(np.asarray(flat), model)
if len(tail) != 0:
raise ValueError("Flattened iterable has more elements than the model.")
return res

def _inv_dict(d):
"""Reverse a dictionary mapping.

Returns multimap where the keys are the former values,
and values are sets of the former keys.

Args:
d (dict[a->b]): mapping to reverse

Returns:
dict[b->set[a]]: reversed mapping
"""
ret = {}
for k, v in d.items():
return ret

def _get_default_args(func):
"""Get the default arguments of a function.

Args:
func (callable): a function

Returns:
dict[str, tuple]: mapping from argument name to (positional idx, default value)
"""
signature = inspect.signature(func)
return {
k: (idx, v.default)
for idx, (k, v) in enumerate(signature.parameters.items())
if v.default is not inspect.Parameter.empty
}

[docs]@functools.lru_cache()
def pauli_eigs(n):
r"""Eigenvalues for :math:A^{\otimes n}, where :math:A is
Pauli operator, or shares its eigenvalues.

As an example if n==2, then the eigenvalues of a tensor product consisting
of two matrices sharing the eigenvalues with Pauli matrices is returned.

Args:
n (int): the number of qubits the matrix acts on
Returns:
list: the eigenvalues of the specified observable
"""
if n == 1:
return np.array([1, -1])
return np.concatenate([pauli_eigs(n - 1), -pauli_eigs(n - 1)])

[docs]def expand_vector(vector, original_wires, expanded_wires):
r"""Expand a vector to more wires.

Args:
vector (array): :math:2^n vector where n = len(original_wires).
original_wires (Sequence[int]): original wires of vector
expanded_wires (Union[Sequence[int], int]): expanded wires of vector, can be shuffled
If a single int m is given, corresponds to list(range(m))

Returns:
array: :math:2^m vector where m = len(expanded_wires).
"""
if isinstance(expanded_wires, numbers.Integral):
expanded_wires = list(range(expanded_wires))

N = len(original_wires)
M = len(expanded_wires)
D = M - N

if not set(expanded_wires).issuperset(original_wires):
raise ValueError("Invalid target subsystems provided in 'original_wires' argument.")

if qml.math.shape(vector) != (2**N,):
raise ValueError("Vector parameter must be of length 2**len(original_wires)")

dims = [2] * N
tensor = qml.math.reshape(vector, dims)

if D > 0:
extra_dims = [2] * D
ones = qml.math.ones(2**D).reshape(extra_dims)
expanded_tensor = qml.math.tensordot(tensor, ones, axes=0)
else:
expanded_tensor = tensor

wire_indices = []
for wire in original_wires:
wire_indices.append(expanded_wires.index(wire))

wire_indices = np.array(wire_indices)

# Order tensor factors according to wires
original_indices = np.array(range(N))
expanded_tensor = qml.math.moveaxis(
expanded_tensor, tuple(original_indices), tuple(wire_indices)
)

return qml.math.reshape(expanded_tensor, 2**M)


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