Source code for pennylane.utils

# 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


# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# See the License for the specific language governing permissions and
# limitations under the License.
This module contains utilities and auxiliary functions which are shared
across the PennyLane submodules.
# pylint: disable=protected-access,too-many-branches
from import Iterable
import functools
import inspect
import itertools
import numbers
from operator import matmul

import numpy as np
import scipy

import pennylane as qml

[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( 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(): ret.setdefault(v, set()).add(k) 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)