def circuit12(params, wires): """Circuit template #12 in 1905.10876 Args: params (array): An array of shapes (4N-4,). wires (Iterable): Wires that the template acts on. """ # Input Checks wires, size = Wires(wires), len(wires) check_shape(params, target_shape=(4 * size - 4, ), msg=f"params must be of shape {(4 * size - 4, )}.") # Define the circuit AngleEmbedding(params[:size], wires, rotation='Y') AngleEmbedding(params[size:2 * size], wires, rotation='Z') pattern = [[i + 1, i] for i in range(0, len(wires) - 1, 2)] qml.broadcast(unitary=qml.CZ, pattern=pattern, wires=wires) AngleEmbedding(params[2 * size:3 * size - 2], wires=wires[1:-1], rotation='Y') AngleEmbedding(params[3 * size - 2:4 * size - 4], wires=wires[1:-1], rotation='Z') pattern = [[i + 1, i] for i in range(1, len(wires) - 1, 2)] qml.broadcast(unitary=qml.CZ, pattern=pattern, wires=wires)
def ArbitraryUnitary(weights, wires): """Implements an arbitrary unitary on the specified wires. An arbitrary unitary on :math:`n` wires is parametrized by :math:`4^n - 1` independent real parameters. This templates uses Pauli word rotations to parametrize the unitary. **Example** ArbitraryUnitary can be used as a building block, e.g. to parametrize arbitrary two-qubit operations in a circuit: .. code-block:: python @qml.template def arbitrary_nearest_neighbour_interaction(weights, wires): qml.broadcast(unitary=ArbitraryUnitary, pattern="double", wires=wires, params=weights) Args: weights (array[float]): The angles of the Pauli word rotations, needs to have length :math:`4^n - 1` where :math:`n` is the number of wires the template acts upon. wires (List[int]): The wires on which the arbitrary unitary acts. """ wires = check_wires(wires) n_wires = len(wires) expected_shape = (4 ** n_wires - 1,) check_shape( weights, expected_shape, msg="'weights' must be of shape {}; got {}." "".format(expected_shape, get_shape(weights)), ) for i, pauli_word in enumerate(_all_pauli_words_but_identity(len(wires))): qml.PauliRot(weights[i], pauli_word, wires=wires)
def _preprocess(weights, initial_layer_weights, wires): """Validate and pre-process inputs as follows: * Check the shapes of the two weights tensors. Args: weights (tensor_like): trainable parameters of the template initial_layer_weights (tensor_like): weight tensor for the initial rotation block, shape ``(M,)`` wires (Wires): wires that template acts on Returns: int: number of times that the ansatz is repeated """ if qml.tape_mode_active(): shape = qml.math.shape(weights) repeat = shape[0] if len(shape) > 1: if shape[1] != len(wires) - 1: raise ValueError( f"Weights tensor must have second dimension of length {len(wires) - 1}; got {shape[1]}" ) if shape[2] != 2: raise ValueError( f"Weights tensor must have third dimension of length 2; got {shape[2]}" ) shape2 = qml.math.shape(initial_layer_weights) if shape2 != (len(wires), ): raise ValueError( f"Initial layer weights must be of shape {(len(wires),)}; got {shape2}" ) else: repeat = check_number_of_layers([weights]) expected_shape_initial = (len(wires), ) check_shape( initial_layer_weights, expected_shape_initial, msg="Initial layer weights must be of shape {}; got {}" "".format(expected_shape_initial, get_shape(initial_layer_weights)), ) if len(wires) in [0, 1]: expected_shape_weights = (0, ) else: expected_shape_weights = (repeat, len(wires) - 1, 2) check_shape( weights, expected_shape_weights, msg="Weights tensor must be of shape {}; got {}" "".format(expected_shape_weights, get_shape(weights)), ) return repeat
def circuit15(params, wires): """Circuit template #15 in 1905.10876 Args: params (array): An array of shapes (2N, ). wires (Iterable): Wires that the template acts on. """ # Input Checks wires, size = Wires(wires), len(wires) check_shape(params, target_shape=(2 * size, ), msg=f"params must be of shape {(2*size,)}.") # Define the circuit AngleEmbedding(params[:size], wires, rotation='Y') pattern = [[i, (i + 1) % len(wires)] for i in reversed(range(len(wires)))] qml.broadcast(unitary=qml.CNOT, pattern=pattern, wires=wires) AngleEmbedding(params[size:], wires, rotation='Y') pattern = [[(i - 1) % len(wires), (i - 2) % len(wires)] for i in range(len(wires))] qml.broadcast(unitary=qml.CNOT, pattern=pattern, wires=wires)
def _preprocess(weights): """Validate and pre-process inputs as follows: * Check that the weights tensor is 2-dimensional. Args: weights (tensor_like): trainable parameters of the template Returns: int: number of times that the ansatz is repeated """ if qml.tape_mode_active(): shape = qml.math.shape(weights) if len(shape) != 2: raise ValueError( f"Weights tensor must be 2-dimensional; got shape {shape}") repeat = shape[0] else: repeat = check_number_of_layers([weights]) n_rots = get_shape(weights)[1] expected_shape = (repeat, n_rots) check_shape( weights, expected_shape, msg="'weights' must be of shape {}; got {}" "".format(expected_shape, get_shape(weights)), ) return repeat
def _preprocess(weight, wires): """Validate and pre-process inputs as follows: * Check the shape of the weights tensor. * Check that there are at least 2 wires. Args: weight (tensor_like): trainable parameters of the template wires (Wires): wires that template acts on """ if len(wires) < 2: raise ValueError("expected at least two wires; got {}".format( len(wires))) if qml.tape_mode_active(): shape = qml.math.shape(weight) if shape != (): raise ValueError( f"Weight must be a scalar tensor {()}; got shape {shape}.") else: expected_shape = () check_shape( weight, expected_shape, msg="Weight must be a scalar; got shape {}".format( expected_shape, get_shape(weight)), )
def _preprocess(weights, wires): """Validate and pre-process inputs as follows: * Check the shape of the weights tensor. Args: weights (tensor_like): trainable parameters of the template wires (Wires): wires that template acts on """ if qml.tape_mode_active(): shape = qml.math.shape(weights) if shape != (4**len(wires) - 1, ): raise ValueError( f"Weights tensor must be of shape {(4 ** len(wires) - 1,)}; got {shape}." ) else: expected_shape = (4**len(wires) - 1, ) check_shape( weights, expected_shape, msg="Weights tensor must be of shape {}; got {}." "".format(expected_shape, get_shape(weights)), )
def circuit06(params, wires): """Circuit template #06 in 1905.10876 Args: params (array): An array of shapes (N^2 + 3N, 1). wires (Iterable): Wires that the template acts on. """ # Input Checks wires, size = Wires(wires), len(wires) check_shape(params, target_shape=(size * (size + 3), ), msg=f"params must be of shape {(size * (size + 3), )}.") # Define the circuit AngleEmbedding(params[:size], wires, rotation='X') AngleEmbedding(params[size:2 * size], wires, rotation='Z') idx_start = 2 * size for cnt, controlled in enumerate(reversed(range(len(wires)))): pattern = [[controlled, j] for j in reversed(range(len(wires))) if j != controlled] qml.broadcast(unitary=qml.CRX, pattern=pattern, wires=wires, parameters=params[idx_start:idx_start + (size - 1)]) idx_start += size - 1 AngleEmbedding(params[idx_start:idx_start + size], wires, rotation='X') AngleEmbedding(params[idx_start + size:idx_start + 2 * size], wires, rotation='Z')
def circuit17(params, wires): """Circuit template #17 in 1905.10876 Args: params (array): An array of shapes (2N + floor(N/2) + floor((N-1)/2), ). wires (Iterable): Wires that the template acts on. """ # Input Checks wires, size = Wires(wires), len(wires) check_shape( params, target_shape=(2 * size + floor(size / 2) + floor((size - 1) / 2), ), msg= f"params must be of shape {(2*size + floor(size/2) + floor((size-1)/2),)}." ) # Define the circuit AngleEmbedding(params[:size], wires, rotation='X') AngleEmbedding(params[size:2 * size], wires, rotation='Z') pattern = [[i + 1, i] for i in range(0, len(wires) - 1, 2)] qml.broadcast(unitary=qml.CRX, pattern=pattern, wires=wires, parameters=params[2 * size:2 * size + floor(size / 2)]) pattern = [[i + 1, i] for i in range(1, len(wires) - 1, 2)] qml.broadcast(unitary=qml.CRX, pattern=pattern, wires=wires, parameters=params[2 * size + floor(size / 2):])
def _preprocess(features, wires): """Validate and pre-process inputs as follows: * Check that the features tensor is one-dimensional. * Check that the first dimension of the features tensor has length :math:`n`, where :math:`n` is the number of qubits. * Check that the entries of the features tensor are zeros and ones. Args: features (tensor_like): input features to pre-process wires (Wires): wires that template acts on Returns: array: numpy array representation of the features tensor """ if qml.tape_mode_active(): shape = qml.math.shape(features) if len(shape) != 1: raise ValueError( f"Features must be one-dimensional; got shape {shape}.") n_features = shape[0] if n_features != len(wires): raise ValueError( f"Features must be of length {len(wires)}; got length {n_features}." ) features = list(qml.math.toarray(features)) if set(features) != {0, 1}: raise ValueError( f"Basis state must only consist of 0s and 1s; got {features}") return features # non-tape mode check_type( features, [Iterable], msg="Features must be iterable; got type {}".format(type(features)), ) expected_shape = (len(wires), ) check_shape( features, expected_shape, msg="Features must be of shape {}; got {}" "".format(expected_shape, get_shape(features)), ) if any([b not in [0, 1] for b in features]): raise ValueError( "Basis state must only consist of 0s and 1s; got {}".format( features)) return features
def _preprocess(weights, wires, init_state): """Validate and pre-process inputs as follows: * Check that the weights tensor has the correct shape. * Extract a wire list for the subroutines of this template. * Cast initial state to a numpy array. Args: weights (tensor_like): trainable parameters of the template wires (Wires): wires that template acts on init_state (tensor_like): shape ``(len(wires),)`` tensor Returns: int, list[Wires], array: number of times that the ansatz is repeated, wires pattern, and preprocessed initial state """ if len(wires) < 2: raise ValueError( "This template requires the number of qubits to be greater than one;" "got a wire sequence with {} elements".format(len(wires))) if qml.tape_mode_active(): shape = qml.math.shape(weights) if len(shape) != 3: raise ValueError( f"Weights tensor must be 3-dimensional; got shape {shape}") if shape[1] != len(wires) - 1: raise ValueError( f"Weights tensor must have second dimension of length {len(wires) - 1}; got {shape[1]}" ) if shape[2] != 2: raise ValueError( f"Weights tensor must have third dimension of length 2; got {shape[2]}" ) repeat = shape[0] else: repeat = get_shape(weights)[0] expected_shape = (repeat, len(wires) - 1, 2) check_shape( weights, expected_shape, msg="Weights tensor must be of shape {}; got {}".format( expected_shape, get_shape(weights)), ) nm_wires = [wires.subset([l, l + 1]) for l in range(0, len(wires) - 1, 2)] nm_wires += [wires.subset([l, l + 1]) for l in range(1, len(wires) - 1, 2)] # we can extract the numpy representation here # since init_state can never be differentiable init_state = qml.math.toarray(init_state) return repeat, nm_wires, init_state
def AngleEmbedding(features, wires, rotation="X"): r""" Encodes :math:`N` features into the rotation angles of :math:`n` qubits, where :math:`N \leq n`. The rotations can be chosen as either :class:`~pennylane.ops.RX`, :class:`~pennylane.ops.RY` or :class:`~pennylane.ops.RZ` gates, as defined by the ``rotation`` parameter: * ``rotation='X'`` uses the features as angles of RX rotations * ``rotation='Y'`` uses the features as angles of RY rotations * ``rotation='Z'`` uses the features as angles of RZ rotations The length of ``features`` has to be smaller or equal to the number of qubits. If there are fewer entries in ``features`` than rotations, the circuit does not apply the remaining rotation gates. Args: features (array): input array of shape ``(N,)``, where N is the number of input features to embed, with :math:`N\leq n` wires (Iterable or Wires): Wires that the template acts on. Accepts an iterable of numbers or strings, or a Wires object. rotation (str): Type of rotations used Raises: ValueError: if inputs do not have the correct format """ ############# # Input checks wires = Wires(wires) check_shape( features, (len(wires),), bound="max", msg="'features' must be of shape {} or smaller; " "got {}.".format((len(wires),), get_shape(features)), ) check_type(rotation, [str], msg="'rotation' must be a string; got {}".format(rotation)) check_is_in_options( rotation, ["X", "Y", "Z"], msg="did not recognize option {} for 'rotation'.".format(rotation), ) ############### if rotation == "X": broadcast(unitary=RX, pattern="single", wires=wires, parameters=features) elif rotation == "Y": broadcast(unitary=RY, pattern="single", wires=wires, parameters=features) elif rotation == "Z": broadcast(unitary=RZ, pattern="single", wires=wires, parameters=features)
def BasisStatePreparation(basis_state, wires): r""" Prepares a basis state on the given wires using a sequence of Pauli X gates. .. warning:: ``basis_state`` influences the circuit architecture and is therefore incompatible with gradient computations. Ensure that ``basis_state`` is not passed to the qnode by positional arguments. Args: basis_state (array): Input array of shape ``(N,)``, where N is the number of wires the state preparation acts on. ``N`` must be smaller or equal to the total number of wires of the device. wires (Iterable or Wires): Wires that the template acts on. Accepts an iterable of numbers or strings, or a Wires object. Raises: ValueError: if inputs do not have the correct format """ ###################### # Input checks wires = Wires(wires) expected_shape = (len(wires),) check_shape( basis_state, expected_shape, msg=" 'basis_state' must be of shape {}; got {}." "".format(expected_shape, get_shape(basis_state)), ) # basis_state cannot be trainable check_no_variable( basis_state, msg="'basis_state' cannot be differentiable; must be passed as a keyword argument " "to the quantum node", ) # basis_state is guaranteed to be a list of binary values if any([b not in [0, 1] for b in basis_state]): raise ValueError( "'basis_state' must only contain values of 0 and 1; got {}".format(basis_state) ) ###################### wires = wires.tolist() # TODO: remove when operator takes Wires object for wire, state in zip(wires, basis_state): if state == 1: qml.PauliX(wire)
def _preprocess(weights, wires, ranges): """Validate and pre-process inputs as follows: * Check the shape of the weights tensor. * If ranges is None, define a default. Args: weights (tensor_like): trainable parameters of the template wires (Wires): wires that template acts on ranges (Sequence[int]): range for each subsequent layer Returns: int, list[int]: number of times that the ansatz is repeated and preprocessed ranges """ if qml.tape_mode_active(): shape = qml.math.shape(weights) repeat = shape[0] if len(shape) != 3: raise ValueError( f"Weights tensor must be 3-dimensional; got shape {shape}") if shape[1] != len(wires): raise ValueError( f"Weights tensor must have second dimension of length {len(wires)}; got {shape[1]}" ) if shape[2] != 3: raise ValueError( f"Weights tensor must have third dimension of length 3; got {shape[2]}" ) else: repeat = check_number_of_layers([weights]) expected_shape = (repeat, len(wires), 3) check_shape( weights, expected_shape, msg="Weights tensor must be of shape {}; got {}" "".format(expected_shape, get_shape(weights)), ) if len(wires) > 1: if ranges is None: # tile ranges with iterations of range(1, n_wires) ranges = [(l % (len(wires) - 1)) + 1 for l in range(repeat)] else: ranges = [0] * repeat return repeat, ranges
def BasisEmbedding(features, wires): r"""Encodes :math:`n` binary features into a basis state of :math:`n` qubits. For example, for ``features=np.array([0, 1, 0])``, the quantum system will be prepared in state :math:`|010 \rangle`. .. warning:: ``BasisEmbedding`` calls a circuit whose architecture depends on the binary features. The ``features`` argument is therefore not differentiable when using the template, and gradients with respect to the argument cannot be computed by PennyLane. Args: features (array): binary input array of shape ``(n, )`` wires (Iterable or Wires): Wires that the template acts on. Accepts an iterable of numbers or strings, or a Wires object. Raises: ValueError: if inputs do not have the correct format """ ############# # Input checks wires = Wires(wires) check_type(features, [Iterable], msg="'features' must be iterable; got type {}".format( type(features))) expected_shape = (len(wires), ) check_shape( features, expected_shape, msg="'features' must be of shape {}; got {}" "".format(expected_shape, get_shape(features)), ) if any([b not in [0, 1] for b in features]): raise ValueError( "'basis_state' must only consist of 0s and 1s; got {}".format( features)) ############### wires = wires.tolist() # TODO: Remove when operators take Wires objects for wire, bit in zip(wires, features): if bit == 1: qml.PauliX(wire)
def _preprocess(basis_state, wires): """Validate and pre-process inputs as follows: * Check the shape of the basis state. * Cast basis state to a numpy array. Args: basis_state (tensor_like): basis state to prepare wires (Wires): wires that template acts on Returns: array: preprocessed basis state """ if qml.tape_mode_active(): shape = qml.math.shape(basis_state) if len(shape) != 1: raise ValueError(f"Basis state must be one-dimensional; got shape {shape}.") n_bits = shape[0] if n_bits != len(wires): raise ValueError(f"Basis state must be of length {len(wires)}; got length {n_bits}.") basis_state = list(qml.math.toarray(basis_state)) if not all(bit in [0, 1] for bit in basis_state): raise ValueError(f"Basis state must only consist of 0s and 1s; got {basis_state}") # we return the input as a list of values, since # it is not differentiable return basis_state expected_shape = (len(wires),) check_shape( basis_state, expected_shape, msg="Basis state must be of shape {}; got {}." "".format(expected_shape, get_shape(basis_state)), ) # basis_state is guaranteed to be a list of binary values if any([b not in [0, 1] for b in basis_state]): raise ValueError( "Basis state must only contain values of 0 and 1; got {}".format(basis_state) ) return basis_state
def circuit01(params, wires): """Circuit template #01 in 1905.10876 Args: params (array): Input array of size (2N, ) that corresponds to rotation parameters. wires (Iterable): Wires that the template acts on. """ # Input Checks wires, size = Wires(wires), len(wires) check_shape(params, target_shape=(2 * size,), msg=f"params must be of shape {(2 * size,)}.") # Define the circuit AngleEmbedding(params[:size], wires, rotation='X') AngleEmbedding(params[size:], wires, rotation='Z')
def ArbitraryStatePreparation(weights, wires): """Implements an arbitrary state preparation on the specified wires. An arbitrary state on :math:`n` wires is parametrized by :math:`2^{n+1} - 2` independent real parameters. This templates uses Pauli word rotations to parametrize the unitary. **Example** ArbitraryStatePreparation can be used to train state preparations, for example using a circuit with some measurement observable ``H``: .. code-block:: python dev = qml.device("default.qubit", wires=4) @qml.qnode(dev) def vqe(weights): qml.ArbitraryStatePreparations(weights, wires=[0, 1, 2, 3]) return qml.expval(qml.Hermitian(H, wires=[0, 1, 2, 3])) Args: weights (array[float]): The angles of the Pauli word rotations, needs to have length :math:`2^(n+1) - 2` where :math:`n` is the number of wires the template acts upon. wires (Iterable or Wires): Wires that the template acts on. Accepts an iterable of numbers or strings, or a Wires object. """ wires = Wires(wires) n_wires = len(wires) expected_shape = (2**(n_wires + 1) - 2, ) check_shape( weights, expected_shape, msg="'weights' must be of shape {}; got {}." "".format(expected_shape, get_shape(weights)), ) wires = wires.tolist() # Todo: remove when ops take Wires object for i, pauli_word in enumerate(_state_preparation_pauli_words(len(wires))): qml.PauliRot(weights[i], pauli_word, wires=wires)
def circuit04(params, wires): """Circuit template #04 in 1905.10876 Args: params (array): An array of shapes (3N-1, 1). wires (Iterable): Wires that the template acts on. """ # Input Checks wires, size = Wires(wires), len(wires) check_shape(params, target_shape=(3 * size - 1,), msg=f"params must be of shape {(3 * size - 1,)}.") # Define the circuit AngleEmbedding(params[:size], wires, rotation='X') AngleEmbedding(params[size:2 * size], wires, rotation='Z') pattern = [[i + 1, i] for i in reversed(range(len(wires) - 1))] qml.broadcast(unitary=qml.CRX, pattern=pattern, wires=wires, parameters=params[2 * size:])
def circuit09(params, wires): """Circuit template #09 in 1905.10876 Args: params (array): An array of shapes (N,). wires (Iterable): Wires that the template acts on. """ # Input Checks wires, size = Wires(wires), len(wires) check_shape(params, target_shape=(size,), msg=f"params must be of shape {(size,)}.") # Define the circuit qml.broadcast(unitary=qml.Hadamard, pattern="single", wires=wires) pattern = [[i + 1, i] for i in reversed(range(len(wires) - 1))] qml.broadcast(unitary=qml.CZ, pattern=pattern, wires=wires) AngleEmbedding(params, wires, rotation='X')
def _preprocess(weights, wires): """Validate and pre-process inputs as follows: * Check the shape of the weights tensor, making sure that the second dimension has length :math:`n`, where :math:`n` is the number of qubits. Args: weights (tensor_like): trainable parameters of the template wires (Wires): wires that template acts on Returns: int: number of times that the ansatz is repeated """ if qml.tape_mode_active(): shape = qml.math.shape(weights) repeat = shape[0] if len(shape) != 2: raise ValueError( f"Weights tensor must be 2-dimensional; got shape {shape}") if shape[1] != len(wires): raise ValueError( f"Weights tensor must have second dimension of length {len(wires)}; got {shape[1]}" ) else: repeat = check_number_of_layers([weights]) expected_shape = (repeat, len(wires)) check_shape( weights, expected_shape, msg= f"Weights tensor must have second dimension of length {len(wires)}; got {get_shape(weights)[1]}", ) return repeat
def _preprocess(weight, wires1, wires2): """Validate and pre-process inputs as follows: * Check the shape of the weights tensor. * Check that both wire sets have at least 2 wires. Args: weight (tensor_like): trainable parameters of the template wires1 (Wires): first set of wires wires2 (Wires): second set of wires """ if len(wires1) < 2: raise ValueError( "expected at least two wires representing the occupied orbitals; " "got {}".format(len(wires1))) if len(wires2) < 2: raise ValueError( "expected at least two wires representing the unoccupied orbitals; " "got {}".format(len(wires2))) if qml.tape_mode_active(): shape = qml.math.shape(weight) if shape != (): raise ValueError(f"Weight must be a scalar; got shape {shape}.") else: expected_shape = () check_shape( weight, expected_shape, msg="Weight must be a scalar; got shape {}".format( expected_shape, get_shape(weight)), )
def _preprocess(features, wires): """Validate and pre-process inputs as follows: * Check that the features tensor is one-dimensional. * Check that the first dimension of the features tensor has length :math:`n` or less, where :math:`n` is the number of qubits. Args: features (tensor_like): input features to pre-process wires (Wires): wires that template acts on Returns: int: number of features """ if qml.tape_mode_active(): shape = qml.math.shape(features) if len(shape) != 1: raise ValueError( f"Features must be a one-dimensional tensor; got shape {shape}." ) n_features = shape[0] if n_features > len(wires): raise ValueError( f"Features must be of length {len(wires)} or less; got length {n_features}." ) else: shp = check_shape( features, (len(wires), ), bound="max", msg="Features must be of shape {} or smaller; " "got {}.".format((len(wires), ), get_shape(features)), ) n_features = shp[0] return n_features
def UCCSD(weights, wires, ph=None, pphh=None, init_state=None): r"""Implements the Unitary Coupled-Cluster Singles and Doubles (UCCSD) ansatz. The UCCSD ansatz calls the :func:`~.SingleExcitationUnitary` and :func:`~.DoubleExcitationUnitary` templates to exponentiate the coupled-cluster excitation operator. UCCSD is a VQE ansatz commonly used to run quantum chemistry simulations. The UCCSD unitary, within the first-order Trotter approximation, is given by: .. math:: \hat{U}(\vec{\theta}) = \prod_{p > r} \mathrm{exp} \Big\{\theta_{pr} (\hat{c}_p^\dagger \hat{c}_r-\mathrm{H.c.}) \Big\} \prod_{p > q > r > s} \mathrm{exp} \Big\{\theta_{pqrs} (\hat{c}_p^\dagger \hat{c}_q^\dagger \hat{c}_r \hat{c}_s-\mathrm{H.c.}) \Big\} where :math:`\hat{c}` and :math:`\hat{c}^\dagger` are the fermionic annihilation and creation operators and the indices :math:`r, s` and :math:`p, q` run over the occupied and unoccupied molecular orbitals, respectively. Using the `Jordan-Wigner transformation <https://arxiv.org/abs/1208.5986>`_ the UCCSD unitary defined above can be written in terms of Pauli matrices as follows (for more details see `arXiv:1805.04340 <https://arxiv.org/abs/1805.04340>`_): .. math:: \hat{U}(\vec{\theta}) = && \prod_{p > r} \mathrm{exp} \Big\{ \frac{i\theta_{pr}}{2} \bigotimes_{a=r+1}^{p-1} \hat{Z}_a (\hat{Y}_r \hat{X}_p - \mathrm{H.c.}) \Big\} \\ && \times \prod_{p > q > r > s} \mathrm{exp} \Big\{ \frac{i\theta_{pqrs}}{8} \bigotimes_{b=s+1}^{r-1} \hat{Z}_b \bigotimes_{a=q+1}^{p-1} \hat{Z}_a (\hat{X}_s \hat{X}_r \hat{Y}_q \hat{X}_p + \hat{Y}_s \hat{X}_r \hat{Y}_q \hat{Y}_p + \hat{X}_s \hat{Y}_r \hat{Y}_q \hat{Y}_p + \hat{X}_s \hat{X}_r \hat{X}_q \hat{Y}_p - \{\mathrm{H.c.}\}) \Big\}. Args: weights (array): length ``len(ph)+len(pphh)`` vector containing the parameters :math:`\theta_{pr}` and :math:`\theta_{pqrs}` entering the Z rotation in :func:`~.SingleExcitationUnitary` and :func:`~.DoubleExcitationUnitary`. These parameters are precisely the coupled-cluster amplitudes that need to be optimized for each configuration generated by the excitation operator. wires (Iterable or Wires): Wires that the template acts on. Accepts an iterable of numbers or strings, or a Wires object. ph (Sequence[list]): Sequence of two-element lists containing the indices ``r, p`` that define the 1particle-1hole (ph) configuration :math:`\vert \mathrm{ph} \rangle = \hat{c}_p^\dagger \hat{c}_r \vert \mathrm{HF} \rangle`, where :math:`\vert \mathrm{HF} \rangle` denotes the Hartee-Fock (HF) reference state. pphh (Sequence[list]): Sequence of four-elements lists containing the indices ``s, r, q, p`` that define the 2particle-2hole configuration (pphh) :math:`\vert \mathrm{pphh} \rangle = \hat{c}_p^\dagger \hat{c}_q^\dagger \hat{c}_r \hat{c}_s \vert \mathrm{HF} \rangle`. init_state (array[int]): Length ``len(wires)`` occupation-number vector representing the HF state. ``init_state`` is used to initialize the qubit register. Raises: ValueError: if inputs do not have the correct format .. UsageDetails:: Notice that: #. The number of wires has to be equal to the number of spin-orbitals included in the active space. #. The ``ph`` and ``pphh`` lists can be generated with the function :func:`~.sd_excitations`. See example below. #. The vector of parameters ``weights`` is a one-dimensional array of size ``len(ph)+len(pphh)`` An example of how to use this template is shown below: .. code-block:: python import pennylane as qml from pennylane.templates import UCCSD # Number of active electrons n_active_electrons = 2 # Number of active spin-orbitals n_active_orbitals = 4 # Set the ``ref_state`` array to encode the HF state ref_state = qml.qchem.hf_state(n_active_electrons, n_active_orbitals) # Spin-projection selection rule to generate the excitations DSz = 0 # Generate the particle-hole configurations ph, pphh = qml.qchem.sd_excitations(n_active_electrons, n_active_orbitals, DS=DSz) dev = qml.device('default.qubit', wires=n_active_orbitals) @qml.qnode(dev) def circuit(weights, wires, ph=ph, pphh=pphh, init_state=ref_state): UCCSD(weights, wires, ph=ph, pphh=pphh, init_state=ref_state) return qml.expval(qml.PauliZ(0)) params = np.random.normal(0, np.pi, len(ph) + len(pphh)) print(circuit(params, wires, ph=ph, pphh=pphh, init_state=ref_state)) """ ############## # Input checks wires = Wires(wires) if (not ph) and (not pphh): raise ValueError( "'ph' and 'pphh' lists can not be both empty; got ph={}, pphh={}".format(ph, pphh) ) check_type(ph, [list], msg="'ph' must be a list; got {}".format(ph)) expected_shape = (2,) for i_ph in ph: check_type(i_ph, [list], msg="Each element of 'ph' must be a list; got {}".format(i_ph)) check_shape( i_ph, expected_shape, msg="Elements of 'ph' must be of shape {}; got {}".format( expected_shape, get_shape(i_ph) ), ) for i in i_ph: check_type( i, [int], msg="Each element of 'ph' must be a list of integers; got {}".format(i_ph) ) check_type(pphh, [list], msg="'pphh' must be a list; got {}".format(pphh)) expected_shape = (4,) for i_pphh in pphh: check_type( i_pphh, [list], msg="Each element of 'pphh' must be a list; got {}".format(i_pphh) ) check_shape( i_pphh, expected_shape, msg="Elements of 'pphh' must be of shape {}; got {}".format( expected_shape, get_shape(i_pphh) ), ) for i in i_pphh: check_type( i, [int], msg="Each element of 'pphh' must be a list of integers; got {}".format(i_pphh), ) check_type( init_state, [np.ndarray], msg="'init_state' must be a Numpy array; got {}".format(init_state), ) for i in init_state: check_type( i, [int, np.int64], msg="Elements of 'init_state' must be integers; got {}".format(init_state), ) expected_shape = (len(ph) + len(pphh),) check_shape( weights, expected_shape, msg="'weights' must be of shape {}; got {}".format(expected_shape, get_shape(weights)), ) expected_shape = (len(wires),) check_shape( init_state, expected_shape, msg="'init_state' must be of shape {}; got {}".format( expected_shape, get_shape(init_state) ), ) ############### wires = wires.tolist() # TODO: Remove when ops accept wires qml.BasisState(np.flip(init_state), wires=wires) for d, i_pphh in enumerate(pphh): DoubleExcitationUnitary(weights[len(ph) + d], wires=i_pphh) for s, i_ph in enumerate(ph): SingleExcitationUnitary(weights[s], wires=i_ph)
def StronglyEntanglingLayers(weights, wires, ranges=None, imprimitive=CNOT): r"""Layers consisting of single qubit rotations and entanglers, inspired by the circuit-centric classifier design `arXiv:1804.00633 <https://arxiv.org/abs/1804.00633>`_. The argument ``weights`` contains the weights for each layer. The number of layers :math:`L` is therefore derived from the first dimension of ``weights``. The 2-qubit gates, whose type is specified by the ``imprimitive`` argument, act chronologically on the :math:`M` wires, :math:`i = 1,...,M`. The second qubit of each gate is given by :math:`(i+r)\mod M`, where :math:`r` is a hyperparameter called the *range*, and :math:`0 < r < M`. If applied to one qubit only, this template will use no imprimitive gates. This is an example of two 4-qubit strongly entangling layers (ranges :math:`r=1` and :math:`r=2`, respectively) with rotations :math:`R` and CNOTs as imprimitives: .. figure:: ../../_static/layer_sec.png :align: center :width: 60% :target: javascript:void(0); Args: weights (array[float]): array of weights of shape ``(L, M, 3)`` wires (Iterable or Wires): Wires that the template acts on. Accepts an iterable of numbers or strings, or a Wires object. ranges (Sequence[int]): sequence determining the range hyperparameter for each subsequent layer; if None using :math:`r=l \mod M` for the :math:`l`th layer and :math:`M` wires. imprimitive (pennylane.ops.Operation): two-qubit gate to use, defaults to :class:`~pennylane.ops.CNOT` Raises: ValueError: if inputs do not have the correct format """ ############# # Input checks wires = Wires(wires) check_no_variable(ranges, msg="'ranges' cannot be differentiable") check_no_variable(imprimitive, msg="'imprimitive' cannot be differentiable") repeat = check_number_of_layers([weights]) expected_shape = (repeat, len(wires), 3) check_shape( weights, expected_shape, msg="'weights' must be of shape {}; got {}" "".format(expected_shape, get_shape(weights)), ) if len(wires) > 1: if ranges is None: # tile ranges with iterations of range(1, n_wires) ranges = [(l % (len(wires) - 1)) + 1 for l in range(repeat)] expected_shape = (repeat, ) check_shape( ranges, expected_shape, msg="'ranges' must be of shape {}; got {}" "".format(expected_shape, get_shape(weights)), ) check_type(ranges, [list], msg="'ranges' must be a list; got {}" "".format(ranges)) for r in ranges: check_type(r, [int], msg="'ranges' must be a list of integers; got {}" "".format(ranges)) if any((r >= len(wires) or r == 0) for r in ranges): raise ValueError( "the range for all layers needs to be smaller than the number of " "qubits; got ranges {}.".format(ranges)) else: ranges = [0] * repeat ############### for l in range(repeat): strongly_entangling_layer(weights=weights[l], wires=wires, r=ranges[l], imprimitive=imprimitive)
def SingleExcitationUnitary(weight, wires=None): r"""Circuit to exponentiate the tensor product of Pauli matrices representing the single-excitation operator entering the Unitary Coupled-Cluster Singles and Doubles (UCCSD) ansatz. UCCSD is a VQE ansatz commonly used to run quantum chemistry simulations. The CC single-excitation operator is given by .. math:: \hat{U}_{pr}(\theta) = \mathrm{exp} \{ \theta_{pr} (\hat{c}_p^\dagger \hat{c}_r -\mathrm{H.c.}) \}, where :math:`\hat{c}` and :math:`\hat{c}^\dagger` are the fermionic annihilation and creation operators and the indices :math:`r` and :math:`p` run over the occupied and unoccupied molecular orbitals, respectively. Using the `Jordan-Wigner transformation <https://arxiv.org/abs/1208.5986>`_ the fermionic operator defined above can be written in terms of Pauli matrices (for more details see `arXiv:1805.04340 <https://arxiv.org/abs/1805.04340>`_). .. math:: \hat{U}_{pr}(\theta) = \mathrm{exp} \Big\{ \frac{i\theta}{2} \bigotimes_{a=r+1}^{p-1}\hat{Z}_a (\hat{Y}_r \hat{X}_p) \Big\} \mathrm{exp} \Big\{ -\frac{i\theta}{2} \bigotimes_{a=r+1}^{p-1} \hat{Z}_a (\hat{X}_r \hat{Y}_p) \Big\}. The quantum circuit to exponentiate the tensor product of Pauli matrices entering the latter equation is shown below (see `arXiv:1805.04340 <https://arxiv.org/abs/1805.04340>`_): | .. figure:: ../../_static/templates/subroutines/single_excitation_unitary.png :align: center :width: 60% :target: javascript:void(0); | As explained in `Seely et al. (2012) <https://arxiv.org/abs/1208.5986>`_, the exponential of a tensor product of Pauli-Z operators can be decomposed in terms of :math:`2(n-1)` CNOT gates and a single-qubit Z-rotation referred to as :math:`U_\theta` in the figure above. If there are :math:`X` or :math:`Y` Pauli matrices in the product, the Hadamard (:math:`H`) or :math:`R_x` gate has to be applied to change to the :math:`X` or :math:`Y` basis, respectively. The latter operations are denoted as :math:`U_1` and :math:`U_2` in the figure above. See the Usage Details section for more information. Args: weight (float): angle :math:`\theta` entering the Z rotation acting on wire ``p`` wires (Iterable or Wires): Wires that the template acts on. The wires represent the subset of orbitals in the interval ``[r, p]``. Must be of minimum length 2. The first wire is interpreted as ``r`` and the last wire as ``p``. Wires in between are acted on with CNOT gates to compute the parity of the set of qubits. Raises: ValueError: if inputs do not have the correct format .. UsageDetails:: Notice that: #. :math:`\hat{U}_{pr}(\theta)` involves two exponentiations where :math:`\hat{U}_1`, :math:`\hat{U}_2`, and :math:`\hat{U}_\theta` are defined as follows, .. math:: [U_1, U_2, U_{\theta}] = \Bigg\{\bigg[R_x(-\pi/2), H, R_z(\theta/2)\bigg], \bigg[H, R_x(-\frac{\pi}{2}), R_z(-\theta/2) \bigg] \Bigg\} #. For a given pair ``[r, p]``, ten single-qubit and ``4*(len(wires)-1)`` CNOT operations are applied. Notice also that CNOT gates act only on qubits ``wires[1]`` to ``wires[-2]``. The operations performed across these qubits are shown in dashed lines in the figure above. An example of how to use this template is shown below: .. code-block:: python import pennylane as qml from pennylane.templates import SingleExcitationUnitary dev = qml.device('default.qubit', wires=3) @qml.qnode(dev) def circuit(weight, wires=None): SingleExcitationUnitary(weight, wires=wires) return qml.expval(qml.PauliZ(0)) weight = 0.56 print(circuit(weight, wires=[0, 1, 2])) """ ############## # Input checks wires = Wires(wires) if len(wires) < 2: raise ValueError("expected at least two wires; got {}".format( len(wires))) expected_shape = () check_shape( weight, expected_shape, msg="'weight' must be of shape {}; got {}".format( expected_shape, get_shape(weight)), ) ############### # Interpret first and last wire as r and p r = wires[0] p = wires[-1] # Sequence of the wires entering the CNOTs between wires 'r' and 'p' set_cnot_wires = [wires.subset([l, l + 1]) for l in range(len(wires) - 1)] # ------------------------------------------------------------------ # Apply the first layer # U_1, U_2 acting on wires 'r' and 'p' RX(-np.pi / 2, wires=r) Hadamard(wires=p) # Applying CNOTs between wires 'r' and 'p' for cnot_wires in set_cnot_wires: CNOT(wires=cnot_wires) # Z rotation acting on wire 'p' RZ(weight / 2, wires=p) # Applying CNOTs in reverse order for cnot_wires in reversed(set_cnot_wires): CNOT(wires=cnot_wires) # U_1^+, U_2^+ acting on wires 'r' and 'p' RX(np.pi / 2, wires=r) Hadamard(wires=p) # ------------------------------------------------------------------ # Apply the second layer # U_1, U_2 acting on wires 'r' and 'p' Hadamard(wires=r) RX(-np.pi / 2, wires=p) # Applying CNOTs between wires 'r' and 'p' for cnot_wires in set_cnot_wires: CNOT(wires=cnot_wires) # Z rotation acting on wire 'p' RZ(-weight / 2, wires=p) # Applying CNOTs in reverse order for cnot_wires in reversed(set_cnot_wires): CNOT(wires=cnot_wires) # U_1^+, U_2^+ acting on wires 'r' and 'p' Hadamard(wires=r) RX(np.pi / 2, wires=p)
def DisplacementEmbedding(features, wires, method="amplitude", c=0.1): r"""Encodes :math:`N` features into the displacement amplitudes :math:`r` or phases :math:`\phi` of :math:`M` modes, where :math:`N\leq M`. The mathematical definition of the displacement gate is given by the operator .. math:: D(\alpha) = \exp(r (e^{i\phi}\ad -e^{-i\phi}\a)), where :math:`\a` and :math:`\ad` are the bosonic creation and annihilation operators. ``features`` has to be an array of at most ``len(wires)`` floats. If there are fewer entries in ``features`` than wires, the circuit does not apply the remaining displacement gates. Args: features (array): Array of features of size (N,) wires (Iterable or Wires): Wires that the template acts on. Accepts an iterable of numbers or strings, or a Wires object. method (str): ``'phase'`` encodes the input into the phase of single-mode displacement, while ``'amplitude'`` uses the amplitude c (float): value of the phase of all displacement gates if ``execution='amplitude'``, or the amplitude of all displacement gates if ``execution='phase'`` Raises: ValueError: if inputs do not have the correct format """ ############# # Input checks wires = Wires(wires) expected_shape = (len(wires), ) check_shape( features, expected_shape, bound="max", msg="'features' must be of shape {} or smaller; got {}." "".format(expected_shape, get_shape(features)), ) check_is_in_options( method, ["amplitude", "phase"], msg="did not recognize option {} for 'method'" "".format(method), ) ############# constants = [c] * len(features) if method == "amplitude": broadcast( unitary=Displacement, pattern="single", wires=wires, parameters=list(zip(features, constants)), ) elif method == "phase": broadcast( unitary=Displacement, pattern="single", wires=wires, parameters=list(zip(constants, features)), )
def RandomLayers(weights, wires, ratio_imprim=0.3, imprimitive=CNOT, rotations=None, seed=42): r"""Layers of randomly chosen single qubit rotations and 2-qubit entangling gates, acting on randomly chosen qubits. .. warning:: This template uses random number generation inside qnodes. Find more details about how to invoke the desired random behaviour in the "Usage Details" section below. The argument ``weights`` contains the weights for each layer. The number of layers :math:`L` is therefore derived from the first dimension of ``weights``. The two-qubit gates of type ``imprimitive`` and the rotations are distributed randomly in the circuit. The number of random rotations is derived from the second dimension of ``weights``. The number of two-qubit gates is determined by ``ratio_imprim``. For example, a ratio of ``0.3`` with ``30`` rotations will lead to the use of ``10`` two-qubit gates. .. note:: If applied to one qubit only, this template will use no imprimitive gates. This is an example of two 4-qubit random layers with four Pauli-Y/Pauli-Z rotations :math:`R_y, R_z`, controlled-Z gates as imprimitives, as well as ``ratio_imprim=0.3``: .. figure:: ../../_static/layer_rnd.png :align: center :width: 60% :target: javascript:void(0); Args: weights (array[float]): array of weights of shape ``(L, k)``, wires (Iterable or Wires): Wires that the template acts on. Accepts an iterable of numbers or strings, or a Wires object. ratio_imprim (float): value between 0 and 1 that determines the ratio of imprimitive to rotation gates imprimitive (pennylane.ops.Operation): two-qubit gate to use, defaults to :class:`~pennylane.ops.CNOT` rotations (list[pennylane.ops.Operation]): List of Pauli-X, Pauli-Y and/or Pauli-Z gates. The frequency determines how often a particular rotation type is used. Defaults to the use of all three rotations with equal frequency. seed (int): seed to generate random architecture, defaults to 42 Raises: ValueError: if inputs do not have the correct format .. UsageDetails:: **Default seed** ``RandomLayers`` always uses a seed to initialize the construction of a random circuit. This means that the template creates the same circuit every time it is called. If no seed is provided, the default seed of ``42`` is used. .. code-block:: python import pennylane as qml import numpy as np from pennylane.templates.layers import RandomLayers dev = qml.device("default.qubit", wires=2) weights = [[0.1, -2.1, 1.4]] @qml.qnode(dev) def circuit1(weights): RandomLayers(weights=weights, wires=range(2)) return qml.expval(qml.PauliZ(0)) @qml.qnode(dev) def circuit2(weights): RandomLayers(weights=weights, wires=range(2)) return qml.expval(qml.PauliZ(0)) >>> np.allclose(circuit1(weights), circuit2(weights)) >>> True You can verify this by drawing the circuits. >>> print(circuit1.draw()) >>> 0: ──RX(0.1)──RX(-2.1)──╭X──╭X───────────┤ ⟨Z⟩ ... 1: ─────────────────────╰C──╰C──RZ(1.4)──┤ >>> print(circuit2.draw()) >>> 0: ──RX(0.1)──RX(-2.1)──╭X──╭X───────────┤ ⟨Z⟩ ... 1: ─────────────────────╰C──╰C──RZ(1.4)──┤ **Changing the seed** To change the randomly generated circuit architecture, you have to change the seed passed to the template. For example, these two calls of ``RandomLayers`` *do not* create the same circuit: .. code-block:: python @qml.qnode(dev) def circuit_9(weights): RandomLayers(weights=weights, wires=range(2), seed=9) return qml.expval(qml.PauliZ(0)) @qml.qnode(dev) def circuit_12(weights): RandomLayers(weights=weights, wires=range(2), seed=12) return qml.expval(qml.PauliZ(0)) >>> np.allclose(circuit_9(weights), circuit_12(weights)) >>> False >>> print(circuit_9.draw()) >>> 0: ──╭X──RY(-2.1)──RX(1.4)──┤ ⟨Z⟩ ... 1: ──╰C──RX(0.1)────────────┤ >>> print(circuit_12.draw()) >>> 0: ──╭X──RX(-2.1)──╭C──╭X──RZ(1.4)──┤ ⟨Z⟩ ... 1: ──╰C──RZ(0.1)───╰X──╰C───────────┤ **Automatically creating random circuits** To automate the process of creating different circuits with ``RandomLayers``, you can set ``seed=None`` to avoid specifying a seed. However, in this case care needs to be taken. In the default setting, a quantum node is **mutable**, which means that the quantum function is re-evaluated every time it is called. This means that the circuit is re-constructed from scratch each time you call the qnode: .. code-block:: python @qml.qnode(dev) def circuit_rnd(weights): RandomLayers(weights=weights, wires=range(2), seed=None) return qml.expval(qml.PauliZ(0)) first_call = circuit_rnd(weights) second_call = circuit_rnd(weights) >>> np.allclose(first_call, second_call) >>> False This can be rectified by making the quantum node **immutable**. .. code-block:: python @qml.qnode(dev, mutable=False) def circuit_rnd(weights): RandomLayers(weights=weights, wires=range(2), seed=None) return qml.expval(qml.PauliZ(0)) first_call = circuit_rnd(weights) second_call = circuit_rnd(weights) >>> np.allclose(first_call, second_call) >>> True """ if seed is not None: np.random.seed(seed) if rotations is None: rotations = [RX, RY, RZ] ############# # Input checks wires = Wires(wires) check_no_variable(ratio_imprim, msg="'ratio_imprim' cannot be differentiable") check_no_variable(imprimitive, msg="'imprimitive' cannot be differentiable") check_no_variable(rotations, msg="'rotations' cannot be differentiable") check_no_variable(seed, msg="'seed' cannot be differentiable") repeat = check_number_of_layers([weights]) n_rots = get_shape(weights)[1] expected_shape = (repeat, n_rots) check_shape( weights, expected_shape, msg="'weights' must be of shape {}; got {}" "".format(expected_shape, get_shape(weights)), ) check_type( ratio_imprim, [float, type(None)], msg="'ratio_imprim' must be a float; got {}".format(ratio_imprim), ) check_type(n_rots, [int, type(None)], msg="'n_rots' must be an integer; got {}".format(n_rots)) # TODO: Check that 'rotations' contains operations check_type( rotations, [list, type(None)], msg="'rotations' must be a list of PennyLane operations; got {}" "".format(rotations), ) check_type(seed, [int, type(None)], msg="'seed' must be an integer; got {}.".format(seed)) ############### for l in range(repeat): random_layer( weights=weights[l], wires=wires, ratio_imprim=ratio_imprim, imprimitive=imprimitive, rotations=rotations, seed=seed, )
def ParticleConservingU1(weights, wires, init_state=None): r"""Implements the heuristic VQE ansatz for quantum chemistry simulations using the particle-conserving gate :math:`U_{1,\mathrm{ex}}` proposed by Barkoutsos *et al.* in `arXiv:1805.04340 <https://arxiv.org/abs/1805.04340>`_. This template prepares :math:`N`-qubit trial states by applying :math:`D` layers of the entangler block :math:`U_\mathrm{ent}(\vec{\phi}, \vec{\theta})` to the Hartree-Fock state .. math:: \vert \Psi(\vec{\phi}, \vec{\theta}) \rangle = \hat{U}^{(D)}_\mathrm{ent}(\vec{\phi}_D, \vec{\theta}_D) \dots \hat{U}^{(2)}_\mathrm{ent}(\vec{\phi}_2, \vec{\theta}_2) \hat{U}^{(1)}_\mathrm{ent}(\vec{\phi}_1, \vec{\theta}_1) \vert \mathrm{HF}\rangle. The circuit implementing the entangler blocks is shown in the figure below: | .. figure:: ../../_static/templates/layers/particle_conserving_u1.png :align: center :width: 50% :target: javascript:void(0); | The repeated units across several qubits are shown in dotted boxes. Each layer contains :math:`N-1` particle-conserving two-parameter exchange gates :math:`U_{1,\mathrm{ex}}(\phi, \theta)` that act on pairs of nearest neighbors qubits. The unitary matrix representing :math:`U_{1,\mathrm{ex}}(\phi, \theta)` is given by (see `arXiv:1805.04340 <https://arxiv.org/abs/1805.04340>`_), .. math:: U_{1, \mathrm{ex}}(\phi, \theta) = \left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & \mathrm{cos}(\theta) & e^{i\phi} \mathrm{sin}(\theta) & 0 \\ 0 & e^{-i\phi} \mathrm{sin}(\theta) & -\mathrm{cos}(\theta) & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right). The figure below shows the circuit decomposing :math:`U_{1, \mathrm{ex}}` in elementary gates. The Pauli matrix :math:`\sigma_z` and single-qubit rotation :math:`R(0, 2 \theta, 0)` apply the Pauli Z operator and an arbitrary rotation on the qubit ``n`` with qubit ``m`` bein the control qubit, | .. figure:: ../../_static/templates/layers/u1_decomposition.png :align: center :width: 80% :target: javascript:void(0); | :math:`U_A(\phi)` is the unitary matrix .. math:: U_A(\phi) = \left(\begin{array}{cc} 0 & e^{-i\phi} \\ e^{-i\phi} & 0 \\ \end{array}\right), which is applied controlled on the state of qubit ``m`` and can be further decomposed in terms of the `quantum operations <https://pennylane.readthedocs.io/en/stable/introduction/operations.html>`_ supported by Pennylane, | .. figure:: ../../_static/templates/layers/ua_decomposition.png :align: center :width: 70% :target: javascript:void(0); | where, | .. figure:: ../../_static/templates/layers/phaseshift_decomposition.png :align: center :width: 65% :target: javascript:void(0); | The quantum circuits above decomposing the unitaries :math:`U_{1,\mathrm{ex}}(\phi, \theta)` and :math:`U_A(\phi)` are implemented by the ``u1_ex_gate`` and ``decompose_ua`` functions, respectively. :math:`R_\phi` refers to the ``PhaseShift`` gate in the circuit diagram. Args: weights (array[float]): Array of weights of shape ``(D, M, 2)``. ``D`` is the number of entangler block layers and :math:`M=N-1` is the number of exchange gates :math:`U_{1,\mathrm{ex}}` per layer. wires (Iterable or Wires): Wires that the template acts on. Accepts an iterable of numbers or strings, or a Wires object. init_state (array[int]): length ``len(wires)`` vector representing the Hartree-Fock state used to initialize the wires Raises: ValueError: if inputs do not have the correct format .. UsageDetails:: #. The number of wires :math:`N` has to be equal to the number of spin orbitals included in the active space. #. The number of trainable parameters scales linearly with the number of layers as :math:`2D(N-1)`. An example of how to use this template is shown below: .. code-block:: python import pennylane as qml from pennylane.templates import ParticleConservingU1 from functools import partial # Build the electronic Hamiltonian from a local .xyz file h, qubits = qml.qchem.molecular_hamiltonian("h2", "h2.xyz") # Define the Hartree-Fock state electrons = 2 ref_state = qml.qchem.hf_state(electrons, qubits) # Define the device dev = qml.device('default.qubit', wires=qubits) # Define the ansatz ansatz = partial(ParticleConservingU1, init_state=ref_state) # Define the cost function cost_fn = qml.ExpvalCost(ansatz, h, dev) # Compute the expectation value of 'h' layers = 2 params = qml.init.particle_conserving_u1_normal(layers, qubits) print(cost_fn(params)) """ wires = Wires(wires) layers = weights.shape[0] if len(wires) < 2: raise ValueError( "This template requires the number of qubits to be greater than one; a wire sequence with {} elements" .format(len(wires))) expected_shape = (layers, len(wires) - 1, 2) check_shape( weights, expected_shape, msg="'weights' must be of shape {}; got {}".format( expected_shape, get_shape(weights)), ) nm_wires = [wires.subset([l, l + 1]) for l in range(0, len(wires) - 1, 2)] nm_wires += [wires.subset([l, l + 1]) for l in range(1, len(wires) - 1, 2)] qml.BasisState(init_state, wires=wires) for l in range(layers): for i, wires_ in enumerate(nm_wires): u1_ex_gate(weights[l, i, 0], weights[l, i, 1], wires=wires_)
def MottonenStatePreparation(state_vector, wires): r""" Prepares an arbitrary state on the given wires using a decomposition into gates developed by Möttönen et al. (Quantum Info. Comput., 2005). The state is prepared via a sequence of "uniformly controlled rotations". A uniformly controlled rotation on a target qubit is composed from all possible controlled rotations on said qubit and can be used to address individual elements of the state vector. In the work of Mottonen et al., the inverse of their state preparation is constructed by first equalizing the phases of the state vector via uniformly controlled Z rotations and then rotating the now real state vector into the direction of the state :math:`|0\rangle` via uniformly controlled Y rotations. This code is adapted from code written by Carsten Blank for PennyLane-Qiskit. Args: state_vector (array): Input array of shape ``(2^N,)``, where N is the number of wires the state preparation acts on. ``N`` must be smaller or equal to the total number of wires. wires (Sequence[int]): sequence of qubit indices that the template acts on Raises: ValueError: if inputs do not have the correct format """ ############### # Input checks wires = check_wires(wires) n_wires = len(wires) expected_shape = (2**n_wires, ) check_shape( state_vector, expected_shape, msg="'state_vector' must be of shape {}; got {}." "".format(expected_shape, get_shape(state_vector)), ) # check if state_vector is normalized if isinstance(state_vector[0], Variable): state_vector_values = [s.val for s in state_vector] norm = np.sum(np.abs(state_vector_values)**2) else: norm = np.sum(np.abs(state_vector)**2) if not np.isclose(norm, 1.0, atol=1e-3): raise ValueError( "'state_vector' has to be of length 1.0, got {}".format(norm)) ####################### # Change ordering of indices, original code was for IBM machines state_vector = np.array(state_vector).reshape( [2] * n_wires).T.flatten()[:, np.newaxis] state_vector = sparse.dok_matrix(state_vector) wires = np.array(wires) a = sparse.dok_matrix(state_vector.shape) omega = sparse.dok_matrix(state_vector.shape) for (i, j), v in state_vector.items(): if isinstance(v, Variable): a[i, j] = np.absolute(v.val) omega[i, j] = np.angle(v.val) else: a[i, j] = np.absolute(v) omega[i, j] = np.angle(v) # This code is directly applying the inverse of Carsten Blank's # code to avoid inverting at the end # Apply y rotations for k in range(n_wires, 0, -1): alpha_y_k = _get_alpha_y(a, n_wires, k) # type: sparse.dok_matrix control = wires[k:] target = wires[k - 1] _uniform_rotation_y_dagger(alpha_y_k, control, target) # Apply z rotations for k in range(n_wires, 0, -1): alpha_z_k = _get_alpha_z(omega, n_wires, k) control = wires[k:] target = wires[k - 1] if len(alpha_z_k) > 0: _uniform_rotation_z_dagger(alpha_z_k, control, target)