qml.kUpCCGSD¶

class
kUpCCGSD
(weights, wires, k=1, delta_sz=0, init_state=None, do_queue=True, id=None)[source]¶ Bases:
pennylane.operation.Operation
Implements the kUnitary Pair CoupledCluster Generalized Singles and Doubles (kUpCCGSD) ansatz.
The kUpCCGSD ansatz calls the
FermionicSingleExcitation()
andFermionicDoubleExcitation()
templates to exponentiate the product of \(k\) generalized singles and pair coupledcluster doubles excitation operators. Here, “generalized” means that the single and double excitation terms do not distinguish between occupied and unoccupied orbitals. Additionally, the term “pair coupledcluster” refers to the fact that the double excitations contain only those twobody excitations that move a pair of electrons from one spatial orbital to another. This kUpCCGSD belongs to the family of Unitary Coupled Cluster (UCC) based ansätze, commonly used to solve quantum chemistry problems on quantum computers.The kUpCCGSD unitary, within the firstorder Trotter approximation for a given integer \(k\), is given by:
\[\hat{U}(\vec{\theta}) = \prod_{l=1}^{k} \bigg(\prod_{p,r}\exp{\Big\{ \theta_{r}^{p}(\hat{c}^{\dagger}_p\hat{c}_r  \text{H.c.})\Big\}} \ \prod_{i,j} \Big\{\exp{\theta_{j_\alpha j_\beta}^{i_\alpha i_\beta} (\hat{c}^{\dagger}_{i_\alpha}\hat{c}^{\dagger}_{i_\beta} \hat{c}_{j_\alpha}\hat{c}_{j_\beta}  \text{H.c.}) \Big\}}\bigg)\]where \(\hat{c}\) and \(\hat{c}^{\dagger}\) are the fermionic annihilation and creation operators. The indices \(p, q\) run over the spin orbitals and \(i, j\) run over the spatial orbitals. The singles and paired doubles amplitudes \(\theta_{r}^{p}\) and \(\theta_{j_\alpha j_\beta}^{i_\alpha i_\beta}\) represent the set of variational parameters.
 Parameters
weights (tensor_like) – Tensor containing the parameters \(\theta_{pr}\) and \(\theta_{pqrs}\) entering the Z rotation in
FermionicSingleExcitation()
andFermionicDoubleExcitation()
. These parameters are the coupledcluster amplitudes that need to be optimized for each generalized single and pair double excitation terms.wires (Iterable) – wires that the template acts on
k (int) – Number of times UpCCGSD unitary is repeated.
delta_sz (int) – Specifies the selection rule
sz[p]  sz[r] = delta_sz
for the spinprojectionsz
of the orbitals involved in the generalized single excitations.delta_sz
can take the values \(0\) and \(\pm 1\).init_state (array[int]) – Length
len(wires)
occupationnumber vector representing the HF state.init_state
is used to initialize the wires.
Usage Details
The number of wires has to be equal to the number of spinorbitals included in the active space, and should be even.
The number of trainable parameters scales linearly with the number of layers as \(2 k n\), where \(n\) is the total number of generalized singles and paired doubles excitation terms.
An example of how to use this template is shown below:
import numpy as np import pennylane as qml # Build the electronic Hamiltonian symbols = ["H", "H"] coordinates = np.array([0.0, 0.0, 0.6614, 0.0, 0.0, 0.6614]) H, qubits = qml.qchem.molecular_hamiltonian(symbols, coordinates) # Define the HartreeFock state electrons = 2 ref_state = qml.qchem.hf_state(electrons, qubits) # Define the device dev = qml.device('default.qubit', wires=qubits) # Define the ansatz @qml.qnode(dev) def ansatz(weights): qml.kUpCCGSD(weights, wires=[0, 1, 2, 3], k=1, delta_sz=0, init_state=ref_state) return qml.expval(H) # Get the shape of the weights for this template layers = 1 shape = qml.kUpCCGSD.shape(k=layers, n_wires=qubits, delta_sz=0) # Initialize the weight tensors np.random.seed(24) weights = np.random.random(size=shape) # Define the optimizer opt = qml.GradientDescentOptimizer(stepsize=0.4) # Store the values of the cost function energy = [ansatz(weights)] # Store the values of the circuit weights angle = [weights] max_iterations = 100 conv_tol = 1e06 for n in range(max_iterations): weights, prev_energy = opt.step_and_cost(ansatz, weights) energy.append(ansatz(weights)) angle.append(weights) conv = np.abs(energy[1]  prev_energy) if n % 4 == 0: print(f"Step = {n}, Energy = {energy[1]:.8f} Ha") if conv <= conv_tol: break print("\n" f"Final value of the groundstate energy = {energy[1]:.8f} Ha") print("\n" f"Optimal value of the circuit parameters = {angle[1]}")
Step = 0, Energy = 0.18046117 Ha Step = 4, Energy = 1.06545723 Ha Step = 8, Energy = 1.13028734 Ha Step = 12, Energy = 1.13528393 Ha Step = 16, Energy = 1.13604954 Ha Step = 20, Energy = 1.13616762 Ha Step = 24, Energy = 1.13618584 Ha Final value of the groundstate energy = 1.13618786 Ha Optimal value of the circuit parameters = [[ 0.97882258 0.46090942 0.98106201 0.45866993 0.91548184 2.01637919]]
Parameter shape
The shape of the weights argument can be computed by the static method
shape()
and used when creating randomly initialised weight tensors:shape = qml.kUpCCGSD.shape(n_layers=2, n_wires=4) weights = np.random.random(size=shape)
>>> weights.shape (2, 6)
Attributes
Arithmetic depth of the operator.
If inverse is requested, this is the name of the original operator to be inverted.
The target operation for controlled gates.
Batch size of the operator if it is used with broadcasted parameters.
Control wires of the operator.
Gradient recipe for the parametershift method.
Integer hash that uniquely represents the operator.
Dictionary of nontrainable variables that this operation depends on.
Custom string to label a specific operator instance.
Boolean determining if the inverse of the operation was requested.
This property determines if an operator is hermitian.
Name of the operator.
Number of dimensions per trainable parameter of the operator.
Number of trainable parameters that the operator depends on.
Returns the frequencies for each operator parameter with respect to an expectation value of the form \(\langle \psi  U(\mathbf{p})^\dagger \hat{O} U(\mathbf{p})\psi\rangle\).
Trainable parameters that the operator depends on.
Wires that the operator acts on.

arithmetic_depth
¶ Arithmetic depth of the operator.

base_name
¶ If inverse is requested, this is the name of the original operator to be inverted.

basis
= None¶ The target operation for controlled gates. target operation. If not
None
, should take a value of"X"
,"Y"
, or"Z"
.For example,
X
andCNOT
havebasis = "X"
, whereasControlledPhaseShift
andRZ
havebasis = "Z"
. Type
str or None

batch_size
¶ Batch size of the operator if it is used with broadcasted parameters.
The
batch_size
is determined based onndim_params
and the provided parameters for the operator. If (some of) the latter have an additional dimension, and this dimension has the same size for all parameters, its size is the batch size of the operator. If no parameter has an additional dimension, the batch size isNone
. Returns
Size of the parameter broadcasting dimension if present, else
None
. Return type
int or None

control_wires
¶ Control wires of the operator.
For operations that are not controlled, this is an empty
Wires
object of length0
. Returns
The control wires of the operation.
 Return type

grad_method
= None¶

grad_recipe
= None¶ Gradient recipe for the parametershift method.
This is a tuple with one nested list per operation parameter. For parameter \(\phi_k\), the nested list contains elements of the form \([c_i, a_i, s_i]\) where \(i\) is the index of the term, resulting in a gradient recipe of
\[\frac{\partial}{\partial\phi_k}f = \sum_{i} c_i f(a_i \phi_k + s_i).\]If
None
, the default gradient recipe containing the two terms \([c_0, a_0, s_0]=[1/2, 1, \pi/2]\) and \([c_1, a_1, s_1]=[1/2, 1, \pi/2]\) is assumed for every parameter. Type
tuple(Union(list[list[float]], None)) or None

has_matrix
= False¶

hash
¶ Integer hash that uniquely represents the operator.
 Type
int

hyperparameters
¶ Dictionary of nontrainable variables that this operation depends on.
 Type
dict

id
¶ Custom string to label a specific operator instance.

inverse
¶ Boolean determining if the inverse of the operation was requested.

is_hermitian
¶ This property determines if an operator is hermitian.

name
¶ Name of the operator.

ndim_params
¶ Number of dimensions per trainable parameter of the operator.
By default, this property returns the numbers of dimensions of the parameters used for the operator creation. If the parameter sizes for an operator subclass are fixed, this property can be overwritten to return the fixed value.
 Returns
Number of dimensions for each trainable parameter.
 Return type
tuple

num_params
¶

num_wires
= 1¶

parameter_frequencies
¶ Returns the frequencies for each operator parameter with respect to an expectation value of the form \(\langle \psi  U(\mathbf{p})^\dagger \hat{O} U(\mathbf{p})\psi\rangle\).
These frequencies encode the behaviour of the operator \(U(\mathbf{p})\) on the value of the expectation value as the parameters are modified. For more details, please see the
pennylane.fourier
module. Returns
Tuple of frequencies for each parameter. Note that only nonnegative frequency values are returned.
 Return type
list[tuple[int or float]]
Example
>>> op = qml.CRot(0.4, 0.1, 0.3, wires=[0, 1]) >>> op.parameter_frequencies [(0.5, 1), (0.5, 1), (0.5, 1)]
For operators that define a generator, the parameter frequencies are directly related to the eigenvalues of the generator:
>>> op = qml.ControlledPhaseShift(0.1, wires=[0, 1]) >>> op.parameter_frequencies [(1,)] >>> gen = qml.generator(op, format="observable") >>> gen_eigvals = qml.eigvals(gen) >>> qml.gradients.eigvals_to_frequencies(tuple(gen_eigvals)) (1.0,)
For more details on this relationship, see
eigvals_to_frequencies()
.

parameters
¶ Trainable parameters that the operator depends on.
Methods
adjoint
()Create an operation that is the adjoint of this one.
compute_decomposition
(weights, wires, …)Representation of the operator as a product of other operators.
compute_diagonalizing_gates
(*params, wires, …)Sequence of gates that diagonalize the operator in the computational basis (static method).
compute_eigvals
(*params, **hyperparams)Eigenvalues of the operator in the computational basis (static method).
compute_matrix
(*params, **hyperparams)Representation of the operator as a canonical matrix in the computational basis (static method).
compute_sparse_matrix
(*params, **hyperparams)Representation of the operator as a sparse matrix in the computational basis (static method).
compute_terms
(*params, **hyperparams)Representation of the operator as a linear combination of other operators (static method).
Representation of the operator as a product of other operators.
Sequence of gates that diagonalize the operator in the computational basis.
eigvals
()Eigenvalues of the operator in the computational basis.
expand
()Returns a tape that has recorded the decomposition of the operator.
Generator of an operator that is in singleparameterform.
get_parameter_shift
(idx)Multiplier and shift for the given parameter, based on its gradient recipe.
inv
()Inverts the operator.
label
([decimals, base_label, cache])A customizable string representation of the operator.
matrix
([wire_order])Representation of the operator as a matrix in the computational basis.
pow
(z)A list of new operators equal to this one raised to the given power.
queue
([context])Append the operator to the Operator queue.
shape
(k, n_wires, delta_sz)Returns the shape of the weight tensor required for this template.
simplify
()Reduce the depth of nested operators to the minimum.
The parameters required to implement a singlequbit gate as an equivalent
Rot
gate, up to a global phase.sparse_matrix
([wire_order])Representation of the operator as a sparse matrix in the computational basis.
terms
()Representation of the operator as a linear combination of other operators.

adjoint
() → pennylane.operation.Operator¶ Create an operation that is the adjoint of this one.
Adjointed operations are the conjugated and transposed version of the original operation. Adjointed ops are equivalent to the inverted operation for unitary gates.
 Returns
The adjointed operation.

static
compute_decomposition
(weights, wires, s_wires, d_wires, k, init_state)[source]¶ Representation of the operator as a product of other operators.
\[O = O_1 O_2 \dots O_n.\]See also
 Parameters
weights (tensor_like) – tensor containing the parameters entering the Z rotation
wires (Any or Iterable[Any]) – wires that the operator acts on
k (int) – number of times UpCCGSD unitary is repeated
s_wires (Iterable[Any]) – single excitation wires
d_wires (Iterable[Any]) – double excitation wires
init_state (array[int]) – Length
len(wires)
occupationnumber vector representing the HF state.
 Returns
decomposition of the operator
 Return type
list[Operator]

static
compute_diagonalizing_gates
(*params, wires, **hyperparams)¶ Sequence of gates that diagonalize the operator in the computational basis (static method).
Given the eigendecomposition \(O = U \Sigma U^{\dagger}\) where \(\Sigma\) is a diagonal matrix containing the eigenvalues, the sequence of diagonalizing gates implements the unitary \(U^{\dagger}\).
The diagonalizing gates rotate the state into the eigenbasis of the operator.
See also
 Parameters
params (list) – trainable parameters of the operator, as stored in the
parameters
attributewires (Iterable[Any], Wires) – wires that the operator acts on
hyperparams (dict) – nontrainable hyperparameters of the operator, as stored in the
hyperparameters
attribute
 Returns
list of diagonalizing gates
 Return type
list[Operator]

static
compute_eigvals
(*params, **hyperparams)¶ Eigenvalues of the operator in the computational basis (static method).
If
diagonalizing_gates
are specified and implement a unitary \(U\), the operator can be reconstructed as\[O = U \Sigma U^{\dagger},\]where \(\Sigma\) is the diagonal matrix containing the eigenvalues.
Otherwise, no particular order for the eigenvalues is guaranteed.
 Parameters
params (list) – trainable parameters of the operator, as stored in the
parameters
attributehyperparams (dict) – nontrainable hyperparameters of the operator, as stored in the
hyperparameters
attribute
 Returns
eigenvalues
 Return type
tensor_like

static
compute_matrix
(*params, **hyperparams)¶ Representation of the operator as a canonical matrix in the computational basis (static method).
The canonical matrix is the textbook matrix representation that does not consider wires. Implicitly, this assumes that the wires of the operator correspond to the global wire order.
 Parameters
params (list) – trainable parameters of the operator, as stored in the
parameters
attributehyperparams (dict) – nontrainable hyperparameters of the operator, as stored in the
hyperparameters
attribute
 Returns
matrix representation
 Return type
tensor_like

static
compute_sparse_matrix
(*params, **hyperparams)¶ Representation of the operator as a sparse matrix in the computational basis (static method).
The canonical matrix is the textbook matrix representation that does not consider wires. Implicitly, this assumes that the wires of the operator correspond to the global wire order.
See also
 Parameters
params (list) – trainable parameters of the operator, as stored in the
parameters
attributehyperparams (dict) – nontrainable hyperparameters of the operator, as stored in the
hyperparameters
attribute
 Returns
sparse matrix representation
 Return type
scipy.sparse._csr.csr_matrix

static
compute_terms
(*params, **hyperparams)¶ Representation of the operator as a linear combination of other operators (static method).
\[O = \sum_i c_i O_i\]See also
 Parameters
params (list) – trainable parameters of the operator, as stored in the
parameters
attributehyperparams (dict) – nontrainable hyperparameters of the operator, as stored in the
hyperparameters
attribute
 Returns
list of coefficients and list of operations
 Return type
tuple[list[tensor_like or float], list[Operation]]

decomposition
()¶ Representation of the operator as a product of other operators.
\[O = O_1 O_2 \dots O_n\]A
DecompositionUndefinedError
is raised if no representation by decomposition is defined.See also
 Returns
decomposition of the operator
 Return type
list[Operator]

diagonalizing_gates
()¶ Sequence of gates that diagonalize the operator in the computational basis.
Given the eigendecomposition \(O = U \Sigma U^{\dagger}\) where \(\Sigma\) is a diagonal matrix containing the eigenvalues, the sequence of diagonalizing gates implements the unitary \(U^{\dagger}\).
The diagonalizing gates rotate the state into the eigenbasis of the operator.
A
DiagGatesUndefinedError
is raised if no representation by decomposition is defined.See also
 Returns
a list of operators
 Return type
list[Operator] or None

eigvals
()¶ Eigenvalues of the operator in the computational basis.
If
diagonalizing_gates
are specified and implement a unitary \(U\), the operator can be reconstructed as\[O = U \Sigma U^{\dagger},\]where \(\Sigma\) is the diagonal matrix containing the eigenvalues.
Otherwise, no particular order for the eigenvalues is guaranteed.
Note
When eigenvalues are not explicitly defined, they are computed automatically from the matrix representation. Currently, this computation is not differentiable.
A
EigvalsUndefinedError
is raised if the eigenvalues have not been defined and cannot be inferred from the matrix representation.See also
 Returns
eigenvalues
 Return type
tensor_like

expand
()¶ Returns a tape that has recorded the decomposition of the operator.
 Returns
quantum tape
 Return type

generator
()¶ Generator of an operator that is in singleparameterform.
For example, for operator
\[U(\phi) = e^{i\phi (0.5 Y + Z\otimes X)}\]we get the generator
>>> U.generator() (0.5) [Y0] + (1.0) [Z0 X1]
The generator may also be provided in the form of a dense or sparse Hamiltonian (using
Hermitian
andSparseHamiltonian
respectively).The default value to return is
None
, indicating that the operation has no defined generator.

get_parameter_shift
(idx)¶ Multiplier and shift for the given parameter, based on its gradient recipe.
 Parameters
idx (int) – parameter index within the operation
 Returns
list of multiplier, coefficient, shift for each term in the gradient recipe
 Return type
list[[float, float, float]]
Note that the default value for
shift
is None, which is replaced by the default shift \(\pi/2\).

inv
()¶ Inverts the operator.
This method concatenates a string to the name of the operation, to indicate that the inverse will be used for computations.
Any subsequent call of this method will toggle between the original operation and the inverse of the operation.
 Returns
operation to be inverted
 Return type
Operator

label
(decimals=None, base_label=None, cache=None)¶ A customizable string representation of the operator.
 Parameters
decimals=None (int) – If
None
, no parameters are included. Else, specifies how to round the parameters.base_label=None (str) – overwrite the nonparameter component of the label
cache=None (dict) – dictionary that carries information between label calls in the same drawing
 Returns
label to use in drawings
 Return type
str
Example:
>>> op = qml.RX(1.23456, wires=0) >>> op.label() "RX" >>> op.label(decimals=2) "RX\n(1.23)" >>> op.label(base_label="my_label") "my_label" >>> op.label(decimals=2, base_label="my_label") "my_label\n(1.23)" >>> op.inv() >>> op.label() "RX⁻¹"
If the operation has a matrixvalued parameter and a cache dictionary is provided, unique matrices will be cached in the
'matrices'
key list. The label will contain the index of the matrix in the'matrices'
list.>>> op2 = qml.QubitUnitary(np.eye(2), wires=0) >>> cache = {'matrices': []} >>> op2.label(cache=cache) 'U(M0)' >>> cache['matrices'] [tensor([[1., 0.], [0., 1.]], requires_grad=True)] >>> op3 = qml.QubitUnitary(np.eye(4), wires=(0,1)) >>> op3.label(cache=cache) 'U(M1)' >>> cache['matrices'] [tensor([[1., 0.], [0., 1.]], requires_grad=True), tensor([[1., 0., 0., 0.], [0., 1., 0., 0.], [0., 0., 1., 0.], [0., 0., 0., 1.]], requires_grad=True)]

matrix
(wire_order=None)¶ Representation of the operator as a matrix in the computational basis.
If
wire_order
is provided, the numerical representation considers the position of the operator’s wires in the global wire order. Otherwise, the wire order defaults to the operator’s wires.If the matrix depends on trainable parameters, the result will be cast in the same autodifferentiation framework as the parameters.
A
MatrixUndefinedError
is raised if the matrix representation has not been defined.See also
 Parameters
wire_order (Iterable) – global wire order, must contain all wire labels from the operator’s wires
 Returns
matrix representation
 Return type
tensor_like

pow
(z) → List[pennylane.operation.Operator]¶ A list of new operators equal to this one raised to the given power.
 Parameters
z (float) – exponent for the operator
 Returns
list[
Operator
]

queue
(context=<class 'pennylane.queuing.QueuingContext'>)¶ Append the operator to the Operator queue.

static
shape
(k, n_wires, delta_sz)[source]¶ Returns the shape of the weight tensor required for this template. :param k: Number of layers :type k: int :param n_wires: Number of qubits :type n_wires: int :param delta_sz: Specifies the selection rules
sz[p]  sz[r] = delta_sz
:type delta_sz: int :param for the spinprojectionsz
of the orbitals involved in the single excitations.: :paramdelta_sz
can take the values \(0\) and \(\pm 1\).: Returns
shape
 Return type
tuple[int]

simplify
() → pennylane.operation.Operator¶ Reduce the depth of nested operators to the minimum.
 Returns
simplified operator
 Return type

single_qubit_rot_angles
()¶ The parameters required to implement a singlequbit gate as an equivalent
Rot
gate, up to a global phase. Returns
A list of values \([\phi, \theta, \omega]\) such that \(RZ(\omega) RY(\theta) RZ(\phi)\) is equivalent to the original operation.
 Return type
tuple[float, float, float]

sparse_matrix
(wire_order=None)¶ Representation of the operator as a sparse matrix in the computational basis.
If
wire_order
is provided, the numerical representation considers the position of the operator’s wires in the global wire order. Otherwise, the wire order defaults to the operator’s wires.A
SparseMatrixUndefinedError
is raised if the sparse matrix representation has not been defined.See also
 Parameters
wire_order (Iterable) – global wire order, must contain all wire labels from the operator’s wires
 Returns
sparse matrix representation
 Return type
scipy.sparse._csr.csr_matrix

terms
()¶ Representation of the operator as a linear combination of other operators.
\[O = \sum_i c_i O_i\]A
TermsUndefinedError
is raised if no representation by terms is defined.See also
 Returns
list of coefficients \(c_i\) and list of operations \(O_i\)
 Return type
tuple[list[tensor_like or float], list[Operation]]