# qml.kUpCCGSD¶

class kUpCCGSD(weights, wires, k=1, delta_sz=0, init_state=None, do_queue=True, id=None)[source]

Implements the k-Unitary Pair Coupled-Cluster Generalized Singles and Doubles (k-UpCCGSD) ansatz.

The k-UpCCGSD ansatz calls the FermionicSingleExcitation() and FermionicDoubleExcitation() templates to exponentiate the product of $$k$$ generalized singles and pair coupled-cluster 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 coupled-cluster” refers to the fact that the double excitations contain only those two-body excitations that move a pair of electrons from one spatial orbital to another. This k-UpCCGSD belongs to the family of Unitary Coupled Cluster (UCC) based ansätze, commonly used to solve quantum chemistry problems on quantum computers.

The k-UpCCGSD unitary, within the first-order 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() and FermionicDoubleExcitation(). These parameters are the coupled-cluster 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 spin-projection sz 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) occupation-number vector representing the HF state. init_state is used to initialize the wires.

1. The number of wires has to be equal to the number of spin-orbitals included in the active space, and should be even.

2. 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 pennylane as qml
from pennylane import numpy as np

# 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 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
@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

# Store the values of the cost function
energy = [ansatz(weights)]

# Store the values of the circuit weights
angle = [weights]
max_iterations = 100
conv_tol = 1e-06
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 ground-state energy = {energy[-1]:.8f} Ha")
print("\n" f"Optimal value of the circuit parameters = {angle[-1]}")

Step = 0,  Energy = -1.08949110 Ha
Step = 4,  Energy = -1.13370605 Ha
Step = 8,  Energy = -1.13581648 Ha
Step = 12,  Energy = -1.13613171 Ha
Step = 16,  Energy = -1.13618030 Ha
Step = 20,  Energy = -1.13618779 Ha

Final value of the ground-state energy = -1.13618779 Ha

Optimal value of the circuit parameters = [[0.97879636 0.46093583 0.98108824
0.45864352 0.65531446 0.44558289]]


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)

 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 The target operation for controlled gates. batch_size Batch size of the operator if it is used with broadcasted parameters. control_wires Control wires of the operator. grad_method grad_recipe Gradient recipe for the parameter-shift method. has_adjoint has_decomposition has_diagonalizing_gates has_matrix hash Integer hash that uniquely represents the operator. hyperparameters Dictionary of non-trainable variables that this operation depends on. 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. num_params Number of trainable parameters that the operator depends on. num_wires parameter_frequencies Returns the frequencies for each operator parameter with respect to an expectation value of the form $$\langle \psi | U(\mathbf{p})^\dagger \hat{O} U(\mathbf{p})|\psi\rangle$$. parameters Trainable parameters that the operator depends on. wires 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 and CNOT have basis = "X", whereas ControlledPhaseShift and RZ have basis = "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 on ndim_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 is None.

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 length 0.

Returns

The control wires of the operation.

Return type

Wires

grad_method = None
grad_recipe = None

Gradient recipe for the parameter-shift 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_adjoint = False
has_decomposition = True
has_diagonalizing_gates = False
has_matrix = False
hash

Integer hash that uniquely represents the operator.

Type

int

hyperparameters

Dictionary of non-trainable 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 non-negative 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)
(1.0,)


For more details on this relationship, see eigvals_to_frequencies().

parameters

Trainable parameters that the operator depends on.

wires

Wires that the operator acts on.

Returns

wires

Return type

Wires

 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). Representation of the operator as a product of other operators. Sequence of gates that diagonalize the operator in the computational basis. Eigenvalues of the operator in the computational basis. Returns a tape that has recorded the decomposition of the operator. Generator of an operator that is in single-parameter-form. Inverts the operator. label([decimals, base_label, cache]) A customizable string representation of the operator. map_wires(wire_map) Returns a copy of the current operator with its wires changed according to the given wire map. matrix([wire_order]) Representation of the operator as a matrix in the computational basis. 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. Reduce the depth of nested operators to the minimum. The parameters required to implement a single-qubit 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. 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

static compute_decomposition(weights, wires, s_wires, d_wires, k, init_state)[source]

Representation of the operator as a product of other operators.

$O = O_1 O_2 \dots O_n.$

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) occupation-number 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.

Parameters
• params (list) – trainable parameters of the operator, as stored in the parameters attribute

• wires (Iterable[Any], Wires) – wires that the operator acts on

• hyperparams (dict) – non-trainable 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 $$U^{dagger}$$, 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 attribute

• hyperparams (dict) – non-trainable 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 attribute

• hyperparams (dict) – non-trainable 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.

Parameters
• params (list) – trainable parameters of the operator, as stored in the parameters attribute

• hyperparams (dict) – non-trainable hyperparameters of the operator, as stored in the hyperparameters attribute

Returns

sparse matrix representation

Return type

scipy.sparse._csr.csr_matrix

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.

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.

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 $$U^{dagger}$$, 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.

Returns

eigenvalues

Return type

tensor_like

expand()

Returns a tape that has recorded the decomposition of the operator.

Returns

quantum tape

Return type

QuantumTape

generator()

Generator of an operator that is in single-parameter-form.

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 and SparseHamiltonian respectively).

The default value to return is None, indicating that the operation has no defined generator.

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 non-parameter 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 matrix-valued 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.],
>>> op3 = qml.QubitUnitary(np.eye(4), wires=(0,1))
>>> op3.label(cache=cache)
'U(M1)'
>>> cache['matrices']
[tensor([[1., 0.],
tensor([[1., 0., 0., 0.],
[0., 1., 0., 0.],
[0., 0., 1., 0.],

map_wires(wire_map: dict)

Returns a copy of the current operator with its wires changed according to the given wire map.

Parameters

wire_map (dict) – dictionary containing the old wires as keys and the new wires as values

Returns

new operator

Return type

Operator

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.

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.QueuingManager'>)

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 spin-projection sz of the orbitals involved in the single excitations.: :param delta_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

Operator

single_qubit_rot_angles()

The parameters required to implement a single-qubit 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.

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.

Returns

list of coefficients $$c_i$$ and list of operations $$O_i$$

Return type

tuple[list[tensor_like or float], list[Operation]]