# 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 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 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 = 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 ground-state 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)

 base_name Get base name of the operator. basis The basis of an operation, or for controlled gates, of the target operation. control_wires Returns the control wires. eigvals Eigenvalues of an instantiated operator. generator Generator of the operation. grad_method Gradient computation method. grad_recipe Gradient recipe for the parameter-shift method. hash returns an integer hash uniquely representing the operator id String for the ID of the operator. inverse Boolean determining if the inverse of the operation was requested. matrix Matrix representation of an instantiated operator in the computational basis. name Get and set the name of the operator. num_params Number of trainable parameters that this operator expects to be fed via the dynamic *params argument. num_wires parameters Current parameter values. single_qubit_rot_angles The parameters required to implement a single-qubit gate as an equivalent Rot gate, up to a global phase. string_for_inverse wires Wires of this operator.
base_name

Get base name of the operator.

basis = None

The basis of an operation, or for controlled gates, of the 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

control_wires

Returns the control wires. 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

eigvals
generator

Generator of the operation.

A length-2 list [generator, scaling_factor], where

• generator is an existing PennyLane operation class or $$2\times 2$$ Hermitian array that acts as the generator of the current operation

• scaling_factor represents a scaling factor applied to the generator operation

For example, if $$U(\theta)=e^{i0.7\theta \sigma_x}$$, then $$\sigma_x$$, with scaling factor $$s$$, is the generator of operator $$U(\theta)$$:

generator = [PauliX, 0.7]


Default is [None, 1], indicating the operation has no generator.

grad_method

• 'A': analytic differentiation using the parameter-shift method.

• 'F': finite difference numerical differentiation.

• None: the operation may not be differentiated.

Default is 'F', or None if the Operation has zero parameters.

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

hash

returns an integer hash uniquely representing the operator

Type

int

id

String for the ID of the operator.

inverse

Boolean determining if the inverse of the operation was requested.

matrix
name

Get and set the name of the operator.

num_params
num_wires = -1
parameters

Current parameter values.

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]

string_for_inverse = '.inv'
wires

Wires of this operator.

Returns

wires

Return type

Wires

 adjoint([do_queue]) Create an operation that is the adjoint of this one. Decomposes this operator into products of other operators. decomposition(*params, wires) Defines a decomposition of this operator into products of other operators. Returns a tape containing the decomposed operations, rather than a list. get_parameter_shift(idx[, shift]) Multiplier and shift for the given parameter, based on its gradient recipe. Inverts the operation, such that the inverse will be used for the computations by the specific device. label([decimals, base_label]) A customizable string representation of the operator. 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.
adjoint(do_queue=False)

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.

Parameters

do_queue – Whether to add the adjointed gate to the context queue.

Returns

decompose()

Decomposes this operator into products of other operators.

Returns

list[Operation]

static decomposition(*params, wires)

Defines a decomposition of this operator into products of other operators.

Parameters
• params (tuple[float, int, array]) – operator parameters

• wires (Union(Sequence[int], Wires)) – wires the operator acts on

Returns

list[Operation]

expand()[source]

Returns a tape containing the decomposed operations, rather than a list.

Returns

Returns a quantum tape that contains the operations decomposition, or if not implemented, simply the operation itself.

Return type

JacobianTape

get_parameter_shift(idx, shift=1.5707963267948966)

Multiplier and shift for the given parameter, based on its gradient recipe.

Parameters

idx (int) – parameter index

Returns

list of multiplier, coefficient, shift for each term in the gradient recipe

Return type

list[[float, float, float]]

inv()

Inverts the operation, such that the inverse will be used for the computations by the specific device.

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)

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

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"

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 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]