# qml.templates.subroutines.UCCSD¶

class UCCSD(weights, wires, s_wires=None, d_wires=None, init_state=None, do_queue=True)[source]

Implements the Unitary Coupled-Cluster Singles and Doubles (UCCSD) ansatz.

The UCCSD ansatz calls the SingleExcitationUnitary() and 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:

$\hat{U}(\vec{\theta}) = \prod_{p > r} \mathrm{exp} \Big\{\theta_{pr} (\hat{c}_p^\dagger \hat{c}_r-\mathrm{H.c.}) \Big\} \prod_{p > q > r > s} \mathrm{exp} \Big\{\theta_{pqrs} (\hat{c}_p^\dagger \hat{c}_q^\dagger \hat{c}_r \hat{c}_s-\mathrm{H.c.}) \Big\}$

where $$\hat{c}$$ and $$\hat{c}^\dagger$$ are the fermionic annihilation and creation operators and the indices $$r, s$$ and $$p, q$$ run over the occupied and unoccupied molecular orbitals, respectively. Using the Jordan-Wigner transformation the UCCSD unitary defined above can be written in terms of Pauli matrices as follows (for more details see arXiv:1805.04340):

$\begin{split}\hat{U}(\vec{\theta}) = && \prod_{p > r} \mathrm{exp} \Big\{ \frac{i\theta_{pr}}{2} \bigotimes_{a=r+1}^{p-1} \hat{Z}_a (\hat{Y}_r \hat{X}_p - \mathrm{H.c.}) \Big\} \\ && \times \prod_{p > q > r > s} \mathrm{exp} \Big\{ \frac{i\theta_{pqrs}}{8} \bigotimes_{b=s+1}^{r-1} \hat{Z}_b \bigotimes_{a=q+1}^{p-1} \hat{Z}_a (\hat{X}_s \hat{X}_r \hat{Y}_q \hat{X}_p + \hat{Y}_s \hat{X}_r \hat{Y}_q \hat{Y}_p + \hat{X}_s \hat{Y}_r \hat{Y}_q \hat{Y}_p + \hat{X}_s \hat{X}_r \hat{X}_q \hat{Y}_p - \{\mathrm{H.c.}\}) \Big\}.\end{split}$
Parameters
• weights (tensor_like) – Size (len(s_wires) + len(d_wires),) tensor containing the parameters $$\theta_{pr}$$ and $$\theta_{pqrs}$$ entering the Z rotation in SingleExcitationUnitary() and DoubleExcitationUnitary(). These parameters are the coupled-cluster amplitudes that need to be optimized for each single and double excitation generated with the excitations() function.

• wires (Iterable) – wires that the template acts on

• s_wires (Sequence[Sequence]) – Sequence of lists containing the wires [r,...,p] resulting from the single excitation $$\vert r, p \rangle = \hat{c}_p^\dagger \hat{c}_r \vert \mathrm{HF} \rangle$$, where $$\vert \mathrm{HF} \rangle$$ denotes the Hartee-Fock reference state. The first (last) entry r (p) is considered the wire representing the occupied (unoccupied) orbital where the electron is annihilated (created).

• d_wires (Sequence[Sequence[Sequence]]) – Sequence of lists, each containing two lists that specify the indices [s, ...,r] and [q,..., p] defining the double excitation $$\vert s, r, q, p \rangle = \hat{c}_p^\dagger \hat{c}_q^\dagger \hat{c}_r \hat{c}_s \vert \mathrm{HF} \rangle$$. The entries s and r are wires representing two occupied orbitals where the two electrons are annihilated while the entries q and p correspond to the wires representing two unoccupied orbitals where the electrons are created. Wires in-between represent the occupied and unoccupied orbitals in the intervals [s, r] and [q, p], respectively.

• init_state (array[int]) – Length len(wires) occupation-number vector representing the HF state. init_state is used to initialize the wires.

Notice that:

1. The number of wires has to be equal to the number of spin orbitals included in the active space.

2. The single and double excitations can be generated be generated with the function excitations(). See example below.

3. The vector of parameters weights is a one-dimensional array of size len(s_wires)+len(d_wires)

An example of how to use this template is shown below:

import pennylane as qml
from pennylane import qchem
from pennylane.templates import UCCSD

from functools import partial

# Build the electronic Hamiltonian
name = "h2"
geo_file = "h2.xyz"
h, qubits = qchem.molecular_hamiltonian(name, geo_file)

# Number of electrons
electrons = 2

# Define the HF state
ref_state = qchem.hf_state(electrons, qubits)

# Generate single and double excitations
singles, doubles = qchem.excitations(electrons, qubits)

# Map excitations to the wires the UCCSD circuit will act on
s_wires, d_wires = qchem.excitations_to_wires(singles, doubles)

# Define the device
dev = qml.device('default.qubit', wires=qubits)

# Define the UCCSD ansatz
ansatz = partial(UCCSD, init_state=ref_state, s_wires=s_wires, d_wires=d_wires)

# Define the cost function
cost_fn = qml.ExpvalCost(ansatz, h, dev)

# Compute the expectation value of 'h' for given set of parameters 'params'
params = np.random.normal(0, np.pi, len(singles) + len(doubles))
print(cost_fn(params))

 base_name Get base name of the operator. 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. 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 num_wires par_domain parameters Current parameter values. string_for_inverse wires Wires of this operator.
base_name

Get base name of the operator.

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

inverse

Boolean determining if the inverse of the operation was requested.

matrix
name

Get and set the name of the operator.

num_params = 1
num_wires = -1
par_domain = 'A'
parameters

Current parameter values.

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. decomposition(*params, wires) Returns a template decomposing the operation into other quantum operations. 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. Append the operator to the Operator queue.
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

static decomposition(*params, wires)

Returns a template decomposing the operation into other quantum operations.

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

multiplier, shift

Return type

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

queue()

Append the operator to the Operator queue.