qml.templates.subroutines.DoubleExcitationUnitary¶

class
DoubleExcitationUnitary
(weight, wires1=None, wires2=None, do_queue=True)[source]¶ Bases:
pennylane.operation.Operation
Circuit to exponentiate the tensor product of Pauli matrices representing the doubleexcitation operator entering the Unitary CoupledCluster Singles and Doubles (UCCSD) ansatz. UCCSD is a VQE ansatz commonly used to run quantum chemistry simulations.
The CC doubleexcitation operator is given by
\[\hat{U}_{pqrs}(\theta) = \mathrm{exp} \{ \theta (\hat{c}_p^\dagger \hat{c}_q^\dagger \hat{c}_r \hat{c}_s  \mathrm{H.c.}) \},\]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 JordanWigner transformation the fermionic operator defined above can be written in terms of Pauli matrices (for more details see arXiv:1805.04340):
\[\hat{U}_{pqrs}(\theta) = \mathrm{exp} \Big\{ \frac{i\theta}{8} \bigotimes_{b=s+1}^{r1} \hat{Z}_b \bigotimes_{a=q+1}^{p1} \hat{Z}_a (\hat{X}_s \hat{X}_r \hat{Y}_q \hat{X}_p + \hat{Y}_s \hat{X}_r \hat{Y}_q \hat{Y}_p + \hat{X}_s \hat{Y}_r \hat{Y}_q \hat{Y}_p + \hat{X}_s \hat{X}_r \hat{X}_q \hat{Y}_p  \mathrm{H.c.} ) \Big\}\]The quantum circuit to exponentiate the tensor product of Pauli matrices entering the latter equation is shown below (see arXiv:1805.04340):
As explained in Seely et al. (2012), the exponential of a tensor product of PauliZ operators can be decomposed in terms of \(2(n1)\) CNOT gates and a singlequbit Zrotation referred to as \(U_\theta\) in the figure above. If there are \(X\) or:math:Y Pauli matrices in the product, the Hadamard (\(H\)) or \(R_x\) gate has to be applied to change to the \(X\) or \(Y\) basis, respectively. The latter operations are denoted as \(U_1\), \(U_2\), \(U_3\) and \(U_4\) in the figure above. See the Usage Details section for more details.
 Parameters
weight (float or tensor_like) – angle \(\theta\) entering the Z rotation acting on wire
p
wires1 (Iterable) – Wires of the qubits representing the subset of occupied orbitals in the interval
[s, r]
. The first wire is interpreted ass
and the last wire asr
. Wires in between are acted on with CNOT gates to compute the parity of the set of qubits.wires2 (Iterable) – Wires of the qubits representing the subset of unoccupied orbitals in the interval
[q, p]
. The first wire is interpreted asq
and the last wire is interpreted asp
. Wires in between are acted on with CNOT gates to compute the parity of the set of qubits.
Usage Details
Notice that:
\(\hat{U}_{pqrs}(\theta)\) involves eight exponentiations where \(\hat{U}_1\), \(\hat{U}_2\), \(\hat{U}_3\), \(\hat{U}_4\) and \(\hat{U}_\theta\) are defined as follows,
\[\begin{split}[U_1, && U_2, U_3, U_4, U_{\theta}] = \\ && \Bigg\{\bigg[H, H, R_x(\frac{\pi}{2}), H, R_z(\theta/8)\bigg], \bigg[R_x(\frac{\pi}{2}), H, R_x(\frac{\pi}{2}), R_x(\frac{\pi}{2}), R_z(\frac{\theta}{8}) \bigg], \\ && \bigg[H, R_x(\frac{\pi}{2}), R_x(\frac{\pi}{2}), R_x(\frac{\pi}{2}), R_z(\frac{\theta}{8}) \bigg], \bigg[H, H, H, R_x(\frac{\pi}{2}), R_z(\frac{\theta}{8}) \bigg], \\ && \bigg[R_x(\frac{\pi}{2}), H, H, H, R_z(\frac{\theta}{8}) \bigg], \bigg[H, R_x(\frac{\pi}{2}), H, H, R_z(\frac{\theta}{8}) \bigg], \\ && \bigg[R_x(\frac{\pi}{2}), R_x(\frac{\pi}{2}), R_x(\frac{\pi}{2}), H, R_z(\frac{\theta}{8}) \bigg], \bigg[R_x(\frac{\pi}{2}), R_x(\frac{\pi}{2}), H, R_x(\frac{\pi}{2}), R_z(\frac{\theta}{8}) \bigg] \Bigg\}\end{split}\]For a given quadruple
[s, r, q, p]
with \(p>q>r>s\), seventytwo singlequbit and16*(len(wires1)1 + len(wires2)1 + 1)
CNOT operations are applied. Consecutive CNOT gates act on qubits with indices betweens
andr
andq
andp
while a single CNOT acts on wiresr
andq
. 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:
import pennylane as qml from pennylane.templates import DoubleExcitationUnitary dev = qml.device('default.qubit', wires=5) @qml.qnode(dev) def circuit(weight, wires1=None, wires2=None): DoubleExcitationUnitary(weight, wires1=wires1, wires2=wires2) return qml.expval(qml.PauliZ(0)) weight = 1.34817 print(circuit(weight, wires1=[0, 1], wires2=[2, 3, 4]))
Attributes
Get base name of the operator.
Eigenvalues of an instantiated operator.
Generator of the operation.
Gradient computation method.
Gradient recipe for the parametershift method.
Boolean determining if the inverse of the operation was requested.
Matrix representation of an instantiated operator in the computational basis.
Get and set the name of the operator.
Current parameter values.
Wires of this operator.

base_name
¶ Get base name of the operator.

eigvals
¶

generator
¶ Generator of the operation.
A length2 list
[generator, scaling_factor]
, wheregenerator
is an existing PennyLane operation class or \(2\times 2\) Hermitian array that acts as the generator of the current operationscaling_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
¶ Gradient computation method.
'A'
: analytic differentiation using the parametershift method.'F'
: finite difference numerical differentiation.None
: the operation may not be differentiated.
Default is
'F'
, orNone
if the Operation has zero parameters.

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

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'¶
Methods
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.
expand
()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.
inv
()Inverts the operation, such that the inverse will be used for the computations by the specific device.
queue
()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
The adjointed operation.

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

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.
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