qml.templates.layers.ParticleConservingU1¶

class
ParticleConservingU1
(weights, wires, init_state=None, do_queue=True)[source]¶ Bases:
pennylane.operation.Operation
Implements the heuristic VQE ansatz for quantum chemistry simulations using the particleconserving gate \(U_{1,\mathrm{ex}}\) proposed by Barkoutsos et al. in arXiv:1805.04340.
This template prepares \(N\)qubit trial states by applying \(D\) layers of the entangler block \(U_\mathrm{ent}(\vec{\phi}, \vec{\theta})\) to the HartreeFock state
\[\vert \Psi(\vec{\phi}, \vec{\theta}) \rangle = \hat{U}^{(D)}_\mathrm{ent}(\vec{\phi}_D, \vec{\theta}_D) \dots \hat{U}^{(2)}_\mathrm{ent}(\vec{\phi}_2, \vec{\theta}_2) \hat{U}^{(1)}_\mathrm{ent}(\vec{\phi}_1, \vec{\theta}_1) \vert \mathrm{HF}\rangle.\]The circuit implementing the entangler blocks is shown in the figure below:
The repeated units across several qubits are shown in dotted boxes. Each layer contains \(N1\) particleconserving twoparameter exchange gates \(U_{1,\mathrm{ex}}(\phi, \theta)\) that act on pairs of nearest neighbors qubits. The unitary matrix representing \(U_{1,\mathrm{ex}}(\phi, \theta)\) is given by (see arXiv:1805.04340),
\[\begin{split}U_{1, \mathrm{ex}}(\phi, \theta) = \left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & \mathrm{cos}(\theta) & e^{i\phi} \mathrm{sin}(\theta) & 0 \\ 0 & e^{i\phi} \mathrm{sin}(\theta) & \mathrm{cos}(\theta) & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right).\end{split}\]The figure below shows the circuit decomposing \(U_{1, \mathrm{ex}}\) in elementary gates. The Pauli matrix \(\sigma_z\) and singlequbit rotation \(R(0, 2 \theta, 0)\) apply the Pauli Z operator and an arbitrary rotation on the qubit
n
with qubitm
bein the control qubit,\(U_A(\phi)\) is the unitary matrix
\[\begin{split}U_A(\phi) = \left(\begin{array}{cc} 0 & e^{i\phi} \\ e^{i\phi} & 0 \\ \end{array}\right),\end{split}\]which is applied controlled on the state of qubit
m
and can be further decomposed in terms of the quantum operations supported by Pennylane,where,
The quantum circuits above decomposing the unitaries \(U_{1,\mathrm{ex}}(\phi, \theta)\) and \(U_A(\phi)\) are implemented by the
u1_ex_gate
anddecompose_ua
functions, respectively. \(R_\phi\) refers to thePhaseShift
gate in the circuit diagram. Parameters
weights (tensor_like) – Array of weights of shape
(D, M, 2)
.D
is the number of entangler block layers and \(M=N1\) is the number of exchange gates \(U_{1,\mathrm{ex}}\) per layer.wires (Iterable) – wires that the template acts on
init_state (tensor_like) – iterable or shape
(len(wires),)
tensor representing the HartreeFock state used to initialize the wires
Usage Details
The number of wires \(N\) has to be equal to the number of spin orbitals included in the active space.
The number of trainable parameters scales linearly with the number of layers as \(2D(N1)\).
An example of how to use this template is shown below:
import pennylane as qml from pennylane.templates import ParticleConservingU1 from functools import partial # Build the electronic Hamiltonian from a local .xyz file h, qubits = qml.qchem.molecular_hamiltonian("h2", "h2.xyz") # 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 ansatz = partial(ParticleConservingU1, init_state=ref_state) # Define the cost function cost_fn = qml.ExpvalCost(ansatz, h, dev) # Compute the expectation value of 'h' layers = 2 params = qml.init.particle_conserving_u1_normal(layers, qubits) print(cost_fn(params))
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 = ParticleConservingU1.shape(n_layers=2, n_wires=2) weights = np.random.random(size=shape)
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.
shape
(n_layers, n_wires)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
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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