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 particle-conserving 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 Hartree-Fock 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:


../../_images/particle_conserving_u1.png

The repeated units across several qubits are shown in dotted boxes. Each layer contains \(N-1\) particle-conserving two-parameter 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 single-qubit rotation \(R(0, 2 \theta, 0)\) apply the Pauli Z operator and an arbitrary rotation on the qubit n with qubit m bein the control qubit,


../../_images/u1_decomposition.png

\(U_A(\phi)\) 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,


../../_images/ua_decomposition.png

where,


../../_images/phaseshift_decomposition.png

The quantum circuits above decomposing the unitaries \(U_{1,\mathrm{ex}}(\phi, \theta)\) and \(U_A(\phi)\) are implemented by the u1_ex_gate and decompose_ua functions, respectively. \(R_\phi\) refers to the PhaseShift 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 \(M=N-1\) 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 Hartree-Fock state used to initialize the wires

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

  2. The number of trainable parameters scales linearly with the number of layers as \(2D(N-1)\).

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 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
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)

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

Gradient computation 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.

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

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.

static shape(n_layers, n_wires)[source]

Returns the shape of the weight tensor required for this template.

Parameters
  • n_layers (int) – number of layers

  • n_wires (int) – number of qubits

Returns

shape

Return type

tuple[int]