qml.templates.layers.ParticleConservingU1

class ParticleConservingU1(weights, wires, init_state=None, do_queue=True, id=None)[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.

basis

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

control_wires

For operations that are controlled, returns the set of 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.

is_composable_rotation

True if composing multiple copies of the operation results in an addition (or alternative accumulation) of parameters.

is_self_inverse

True if the operation is its own inverse.

is_symmetric_over_all_wires

True if the operation is the same if you exchange the order of wires.

is_symmetric_over_control_wires

True if the operation is the same if you exchange the order of all but the last wire.

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.

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

For operations that are controlled, returns the set of control wires.

Returns

The set of 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

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

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.

is_composable_rotation = None

True if composing multiple copies of the operation results in an addition (or alternative accumulation) of parameters.

For example, qml.RZ is a composable rotation. Applying qml.RZ(0.1, wires=0) followed by qml.RZ(0.2, wires=0) is equivalent to performing a single rotation qml.RZ(0.3, wires=0).

If set to None, the operation will be ignored during compilation transforms that merge adjacent rotations.

Type

bool or None

is_self_inverse = None

True if the operation is its own inverse.

If None, all instances of the given operation will be ignored during compilation transforms involving inverse cancellation.

Type

bool or None

is_symmetric_over_all_wires = None

True if the operation is the same if you exchange the order of wires.

For example, qml.CZ(wires=[0, 1]) has the same effect as qml.CZ(wires=[1, 0]) due to symmetry of the operation.

If None, all instances of the operation will be ignored during compilation transforms that check for wire symmetry.

Type

bool or None

is_symmetric_over_control_wires = None

True if the operation is the same if you exchange the order of all but the last wire.

For example, qml.Toffoli(wires=[0, 1, 2]) has the same effect as qml.Toffoli(wires=[1, 0, 2]), but neither are the same as qml.Toffoli(wires=[0, 2, 1]).

If None, all instances of the operation will be ignored during compilation transforms that check for control-wire symmetry.

Type

bool or None

matrix
name

Get and set the name of the operator.

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

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([context])

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

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

queue(context=<class 'pennylane.queuing.QueuingContext'>)

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]