qml.ParticleConservingU1

class ParticleConservingU1(weights, wires, init_state=None, id=None)

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:

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,

$U_A(\phi)$ 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 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. If None, a tuple of zeros is selected as initial state.

Usage Details

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
import numpy as np
from functools import partial

# Build the electronic Hamiltonian
symbols, coordinates = (['H', 'H'], np.array([0., 0., -0.66140414, 0., 0., 0.66140414]))
h, qubits = qml.qchem.molecular_hamiltonian(symbols, coordinates)

# 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(qml.ParticleConservingU1, init_state=ref_state, wires=dev.wires)

# Define the cost function
@qml.qnode(dev)
def cost_fn(params):
    ansatz(params)
    return qml.expval(h)

# Compute the expectation value of 'h'
layers = 2
shape = qml.ParticleConservingU1.shape(layers, qubits)
params = np.random.random(shape)
print(cost_fn(params))

Parameter shape

The shape of the trainable weights tensor can be computed by the static method shape() and used when creating randomly initialised weight tensors:

shape = qml.ParticleConservingU1.shape(n_layers=2, n_wires=2)
params = np.random.random(size=shape)

Attributes

arithmetic_depth	Arithmetic depth of the operator.
basis	The basis of an operation, or for controlled gates, of the target operation.
batch_size	Batch size of the operator if it is used with broadcasted parameters.
control_wires	Control wires of the operator.
grad_method
grad_recipe	Gradient recipe for the parameter-shift method.
has_adjoint
has_decomposition
has_diagonalizing_gates
has_generator
has_matrix
has_sparse_matrix
hash	Integer hash that uniquely represents the operator.
hyperparameters	Dictionary of non-trainable variables that this operation depends on.
id	Custom string to label a specific operator instance.
is_hermitian	This property determines if an operator is hermitian.
name	String for the name of the operator.
ndim_params	Number of dimensions per trainable parameter of the operator.
num_params	Number of trainable parameters that the operator depends on.
num_wires	Number of wires the operator acts on.
parameter_frequencies	Returns the frequencies for each operator parameter with respect to an expectation value of the form $\langle \psi | U(\mathbf{p})^\dagger \hat{O} U(\mathbf{p})|\psi\rangle$.
parameters	Trainable parameters that the operator depends on.
pauli_rep	A PauliSentence representation of the Operator, or None if it doesn't have one.
wires	Wires that the operator acts on.

arithmetic_depth

Arithmetic depth of the operator.

basis

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

batch_size

Batch size of the operator if it is used with broadcasted parameters.

The batch_size is determined based on ndim_params and the provided parameters for the operator. If (some of) the latter have an additional dimension, and this dimension has the same size for all parameters, its size is the batch size of the operator. If no parameter has an additional dimension, the batch size is None.

Returns

Size of the parameter broadcasting dimension if present, else None.

Return type

int or None

control_wires

Control wires of the operator.

For operations that are not controlled, this is an empty Wires object of length 0.

Returns

The control wires of the operation.

Return type

Wires

grad_method = None

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

has_adjoint = False

has_decomposition = True

has_diagonalizing_gates = False

has_generator = False

has_matrix = False

has_sparse_matrix = False

hash

Integer hash that uniquely represents the operator.

Type

int

hyperparameters

Dictionary of non-trainable variables that this operation depends on.

Type

dict

id

Custom string to label a specific operator instance.

is_hermitian

This property determines if an operator is hermitian.

name

String for the name of the operator.

ndim_params

Number of dimensions per trainable parameter of the operator.

By default, this property returns the numbers of dimensions of the parameters used for the operator creation. If the parameter sizes for an operator subclass are fixed, this property can be overwritten to return the fixed value.

Returns

Number of dimensions for each trainable parameter.

Return type

tuple

num_params

num_wires = -1

Number of wires the operator acts on.

parameter_frequencies

Returns the frequencies for each operator parameter with respect to an expectation value of the form $\langle \psi | U(\mathbf{p})^\dagger \hat{O} U(\mathbf{p})|\psi\rangle$.

These frequencies encode the behaviour of the operator $U(\mathbf{p})$ on the value of the expectation value as the parameters are modified. For more details, please see the pennylane.fourier module.

Returns

Tuple of frequencies for each parameter. Note that only non-negative frequency values are returned.

Return type

list[tuple[int or float]]

Example

>>> op = qml.CRot(0.4, 0.1, 0.3, wires=[0, 1])
>>> op.parameter_frequencies
[(0.5, 1), (0.5, 1), (0.5, 1)]

For operators that define a generator, the parameter frequencies are directly related to the eigenvalues of the generator:

>>> op = qml.ControlledPhaseShift(0.1, wires=[0, 1])
>>> op.parameter_frequencies
[(1,)]
>>> gen = qml.generator(op, format="observable")
>>> gen_eigvals = qml.eigvals(gen)
>>> qml.gradients.eigvals_to_frequencies(tuple(gen_eigvals))
(1.0,)

For more details on this relationship, see eigvals_to_frequencies().

parameters

Trainable parameters that the operator depends on.

pauli_rep

A PauliSentence representation of the Operator, or None if it doesn’t have one.

wires

Wires that the operator acts on.

Returns

wires

Return type

Wires

Methods

adjoint()	Create an operation that is the adjoint of this one.
compute_decomposition(weights, wires, init_state)	Representation of the ParticleConservingU1operator as a product of other operators.
compute_diagonalizing_gates(*params, wires, ...)	Sequence of gates that diagonalize the operator in the computational basis (static method).
compute_eigvals(*params, **hyperparams)	Eigenvalues of the operator in the computational basis (static method).
compute_matrix(*params, **hyperparams)	Representation of the operator as a canonical matrix in the computational basis (static method).
compute_sparse_matrix(*params, **hyperparams)	Representation of the operator as a sparse matrix in the computational basis (static method).
decomposition()	Representation of the operator as a product of other operators.
diagonalizing_gates()	Sequence of gates that diagonalize the operator in the computational basis.
eigvals()	Eigenvalues of the operator in the computational basis.
generator()	Generator of an operator that is in single-parameter-form.
label([decimals, base_label, cache])	A customizable string representation of the operator.
map_wires(wire_map)	Returns a copy of the current operator with its wires changed according to the given wire map.
matrix([wire_order])	Representation of the operator as a matrix in the computational basis.
pow(z)	A list of new operators equal to this one raised to the given power.
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.
simplify()	Reduce the depth of nested operators to the minimum.
single_qubit_rot_angles()	The parameters required to implement a single-qubit gate as an equivalent Rot gate, up to a global phase.
sparse_matrix([wire_order])	Representation of the operator as a sparse matrix in the computational basis.
terms()	Representation of the operator as a linear combination of other operators.

adjoint()

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.

Returns

The adjointed operation.

static compute_decomposition(weights, wires, init_state)

Representation of the ParticleConservingU1operator as a product of other operators.

$$O = O_1 O_2 \dots O_n.$$

See also

decomposition().

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 (Any or Iterable[Any]) – wires that the operator acts on

• init_state (tensor_like) – iterable or shape (len(wires),) tensor representing the Hartree-Fock state used to initialize the wires

Returns

decomposition of the operator

Return type

list[Operator]

Example

>>> weights = torch.tensor([[[0.3, 1.]]])
>>> qml.ParticleConservingU1.compute_decomposition(weights, wires=["a", "b"], init_state=[0, 1])
[BasisEmbedding(wires=['a', 'b']),
 CZ(wires=['a', 'b']),
 CRot(tensor(-0.3000), 3.141592653589793, tensor(0.3000), wires=['a', 'b']),
 PhaseShift(tensor(-0.3000), wires=['b']), CNOT(wires=['a', 'b']),
 PhaseShift(tensor(0.3000), wires=['b']), CNOT(wires=['a', 'b']),
 PhaseShift(tensor(-0.3000), wires=['a']), CZ(wires=['b', 'a']),
 CRot(0, tensor(2.), 0, wires=['b', 'a']), CZ(wires=['a', 'b']),
 CRot(tensor(0.3000), 3.141592653589793, tensor(-0.3000), wires=['a', 'b']),
 PhaseShift(tensor(0.3000), wires=['b']),
 CNOT(wires=['a', 'b']),
 PhaseShift(tensor(-0.3000), wires=['b']),
 CNOT(wires=['a', 'b']),
 PhaseShift(tensor(0.3000), wires=['a'])]

static compute_diagonalizing_gates(*params, wires, **hyperparams)

Sequence of gates that diagonalize the operator in the computational basis (static method).

Given the eigendecomposition $O = U \Sigma U^{\dagger}$ where $\Sigma$ is a diagonal matrix containing the eigenvalues, the sequence of diagonalizing gates implements the unitary $U^{\dagger}$.

The diagonalizing gates rotate the state into the eigenbasis of the operator.

See also

diagonalizing_gates().

Parameters

• params (list) – trainable parameters of the operator, as stored in the parameters attribute

• wires (Iterable[Any], Wires) – wires that the operator acts on

• hyperparams (dict) – non-trainable hyperparameters of the operator, as stored in the hyperparameters attribute

Returns

list of diagonalizing gates

Return type

list[Operator]

static compute_eigvals(*params, **hyperparams)

Eigenvalues of the operator in the computational basis (static method).

If diagonalizing_gates are specified and implement a unitary $U^{\dagger}$, the operator can be reconstructed as

$$O = U \Sigma U^{\dagger},$$

where $\Sigma$ is the diagonal matrix containing the eigenvalues.

Otherwise, no particular order for the eigenvalues is guaranteed.

See also

Operator.eigvals() and qml.eigvals()

Parameters

• *params (list) – trainable parameters of the operator, as stored in the parameters attribute

• **hyperparams (dict) – non-trainable hyperparameters of the operator, as stored in the hyperparameters attribute

Returns

eigenvalues

Return type

tensor_like

static compute_matrix(*params, **hyperparams)

Representation of the operator as a canonical matrix in the computational basis (static method).

The canonical matrix is the textbook matrix representation that does not consider wires. Implicitly, this assumes that the wires of the operator correspond to the global wire order.

See also

Operator.matrix() and qml.matrix()

Parameters

• *params (list) – trainable parameters of the operator, as stored in the parameters attribute

• **hyperparams (dict) – non-trainable hyperparameters of the operator, as stored in the hyperparameters attribute

Returns

matrix representation

Return type

tensor_like

static compute_sparse_matrix(*params, **hyperparams)

Representation of the operator as a sparse matrix in the computational basis (static method).

The canonical matrix is the textbook matrix representation that does not consider wires. Implicitly, this assumes that the wires of the operator correspond to the global wire order.

See also

sparse_matrix()

Parameters

• *params (list) – trainable parameters of the operator, as stored in the parameters attribute

• **hyperparams (dict) – non-trainable hyperparameters of the operator, as stored in the hyperparameters attribute

Returns

sparse matrix representation

Return type

scipy.sparse._csr.csr_matrix

decomposition()

Representation of the operator as a product of other operators.

$$O = O_1 O_2 \dots O_n$$

A DecompositionUndefinedError is raised if no representation by decomposition is defined.

See also

compute_decomposition().

Returns

decomposition of the operator

Return type

list[Operator]

diagonalizing_gates()

Sequence of gates that diagonalize the operator in the computational basis.

Given the eigendecomposition $O = U \Sigma U^{\dagger}$ where $\Sigma$ is a diagonal matrix containing the eigenvalues, the sequence of diagonalizing gates implements the unitary $U^{\dagger}$.

The diagonalizing gates rotate the state into the eigenbasis of the operator.

A DiagGatesUndefinedError is raised if no representation by decomposition is defined.

See also

compute_diagonalizing_gates().

Returns

a list of operators

Return type

list[Operator] or None

eigvals()

Eigenvalues of the operator in the computational basis.

If diagonalizing_gates are specified and implement a unitary $U^{\dagger}$, the operator can be reconstructed as

$$O = U \Sigma U^{\dagger},$$

where $\Sigma$ is the diagonal matrix containing the eigenvalues.

Otherwise, no particular order for the eigenvalues is guaranteed.

Note

When eigenvalues are not explicitly defined, they are computed automatically from the matrix representation. Currently, this computation is not differentiable.

A EigvalsUndefinedError is raised if the eigenvalues have not been defined and cannot be inferred from the matrix representation.

See also

compute_eigvals()

Returns

eigenvalues

Return type

tensor_like

generator()

Generator of an operator that is in single-parameter-form.

For example, for operator

$$U(\phi) = e^{i\phi (0.5 Y + Z\otimes X)}$$

we get the generator

>>> U.generator()
  0.5 * Y(0) + Z(0) @ X(1)

The generator may also be provided in the form of a dense or sparse Hamiltonian (using LinearCombination and SparseHamiltonian respectively).

The default value to return is None, indicating that the operation has no defined generator.

label(decimals=None, base_label=None, cache=None)

A customizable string representation of the operator.

Parameters

• decimals=None (int) – If None, no parameters are included. Else, specifies how to round the parameters.

• base_label=None (str) – overwrite the non-parameter component of the label

• cache=None (dict) – dictionary that carries information between label calls in the same drawing

Returns

label to use in drawings

Return type

str

Example:

>>> op = qml.RX(1.23456, wires=0)
>>> op.label()
"RX"
>>> op.label(base_label="my_label")
"my_label"
>>> op = qml.RX(1.23456, wires=0, id="test_data")
>>> op.label()
"RX("test_data")"
>>> op.label(decimals=2)
"RX\n(1.23,"test_data")"
>>> op.label(base_label="my_label")
"my_label("test_data")"
>>> op.label(decimals=2, base_label="my_label")
"my_label\n(1.23,"test_data")"

If the operation has a matrix-valued parameter and a cache dictionary is provided, unique matrices will be cached in the 'matrices' key list. The label will contain the index of the matrix in the 'matrices' list.

>>> op2 = qml.QubitUnitary(np.eye(2), wires=0)
>>> cache = {'matrices': []}
>>> op2.label(cache=cache)
'U(M0)'
>>> cache['matrices']
[tensor([[1., 0.],
 [0., 1.]], requires_grad=True)]
>>> op3 = qml.QubitUnitary(np.eye(4), wires=(0,1))
>>> op3.label(cache=cache)
'U(M1)'
>>> cache['matrices']
[tensor([[1., 0.],
        [0., 1.]], requires_grad=True),
tensor([[1., 0., 0., 0.],
        [0., 1., 0., 0.],
        [0., 0., 1., 0.],
        [0., 0., 0., 1.]], requires_grad=True)]

map_wires(wire_map)

Returns a copy of the current operator with its wires changed according to the given wire map.

Parameters

wire_map (dict) – dictionary containing the old wires as keys and the new wires as values

Returns

new operator

Return type

Operator

matrix(wire_order=None)

Representation of the operator as a matrix in the computational basis.

If wire_order is provided, the numerical representation considers the position of the operator’s wires in the global wire order. Otherwise, the wire order defaults to the operator’s wires.

If the matrix depends on trainable parameters, the result will be cast in the same autodifferentiation framework as the parameters.

A MatrixUndefinedError is raised if the matrix representation has not been defined.

See also

compute_matrix()

Parameters

wire_order (Iterable) – global wire order, must contain all wire labels from the operator’s wires

Returns

matrix representation

Return type

tensor_like

pow(z)

A list of new operators equal to this one raised to the given power.

Parameters

z (float) – exponent for the operator

Returns

list[Operator]

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

Append the operator to the Operator queue.

static shape(n_layers, n_wires)

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]

simplify()

Reduce the depth of nested operators to the minimum.

Returns

simplified operator

Return type

Operator

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]

sparse_matrix(wire_order=None)

Representation of the operator as a sparse matrix in the computational basis.

If wire_order is provided, the numerical representation considers the position of the operator’s wires in the global wire order. Otherwise, the wire order defaults to the operator’s wires.

A SparseMatrixUndefinedError is raised if the sparse matrix representation has not been defined.

See also

compute_sparse_matrix()

Parameters

wire_order (Iterable) – global wire order, must contain all wire labels from the operator’s wires

Returns

sparse matrix representation

Return type

scipy.sparse._csr.csr_matrix

terms()

Representation of the operator as a linear combination of other operators.

$$O = \sum_i c_i O_i$$

A TermsUndefinedError is raised if no representation by terms is defined.

Returns

list of coefficients $c_i$ and list of operations $O_i$

Return type

tuple[list[tensor_like or float], list[Operation]]

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