# qml.PCPhase¶

class PCPhase(phi, dim, wires)[source]

A projector-controlled phase gate.

This gate applies a complex phase $$e^{i\phi}$$ to the first $$dim$$ basis vectors of the input state while applying a complex phase $$e^{-i \phi}$$ to the remaining basis vectors. For example, consider the 2-qubit case where $$dim = 3$$:

$\begin{split}\Pi(\phi) = \begin{bmatrix} e^{i\phi} & 0 & 0 & 0 \\ 0 & e^{i\phi} & 0 & 0 \\ 0 & 0 & e^{i\phi} & 0 \\ 0 & 0 & 0 & e^{-i\phi} \end{bmatrix}.\end{split}$

Details:

• Number of wires: Any (the operation can act on any number of wires)

• Number of parameters: 1

• Number of dimensions per parameter: (0,)

Parameters
• phi (float) – rotation angle $$\phi$$

• dim (int) – the dimension of the subspace

• wires (Iterable[int, str], Wires) – the wires the operation acts on

• id (str or None) – String representing the operation (optional)

Example:

We can define a circuit using PCPhase as follows:

>>> dev = qml.device('default.qubit', wires=2)
>>> @qml.qnode(dev)
>>> def example_circuit():
...     qml.PCPhase(0.27, dim = 2, wires=range(2))
...     return qml.state()


The resulting operation applies a complex phase $$e^{0.27i}$$ to the first $$dim = 2$$ basis vectors and $$e^{-0.27i}$$ to the remaining basis vectors.

>>> print(np.round(qml.matrix(example_circuit)(),2))
[[0.96+0.27j 0.  +0.j   0.  +0.j   0.  +0.j  ]
[0.  +0.j   0.96+0.27j 0.  +0.j   0.  +0.j  ]
[0.  +0.j   0.  +0.j   0.96-0.27j 0.  +0.j  ]
[0.  +0.j   0.  +0.j   0.  +0.j   0.96-0.27j]]


We can also choose a different $$dim$$ value to apply the phase shift to a different set of basis vectors as follows:

>>> pc_op = qml.PCPhase(1.23, dim=3, wires=[1, 2])
>>> print(np.round(qml.matrix(pc_op),2))
[[0.33+0.94j 0.  +0.j   0.  +0.j   0.  +0.j  ]
[0.  +0.j   0.33+0.94j 0.  +0.j   0.  +0.j  ]
[0.  +0.j   0.  +0.j   0.33+0.94j 0.  +0.j  ]
[0.  +0.j   0.  +0.j   0.  +0.j   0.33-0.94j]]

 arithmetic_depth Arithmetic depth of the operator. basis 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 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 that the operator depends on. num_params Number of trainable parameters that the operator depends on. num_wires Number of wires the operator acts on. parameter_frequencies parameters Trainable parameters that the operator depends on. wires Wires that the operator acts on.
arithmetic_depth

Arithmetic depth of the operator.

basis = 'Z'
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 = 'A'
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 = True
has_decomposition = True
has_diagonalizing_gates = False
has_generator = True
has_matrix = True
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 = (0,)

Number of dimensions per trainable parameter that the operator depends on.

Type

tuple[int]

num_params = 1

Number of trainable parameters that the operator depends on.

Type

int

num_wires = -1

Number of wires the operator acts on.

parameter_frequencies = [(2,)]
parameters

Trainable parameters that the operator depends on.

wires

Wires that the operator acts on.

Returns

wires

Return type

Wires

 Computes the adjoint of the operator. compute_decomposition(*params[, wires]) Representation of the operator as a product of other operators (static method). compute_diagonalizing_gates(*params, wires, …) Sequence of gates that diagonalize the operator in the computational basis (static method). compute_eigvals(*params, **hyperparams) Get the eigvals for the Pi-controlled phase unitary. compute_matrix(*params, **hyperparams) Get the matrix representation of Pi-controlled phase unitary. compute_sparse_matrix(*params, **hyperparams) Representation of the operator as a sparse matrix in the computational basis (static method). Representation of the operator as a product of other operators. Sequence of gates that diagonalize the operator in the computational basis. Eigenvalues of the operator in the computational basis. Returns a tape that contains the decomposition of the operator. Generator of an operator that is in single-parameter-form. label([decimals, base_label, cache]) The label of the operator when displayed in a circuit. 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. Computes the operator raised to z. queue([context]) Append the operator to the Operator queue. Simplifies the operator if possible. 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. Representation of the operator as a linear combination of other operators. validate_subspace(subspace) Validate the subspace for qutrit operations.
adjoint()[source]

Computes the adjoint of the operator.

static compute_decomposition(*params, wires=None, **hyperparams)[source]

Representation of the operator as a product of other operators (static method).

$O = O_1 O_2 \dots O_n.$

Note

Operations making up the decomposition should be queued within the compute_decomposition method.

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 hyper-parameters of the operator, as stored in the hyperparameters attribute

Returns

decomposition of the operator

Return type

list[Operator]

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.

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)[source]

Get the eigvals for the Pi-controlled phase unitary.

static compute_matrix(*params, **hyperparams)[source]

Get the matrix representation of Pi-controlled phase unitary.

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.

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.

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.

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.

Returns

eigenvalues

Return type

tensor_like

expand()

Returns a tape that contains the decomposition of the operator.

Returns

quantum tape

Return type

QuantumTape

generator()[source]

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) [Y0]
+ (1.0) [Z0 X1]


The generator may also be provided in the form of a dense or sparse Hamiltonian (using Hermitian 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)[source]

The label of the operator when displayed in a circuit.

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.

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)[source]

Computes the operator raised to z.

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

Append the operator to the Operator queue.

simplify()[source]

Simplifies the operator if possible.

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.

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]]

static validate_subspace(subspace)

Validate the subspace for qutrit operations.

This method determines whether a given subspace for qutrit operations is defined correctly or not. If not, a ValueError is thrown.

Parameters

subspace (tuple[int]) – Subspace to check for correctness