qml.PCPhase

class PCPhase(phi, dim, wires)

Bases: Operation

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:

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

This can also be written as \(\Pi(\phi) = \exp(i\phi(2\Pi-\mathbb{I}_N))\), where \(N=2^n\) is the Hilbert space dimension for \(n\) qubits and \(\Pi\) is the diagonal projector with dim ones and N-dim zeros.

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:

>>> op_3 = qml.PCPhase(0.27, dim = 3, wires=range(3))

The resulting operation applies a complex phase \(e^{0.27i}\) to the first \(dim = 3\) basis vectors and \(e^{-0.27i}\) to the remaining basis vectors, as we can see from the diagonal of the matrix for this circuit.

>>> print(np.round(np.diag(qml.matrix(op_3)),2))
[0.96+0.27j 0.96+0.27j 0.96+0.27j 0.96-0.27j 0.96-0.27j 0.96-0.27j
 0.96-0.27j 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:

>>> op_7 = qml.PCPhase(1.23, dim=7, wires=[1, 2, 3])
>>> print(np.round(np.diag(qml.matrix(op_7)),2))
[0.33+0.94j 0.33+0.94j 0.33+0.94j 0.33+0.94j 0.33+0.94j 0.33+0.94j
 0.33+0.94j 0.33-0.94j]

PCPhase operations are decomposed into (multi-)controlled PhaseShift operations which share the same control values on common control wires, and Pauli-X operations, possibly complemented by a global phase.

>>> op_13 = qml.PCPhase(1.23, dim=13, wires=[1, 2, 3, 4])
>>> print(qml.draw(op_13.decomposition)())
1: ──GlobalPhase(-1.23)─╭●─────────╭●───────────┤
2: ──GlobalPhase(-1.23)─╰Rϕ(-2.46)─├●───────────┤
3: ──GlobalPhase(-1.23)────────────├○───────────┤
4: ──GlobalPhase(-1.23)──X─────────╰Rϕ(2.46)──X─┤

If dim is a power of two, a single (multi-controlled) PhaseShift gate is sufficient:

>>> op_16 = qml.PCPhase(1.23, dim=16, wires=range(6))
>>> print(qml.draw(op_16.decomposition, wire_order=range(6), show_all_wires=True)())
0: ──GlobalPhase(1.23)────╭○───────────┤
1: ──GlobalPhase(1.23)──X─╰Rϕ(2.46)──X─┤
2: ──GlobalPhase(1.23)─────────────────┤
3: ──GlobalPhase(1.23)─────────────────┤
4: ──GlobalPhase(1.23)─────────────────┤
5: ──GlobalPhase(1.23)─────────────────┤

Attributes

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
has_qfunc_decomposition
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 likely 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.
pauli_rep	A PauliSentence representation of the Operator, or None if it doesn't have one.
resource_keys
resource_params	A dictionary containing the minimal information needed to compute a resource estimate of the operator's decomposition.
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

has_qfunc_decomposition = 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 likely hermitian.

Note

It is recommended to use the is_hermitian() function. Although this function may be expensive to calculate, the op.is_hermitian property can lead to technically incorrect results.

If this property returns True, the operator is guaranteed to be hermitian, but if it returns False, the operator may still be hermitian.

As an example, consider the following edge case:

>>> op = (qml.X(0) @ qml.Y(0) - qml.X(0) @ qml.Z(0)) * 1j
>>> op.is_hermitian
False

On the contrary, the is_hermitian() function will give the correct answer:

>>> qml.is_hermitian(op)
True

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 = None

Number of wires the operator acts on.

parameter_frequencies = [(2,)]

parameters

Trainable parameters that the operator depends on.

pauli_rep

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

resource_keys = {'dim', 'num_wires'}

resource_params

wires

Wires that the operator acts on.

Returns:

wires

Return type:

Wires

Methods

adjoint()	Computes the adjoint of the operator.
compute_decomposition(*params, wires, ...)	Representation of the PCPhase 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(phi, dimension)	Get the matrix representation of Pi-controlled phase unitary.
compute_qfunc_decomposition(*args, ...)	Experimental method to compute the dynamic decomposition of the operator with program capture enabled.
compute_sparse_matrix(*params[, format])	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 the PCPhase operator, which 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.
pow(z)	Computes the operator raised to z.
queue([context])	Append the operator to the Operator queue.
simplify()	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.
sparse_matrix([wire_order, format])	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()

Computes the adjoint of the operator.

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

Representation of the PCPhase operator as a product of other operators (static 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]

In short, this decomposition relies on decomposing the generator (see generator()) of the PCPhase gate into generators of multicontrolled PhaseShift gates, potentially complemented with (non-controlled) Pauli-X gates and/or a global phase. For example, for dim=13 on four qubits:

>>> op_13 = qml.PCPhase(1.23, dim=13, wires=[1, 2, 3, 4])
>>> print(qml.draw(op_13.decomposition)())
1: ──GlobalPhase(-1.23)─╭●─────────╭●───────────┤
2: ──GlobalPhase(-1.23)─╰Rϕ(-2.46)─├●───────────┤
3: ──GlobalPhase(-1.23)────────────├○───────────┤
4: ──GlobalPhase(-1.23)──X─────────╰Rϕ(2.46)──X─┤

In the following we provide a detailed example for illustration purposes.

Detailed example

Consider the projector-controlled phase gate on \(n=4\) qubits and with \(d=\texttt{dim}=3\), i.e,

>>> op_3 = qml.PCPhase(1.23, dim=3, wires=[0, 1, 2, 3])

It acts on \(N=2^n=16\)-dimensional vectors and is described by

\[\Pi(\phi) = \exp(i\phi G) = \exp(i\phi(2\Pi-\mathbb{I}_N)),\]

where \(G\) is a diagonal matrix with \(d=3\) ones, followed by \(2^n-d = 16 - 3=13\) negative ones. Accordingly, \(\Pi\) is diagonal with \(3\) ones and \(13\) zeros.

First, we implement the global phase generated by \(\mathbb{I}_N\) with a GlobalPhase gate with angle \(-\phi\). Then we decompose \(d\) into powers of two with positive or negative sign, via \(d=3=4-1 = 2^2-2^0\). This decomposition tells us that we can write the target gate with two (multi-)controlled phase shift gates. For this, we rewrite the projector \(\Pi\) according to the decomposition as

\[\begin{split}\Pi &= \text{diag}(1, 1, 1, 0, 0, \dots, 0)\\ &=\text{diag}(1, 1, 1, 1, 0, \dots, 0) -\text{diag}(0, 0, 0, 1, 0, \dots, 0)\end{split}\]

where \(0,\dots, 0\) indicates \(12\) zeros each time. How do we realize this projector decomposition on the gate level?

A singly-controlled phase shift gate applies a phase to a quarter of all computational basis states (the control filters by the state of one qubit, and the phase shift gate itself filters by the \(|1\rangle\) state of the target qubit, cutting the number of states we are acting on in half each time). For \(n=4\), this amounts to \(2^4/4=4\) states, which is exactly what we need for the first term above. To apply the phase to the first four states, \(|0000\rangle\), \(|0001\rangle\), \(|0010\rangle\), and \(|0011\rangle\), we want to “filter by” the first two qubits being in the \(|0\rangle\) state. For qubit \(0\), we do this by controlling on the \(|0\rangle\) state. For qubit \(1\), we pick it as the target of the controlled phase shift operation. Generically, this would make it act on the \(|1\rangle\) state, so we simply flip qubit \(1\) before and after the operation to apply the phase to the \(|0\rangle\) state instead. Thus, we conclude this first step by applying the gates qml.X(1), qml.ctrl(qml.PhaseShift(2 * phi, 1), control=[0], control_values=[0]), and qml.X(1).

Next, we implement the second term in the projector decomposition, applying a phase to a single computational basis state. This requires us to fully control a phase shift gate, i.e., we use the last qubit as target and the other three as controls (there is some freedom of choice here, but this is a convenient choice). We want to apply the phase to the state \(|3\rangle=|0011\rangle\). So the controls \(0\) and \(1\) are set to zero and the control \(2\) is set to one. As we want to effect the phase onto the \(|1\rangle\) state of qubit \(3\), we don’t need to flip the target bit as we did before. However, given the negative sign in the projector decomposition, we need to multiply the phase with \(-1\). Overall, we apply the gate qml.ctrl(qml.PhaseShift(-2 * phi, 3), control=[0, 1, 2], control_values=[0, 0, 1]), which concludes the decomposition, now reading:

>>> print(qml.draw(op_3.decomposition)())
0: ──GlobalPhase(1.23)────╭○───────────╭○─────────┤
1: ──GlobalPhase(1.23)──X─╰Rϕ(2.46)──X─├○─────────┤
2: ──GlobalPhase(1.23)─────────────────├●─────────┤
3: ──GlobalPhase(1.23)─────────────────╰Rϕ(-2.46)─┤

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)

Get the eigvals for the Pi-controlled phase unitary.

static compute_matrix(phi, dimension)

Get the matrix representation of Pi-controlled phase unitary.

static compute_qfunc_decomposition(*args, **hyperparameters)

Experimental method to compute the dynamic decomposition of the operator with program capture enabled.

When the program capture feature is enabled with qml.capture.enable(), the decomposition of the operator is computed with this method if it is defined. Otherwise, the compute_decomposition() method is used.

The exception to this rule is when the operator is returned from the compute_decomposition() method of another operator, in which case the decomposition is performed with compute_decomposition() (even if this method is defined), and not with this method.

When compute_qfunc_decomposition is defined for an operator, the control flow operations within the method (specifying the decomposition of the operator) are recorded in the JAX representation.

Note

This method is experimental and subject to change.

See also

compute_decomposition().

Parameters:

• *args (list) – positional arguments passed to the operator, including trainable parameters and wires

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

static compute_sparse_matrix(*params, format='csr', **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

• format (str) – format of the returned scipy sparse matrix, for example ‘csr’

• **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() and qml.eigvals()

Returns:

eigenvalues

Return type:

tensor_like

generator()

Generator of the PCPhase operator, which is in single-parameter-form. The operator reads

\[\Pi(\phi) = e^{i\phi (2\Pi - \mathbb{I}_N)},\]

where \(\Pi\) is the projector onto the first :math`d` (dim) computational basis states and \(N=2^n\) is the Hilbert space dimension for \(n\) qubits.

Correspondingly, the generator is \(2\Pi - \mathbb{I}_N=\text{diag}(\underset{d\text{ times}}{\underbrace{1, \dots, 1}},\underset{(N-d)\text{ times}}{\underbrace{-1, \dots, -1}})\):

>>> qml.PCPhase(0.5, dim=3, wires=[0, 1]).generator()
Hermitian(array([[ 1,  0,  0,  0],
   [ 0,  1,  0,  0],
   [ 0,  0,  1,  0],
   [ 0,  0,  0, -1]]), wires=[0, 1])

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

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.

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)

Computes the operator raised to z.

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

Append the operator to the Operator queue.

simplify()

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, format='csr')

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

• format (str) – format of the returned scipy sparse matrix, for example ‘csr’

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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