qml.PolyXP

class PolyXP(q, wires)

Bases: pennylane.operation.CVObservable

An arbitrary second-order polynomial observable.

Represents an arbitrary observable \(P(\x,\p)\) that is a second order polynomial in the basis \(\mathbf{r} = (\I, \x_0, \p_0, \x_1, \p_1, \ldots)\).

For first-order observables the representation is a real vector \(\mathbf{d}\) such that \(P(\x,\p) = \mathbf{d}^T \mathbf{r}\).

For second-order observables the representation is a real symmetric matrix \(A\) such that \(P(\x,\p) = \mathbf{r}^T A \mathbf{r}\).

Used for evaluating arbitrary order-2 CV expectation values of CVParamShiftTape.

Details:

• Number of wires: Any

• Number of parameters: 1

• Observable order: 2nd order in the quadrature operators

• Heisenberg representation: \(A\)

Parameters

q (array[float]) – expansion coefficients

Attributes

eigvals	Eigenvalues of an instantiated observable.
ev_order	if not None, the observable is a polynomial of the given order in (x, p).
grad_method
id	String for the ID of the operator.
matrix	Matrix representation of an instantiated operator in the computational basis.
name	String for the name of the operator.
num_params
num_wires
par_domain
parameters	Current parameter values.
return_type
supports_heisenberg
wires	Wires of this operator.

eigvals

Eigenvalues of an instantiated observable.

The order of the eigenvalues needs to match the order of the computational basis vectors when the observable is diagonalized using diagonalizing_gates. This is a requirement for using qubit observables in quantum functions.

Example:

>>> U = qml.PauliZ(wires=1)
>>> U.eigvals
>>> array([1, -1])

Returns

eigvals representation

Return type

array

ev_order = 2

if not None, the observable is a polynomial of the given order in (x, p).

Type

None, int

grad_method = 'F'

id

String for the ID of the operator.

matrix

Matrix representation of an instantiated operator in the computational basis.

Example:

>>> U = qml.RY(0.5, wires=1)
>>> U.matrix
>>> array([[ 0.96891242+0.j, -0.24740396+0.j],
           [ 0.24740396+0.j,  0.96891242+0.j]])

Returns

matrix representation

Return type

array

name

String for the name of the operator.

num_params = 1

num_wires = -1

par_domain = 'A'

parameters

Current parameter values.

return_type = None

supports_heisenberg = True

wires

Wires of this operator.

Returns

wires

Return type

Wires

Methods

compare(other)	Compares with another Hamiltonian, Tensor, or Observable, to determine if they are equivalent.
diagonalizing_gates()	Returns the list of operations such that they diagonalize the observable in the computational basis.
heisenberg_expand(U, wires)	Expand the given local Heisenberg-picture array into a full-system one.
heisenberg_obs(wires)	Representation of the observable in the position/momentum operator basis.
queue()	Append the operator to the Operator queue.

compare(other)

Compares with another Hamiltonian, Tensor, or Observable, to determine if they are equivalent.

Observables/Hamiltonians are equivalent if they represent the same operator (their matrix representations are equal), and they are defined on the same wires.

Warning

The compare method does not check if the matrix representation of a Hermitian observable is equal to an equivalent observable expressed in terms of Pauli matrices. To do so would require the matrix form of Hamiltonians and Tensors be calculated, which would drastically increase runtime.

Returns

True if equivalent.

Return type

(bool)

Examples

>>> ob1 = qml.PauliX(0) @ qml.Identity(1)
>>> ob2 = qml.Hamiltonian([1], [qml.PauliX(0)])
>>> ob1.compare(ob2)
True
>>> ob1 = qml.PauliX(0)
>>> ob2 = qml.Hermitian(np.array([[0, 1], [1, 0]]), 0)
>>> ob1.compare(ob2)
False

diagonalizing_gates()

Returns the list of operations such that they diagonalize the observable in the computational basis.

Returns

A list of gates that diagonalize the observable in the computational basis.

Return type

list(qml.Operation)

heisenberg_expand(U, wires)

Expand the given local Heisenberg-picture array into a full-system one.

Parameters

• U (array[float]) – array to expand (expected to be of the dimension 1+2*self.num_wires)

• wires (Wires) – wires on the device the array U should be expanded to apply to

Raises

ValueError – if the size of the input matrix is invalid or num_wires is incorrect

Returns

expanded array, dimension 1+2*num_wires

Return type

array[float]

heisenberg_obs(wires)

Representation of the observable in the position/momentum operator basis.

Returns the expansion \(q\) of the observable, \(Q\), in the basis \(\mathbf{r} = (\I, \x_0, \p_0, \x_1, \p_1, \ldots)\).

• For first-order observables returns a real vector such that \(Q = \sum_i q_i \mathbf{r}_i\).

• For second-order observables returns a real symmetric matrix such that \(Q = \sum_{ij} q_{ij} \mathbf{r}_i \mathbf{r}_j\).

Parameters

wires (Wires) – wires on the device that the observable gets applied to

Returns

\(q\)

Return type

array[float]

queue()

Append the operator to the Operator queue.