qml.Interferometer

class Interferometer(theta, phi, varphi, wires, mesh='rectangular', beamsplitter='pennylane', id=None)[source]

Bases: pennylane.operation.CVOperation

General linear interferometer, an array of beamsplitters and phase shifters.

For \(M\) wires, the general interferometer is specified by providing \(M(M-1)/2\) transmittivity angles \(\theta\) and the same number of phase angles \(\phi\), as well as \(M-1\) additional rotation parameters \(\varphi\).

By specifying the keyword argument mesh, the scheme used to implement the interferometer may be adjusted:

  • mesh='rectangular' (default): uses the scheme described in Clements et al., resulting in a rectangular array of \(M(M-1)/2\) beamsplitters arranged in \(M\) slices and ordered from left to right and top to bottom in each slice. The first beamsplitter acts on wires \(0\) and \(1\):

    ../../_images/clements.png
  • mesh='triangular': uses the scheme described in Reck et al., resulting in a triangular array of \(M(M-1)/2\) beamsplitters arranged in \(2M-3\) slices and ordered from left to right and top to bottom. The first and fourth beamsplitters act on wires \(M-1\) and \(M\), the second on \(M-2\) and \(M-1\), and the third on \(M-3\) and \(M-2\), and so on.

    ../../_images/reck.png

In both schemes, the network of Beamsplitter operations is followed by \(M\) local Rotation Operations.

The rectangular decomposition is generally advantageous, as it has a lower circuit depth (\(M\) vs \(2M-3\)) and optical depth than the triangular decomposition, resulting in reduced optical loss.

This is an example of a 4-mode interferometer with beamsplitters \(B\) and rotations \(R\), using mesh='rectangular':

../../_images/layer_interferometer.png

Note

The decomposition as formulated in Clements et al. uses a different convention for a beamsplitter \(T(\theta, \phi)\) than PennyLane, namely:

\[T(\theta, \phi) = BS(\theta, 0) R(\phi)\]

For the universality of the decomposition, the used convention is irrelevant, but for a given set of angles the resulting interferometers will be different.

If an interferometer consistent with the convention from Clements et al. is needed, the optional keyword argument beamsplitter='clements' can be specified. This will result in each Beamsplitter being preceded by a Rotation and thus increase the number of elementary operations in the circuit.

Parameters
  • theta (tensor_like) – size \((M(M-1)/2,)\) tensor of transmittivity angles \(\theta\)

  • phi (tensor_like) – size \((M(M-1)/2,)\) tensor of phase angles \(\phi\)

  • varphi (tensor_like) – size \((M,)\) tensor of rotation angles \(\varphi\)

  • wires (Iterable or Wires) – Wires that the template acts on. Accepts an iterable of numbers or strings, or a Wires object.

  • mesh (string) – the type of mesh to use

  • beamsplitter (str) – if clements, the beamsplitter convention from Clements et al. 2016 (https://dx.doi.org/10.1364/OPTICA.3.001460) is used; if pennylane, the beamsplitter is implemented via PennyLane’s Beamsplitter operation.

Raises

ValueError – if inputs do not have the correct format

Example

The template requires \(3\) sets of parameters. The mesh and beamsplitter keyword arguments are optional and have 'rectangular' and 'pennylane' as default values.

dev = qml.device('default.gaussian', wires=4)

@qml.qnode(dev)
def circuit(params):
    qml.Interferometer(*params, wires=range(4))
    return qml.expval(qml.Identity(0))

shapes = [[6, ], [6, ], [4, ]]
params = []
for shape in shapes:
    params.append(np.random.random(shape))

Using these random parameters, the resulting circuit is:

>>> print(qml.draw(circuit, expansion_strategy="device")(params))
0: ─╭BS(0.97,0.09)────────────────╭BS(0.89,0.33)──R(0.83)────────────────┤  <I>
1: ─╰BS(0.97,0.09)─╭BS(0.94,0.05)─╰BS(0.89,0.33)─╭BS(0.92,0.27)──R(0.36)─┤
2: ─╭BS(0.78,0.20)─╰BS(0.94,0.05)─╭BS(0.60,0.39)─╰BS(0.92,0.27)──R(0.28)─┤
3: ─╰BS(0.78,0.20)────────────────╰BS(0.60,0.39)──R(0.54)────────────────┤

Using different values for optional arguments:

@qml.qnode(dev)
def circuit(params):
    qml.Interferometer(*params, wires=range(4), mesh='triangular', beamsplitter='clements')
    return qml.expval(qml.Identity(0))

shapes = [[6, ], [6, ], [4, ]]
params = []
for shape in shapes:
    params.append(np.random.random(shape))

print(qml.draw(circuit, expansion_strategy="device")(params))
0: ──R(0.71)───────────────────────────────╭BS(0.07,0.00)──R(0.36)──────────────────────────────
1: ──R(0.82)────────────────╭BS(0.80,0.00)─╰BS(0.07,0.00)──R(0.77)───────╭BS(0.77,0.00)──R(0.12)
2: ──R(0.01)─╭BS(0.14,0.00)─╰BS(0.80,0.00)──R(0.73)───────╭BS(0.99,0.00)─╰BS(0.77,0.00)──R(0.07)
3: ──────────╰BS(0.14,0.00)───────────────────────────────╰BS(0.99,0.00)────────────────────────

──────────────────────────┤  <I>
──────────────────────────┤
──╭BS(0.20,0.00)──R(0.86)─┤
──╰BS(0.20,0.00)──R(0.62)─┤

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

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.

supports_heisenberg

supports_parameter_shift

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
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

Number of trainable parameters that the operator depends on.

By default, this property returns as many parameters as were used for the operator creation. If the number of parameters for an operator subclass is fixed, this property can be overwritten to return the fixed value.

Returns

number of parameters

Return type

int

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.

supports_heisenberg = False
supports_parameter_shift = False
wires

Wires that the operator acts on.

Returns

wires

Return type

Wires

adjoint()

Create an operation that is the adjoint of this one.

compute_decomposition(theta, phi, varphi, …)

Representation of the operator 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.

expand()

Returns a tape that contains the decomposition of the operator.

generator()

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

heisenberg_expand(U, wire_order)

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

heisenberg_pd(idx)

Partial derivative of the Heisenberg picture transform matrix.

heisenberg_tr(wire_order[, inverse])

Heisenberg picture representation of the linear transformation carried out by the gate at current parameter values.

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_wires)

Returns a list of shapes for the 3 parameter tensors.

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.

validate_subspace(subspace)

Validate the subspace for qutrit operations.

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(theta, phi, varphi, wires, mesh, beamsplitter)[source]

Representation of the operator as a product of other operators.

\[O = O_1 O_2 \dots O_n.\]

See also

decomposition().

Parameters
  • theta (tensor_like) – size \((M(M-1)/2,)\) tensor of transmittivity angles

  • phi (tensor_like) – size \((M(M-1)/2,)\) tensor of phase angles

  • varphi (tensor_like) – size \((M,)\) tensor of rotation angles

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

  • mesh (string) – the type of mesh to use

  • beamsplitter (str) – beamsplitter convention

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)

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.

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.

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.

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()

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.

heisenberg_expand(U, wire_order)

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)

  • wire_order (Wires) – global wire order defining which subspace the operator acts on

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_pd(idx)

Partial derivative of the Heisenberg picture transform matrix.

Computed using grad_recipe.

Parameters

idx (int) – index of the parameter with respect to which the partial derivative is computed.

Returns

partial derivative

Return type

array[float]

heisenberg_tr(wire_order, inverse=False)

Heisenberg picture representation of the linear transformation carried out by the gate at current parameter values.

Given a unitary quantum gate \(U\), we may consider its linear transformation in the Heisenberg picture, \(U^\dagger(\cdot) U\).

If the gate is Gaussian, this linear transformation preserves the polynomial order of any observables that are polynomials in \(\mathbf{r} = (\I, \x_0, \p_0, \x_1, \p_1, \ldots)\). This also means it maps \(\text{span}(\mathbf{r})\) into itself:

\[U^\dagger \mathbf{r}_i U = \sum_j \tilde{U}_{ij} \mathbf{r}_j\]

For Gaussian CV gates, this method returns the transformation matrix for the current parameter values of the Operation. The method is not defined for non-Gaussian (and non-CV) gates.

Parameters
  • wire_order (Wires) – global wire order defining which subspace the operator acts on

  • inverse (bool) – if True, return the inverse transformation instead

Raises

RuntimeError – if the specified operation is not Gaussian or is missing the _heisenberg_rep method

Returns

\(\tilde{U}\), the Heisenberg picture representation of the linear transformation

Return type

array[float]

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_wires)[source]

Returns a list of shapes for the 3 parameter tensors.

Parameters

n_wires (int) – number of wires

Returns

list of shapes

Return type

list[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.

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

Warning

Operator.validate_subspace(subspace) has been relocated to the qml.ops.qutrit.parametric_ops module and will be removed from the Operator class in an upcoming release.

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