qml.ftqc.GraphStatePrep¶

class GraphStatePrep(graph, one_qubit_ops=<class 'pennylane.ops.qubit.non_parametric_ops.Hadamard'>, two_qubit_ops=<class 'pennylane.ops.op_math.controlled_ops.CZ'>, wires=None)[source]¶

Bases: pennylane.operation.Operation

Encode a graph state with a single graph operation applied on each qubit, and an entangling operation applied on nearest-neighbor qubits defined by the graph connectivity. The initial graph is $|0\rangle^{\otimes V}$ , given each qubit or graph vertex node ( $V$ ) in the graph is in the $|0\rangle$ state and is not entangled with any other qubit. The target graph state $| \psi \rangle$ is: $| \psi \rangle = \prod\limits_{\{a, b\} \in E} U_{ab}|+\rangle^{\otimes V}$ where $U_{ab}$ is a phase gate applied to the vertices $a$ , $b$ of a edge $E$ in the graph as illustrated in eq. (24) in arxiv:quant-ph/0602096.

The target graph state can be prepared as below:

1. Each qubit is prepared as $|+\rangle^{\otimes V}$ state by applying the one_qubit_ops (H gate) operation.

2. Entangle every nearest qubit pair in the graph with two_qubit_ops (CZ gate) operation.

Parameters

graph (Union[QubitGraph, networkx.Graph]) – QubitGraph or networkx.Graph object mapping qubit to wires. The node labels of graph must be sortable.
one_qubit_ops (Operation) – Operator to prepare the initial state of each qubit. Defaults to Hadamard.
two_qubit_ops (Operation) – Operator to entangle nearest qubits. Defaults to CZ.
wires (Optional[Wires]) – Wires the operator applies on. Wires are be mapped 1:1 to the graph nodes sorted in ascending order. Optional only graph is a QubitGraph. If no wires are provided, the children of the provided QubitGraph will be used as wires.

Example:

The graph state preparation layer can be customized by the user.

import pennylane as qml
from pennylane.ftqc import generate_lattice, GraphStatePrep, QubitGraph

dev = qml.device('default.qubit')

@qml.qnode(dev)
def circuit(q, one_qubit_ops, two_qubit_ops, wires = None):
    GraphStatePrep(graph=q, one_qubit_ops=one_qubit_ops, two_qubit_ops=two_qubit_ops, wires = wires)
    return qml.probs()

lattice = generate_lattice([2, 2], "square")
q = QubitGraph(lattice.graph, id="square")

one_qubit_ops = qml.Y
two_qubit_ops = qml.CNOT

If the wires argument is not explicitly passed to the circuit, the child nodes of the QubitGraph are used as the wires. The resulting circuit after applying the GraphStatePrep template is:

>>> print(qml.draw(circuit, level="device")(q, one_qubit_ops, two_qubit_ops))
QubitGraph<id=(0, 0), loc=[square]>: ──Y─╭●─╭●───────┤  Probs
QubitGraph<id=(0, 1), loc=[square]>: ──Y─│──╰X─╭●────┤  Probs
QubitGraph<id=(1, 0), loc=[square]>: ──Y─╰X────│──╭●─┤  Probs
QubitGraph<id=(1, 1), loc=[square]>: ──Y───────╰X─╰X─┤  Probs

The circuit wires can also be customized by passing a wires argument to the circuit as follows:

>>> print(qml.draw(circuit, level="device")(q, one_qubit_ops, two_qubit_ops, wires=[0, 1, 2, 3]))
──Y─╭●─╭●───────┤  Probs
──Y─│──╰X─╭●────┤  Probs
──Y─╰X────│──╭●─┤  Probs
──Y───────╰X─╰X─┤  Probs

A Note on Node Ordering

The graph structures used for defining qubit connectivity are inherently unordered data structures. Mapping the nodes in the graph to the ordered sequence of wires can therefore result in ambiguity. To ensure this mapping is reliable and deterministic, the sequence of wires is mapped to the list of graph nodes sorted in ascending order.

Consider the following example:

import networkx as nx
import pennylane as qml
from pennylane.ftqc import GraphStatePrep

dev = qml.device("default.qubit")

@qml.qnode(dev)
def circuit(graph, wires):
    GraphStatePrep(graph=graph, one_qubit_ops=qml.H, two_qubit_ops=qml.CZ, wires=wires)
    return qml.state()

Defining a graph structure and drawing the circuit shows how the graph node labels have been mapped to wires:

>>> g1 = nx.Graph([("a", "b"), ("b", "c"), ("c", "d")])  # (a) -- (b) -- (c) -- (d)
>>> print(qml.draw(circuit, level="device")(g1, wires=range(4)))
0: ──H─╭●───────┤  State
1: ──H─╰Z─╭●────┤  State
2: ──H────╰Z─╭●─┤  State
3: ──H───────╰Z─┤  State

In other words, GraphStatePrep has defined the node-label-to-wire mapping:

{"a": 0, "b": 1, "c": 2, "d": 3}

which corresponds to the graph structure with wire indices:

(0) -- (1) -- (2) -- (3)

as shown in the circuit diagram with CZ operations applied along each edge in the graph.

Drawing the circuit for a graph with the same structure but with different node labels gives:

>>> g2 = nx.Graph([("b", "a"), ("a", "c"), ("c", "d")])  # (b) -- (a) -- (c) -- (d)
>>> print(qml.draw(circuit, level="device")(g2, wires=range(4)))
0: ──H─╭Z─╭●────┤  State
1: ──H─╰●─│─────┤  State
2: ──H────╰Z─╭●─┤  State
3: ──H───────╰Z─┤  State

As before, GraphStatePrep defined the node-label-to-wire mapping to be:

{"a": 0, "b": 1, "c": 2, "d": 3}

but now, this corresponds to the graph structure with wire indices:

(1) -- (0) -- (2) -- (3)

as shown in the circuit diagram.

While these two circuit might appear to be the same, they are indeed distinct for this sequence of wires, and result in different state vectors. It is therefore important to remember that the node labels influence how nearest-neighbour wires are interpreted.

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`	Gradient computation method.
`grad_recipe`	Gradient recipe for the parameter-shift method.
`has_adjoint`
`has_decomposition`
`has_diagonalizing_gates`
`has_generator`
`has_matrix`
`has_plxpr_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 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.
`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¶

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¶

Gradient computation method.

'A': analytic differentiation using the parameter-shift method.
'F': finite difference numerical differentiation.
None: the operation may not be differentiated.

Default is 'F', or None if the Operation has zero parameters.

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_plxpr_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 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.

resource_keys = {}¶

resource_params¶

A dictionary containing the minimal information needed to compute a resource estimate of the operator’s decomposition.

The keys of this dictionary should match the resource_keys attribute of the operator class. Two instances of the same operator type should have identical resource_params iff their decompositions exhibit the same counts for each gate type, even if the individual gate parameters differ.

Examples

The MultiRZ has non-empty resource_keys:

>>> qml.MultiRZ.resource_keys
{"num_wires"}

The resource_params of an instance of MultiRZ will contain the number of wires:

>>> op = qml.MultiRZ(0.5, wires=[0, 1])
>>> op.resource_params
{"num_wires": 2}

Note that another MultiRZ may have different parameters but the same resource_params:

>>> op2 = qml.MultiRZ(0.7, wires=[1, 2])
>>> op2.resource_params
{"num_wires": 2}

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`(wires, graph[, ...])	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)	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_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 an operator that is in single-parameter-form.
`label`(args, *kwargs)	Defines how the graph state preparation is represented in diagrams and drawings.
`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.
`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, 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()¶

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(wires, graph, one_qubit_ops=<class 'pennylane.ops.qubit.non_parametric_ops.Hadamard'>, two_qubit_ops=<class 'pennylane.ops.op_math.controlled_ops.CZ'>)[source]¶

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

Note

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

See also

decomposition().

Parameters

wires (Wires) – Wires the decomposition applies on. Wires will be mapped 1:1 to graph nodes.
graph (Union[nx.Graph, QubitGraph]) – QubitGraph or nx.Graph object mapping qubit to wires.
one_qubit_ops (Operation) – Operator to prepare the initial state of each qubit. Default to H.
two_qubit_ops (Operation) – Operator to entangle nearest qubits. Default to CZ.

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.

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.

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_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()

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).

label(*args, **kwargs)[source]¶

Defines how the graph state preparation is represented in diagrams and drawings.

Parameters

*args (Optional[Union[int, str]]) – positional arguments for decimals and base_label.
**kwargs (Optional[dict]) – keyword arguments for cache.

Returns

label to use in drawings

Return type

str

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.

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, 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]]

qml.ftqc.GraphStatePrep¶

A Note on Node Ordering

Attributes

Methods

Contents

Downloads

qml.ftqc.GraphStatePrep¶

A Note on Node Ordering

Attributes

Methods

Contents

Downloads

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