qml.QuantumPhaseEstimation¶

class
QuantumPhaseEstimation
(unitary, target_wires=None, estimation_wires=None, id=None)[source]¶ Bases:
pennylane.operation.Operation
Performs the quantum phase estimation circuit.
Given a unitary matrix \(U\), this template applies the circuit for quantum phase estimation. The unitary is applied to the qubits specified by
target_wires
and \(n\) qubits are used for phase estimation as specified byestimation_wires
. Parameters
unitary (array or Operator) – the phase estimation unitary, specified as a matrix or an
Operator
target_wires (Union[Wires, Sequence[int], or int]) – the target wires to apply the unitary. If the unitary is specified as an operator, the target wires should already have been defined as part of the operator. In this case, target_wires should not be specified.
estimation_wires (Union[Wires, Sequence[int], or int]) – the wires to be used for phase estimation
 Raises
QuantumFunctionError – if the
target_wires
andestimation_wires
share a common element, or iftarget_wires
are specified for an operator unitary.
Usage Details
This circuit can be used to perform the standard quantum phase estimation algorithm, consisting of the following steps:
Prepare
target_wires
in a given state. Iftarget_wires
are prepared in an eigenstate of \(U\) that has corresponding eigenvalue \(e^{2 \pi i \theta}\) with phase \(\theta \in [0, 1)\), this algorithm will measure \(\theta\). Other input states can be prepared more generally.Apply the
QuantumPhaseEstimation
circuit.Measure
estimation_wires
usingprobs()
, giving a probability distribution over measurement outcomes in the computational basis.Find the index of the largest value in the probability distribution and divide that number by \(2^{n}\). This number will be an estimate of \(\theta\) with an error that decreases exponentially with the number of qubits \(n\).
Note that if \(\theta \in (1, 0]\), we can estimate the phase by again finding the index \(i\) found in step 4 and calculating \(\theta \approx \frac{1  i}{2^{n}}\). An example of this case is below.
Consider the matrix corresponding to a rotation from an
RX
gate:import pennylane as qml from pennylane.templates import QuantumPhaseEstimation from pennylane import numpy as np phase = 5 target_wires = [0] unitary = qml.RX(phase, wires=0).matrix()
The
phase
parameter can be estimated usingQuantumPhaseEstimation
. An example is shown below using a register of five phaseestimation qubits:n_estimation_wires = 5 estimation_wires = range(1, n_estimation_wires + 1) dev = qml.device("default.qubit", wires=n_estimation_wires + 1) @qml.qnode(dev) def circuit(): # Start in the +> eigenstate of the unitary qml.Hadamard(wires=target_wires) QuantumPhaseEstimation( unitary, target_wires=target_wires, estimation_wires=estimation_wires, ) return qml.probs(estimation_wires) phase_estimated = np.argmax(circuit()) / 2 ** n_estimation_wires # Need to rescale phase due to convention of RX gate phase_estimated = 4 * np.pi * (1  phase_estimated)
We can also perform phase estimation on an operator. Note that since operators are defined with target wires, the target wires should not be provided for the QPE.
# use the product to specify compound operators unitary = qml.RX(np.pi / 2, wires=[0]) @ qml.CNOT(wires=[0, 1]) eigenvector = np.array([1/2, 1/2, 1/2, 1/2]) n_estimation_wires = 5 estimation_wires = range(2, n_estimation_wires + 2) target_wires = [0, 1] dev = qml.device("default.qubit", wires=n_estimation_wires + 2) @qml.qnode(dev) def circuit(): qml.StatePrep(eigenvector, wires=target_wires) QuantumPhaseEstimation( unitary, estimation_wires=estimation_wires, ) return qml.probs(estimation_wires) phase_estimated = np.argmax(circuit()) / 2 ** n_estimation_wires
Attributes
Arithmetic depth of the operator.
The basis of an operation, or for controlled gates, of the target operation.
Batch size of the operator if it is used with broadcasted parameters.
Control wires of the operator.
The estimation wires of the QPE
Gradient recipe for the parametershift method.
Integer hash that uniquely represents the operator.
Dictionary of nontrainable variables that this operation depends on.
Custom string to label a specific operator instance.
This property determines if an operator is hermitian.
String for the name of the operator.
Number of dimensions per trainable parameter of the operator.
Number of trainable parameters that the operator depends on.
Number of wires the operator acts on.
Returns the frequencies for each operator parameter with respect to an expectation value of the form \(\langle \psi  U(\mathbf{p})^\dagger \hat{O} U(\mathbf{p})\psi\rangle\).
Trainable parameters that the operator depends on.
The target wires of the QPE
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
andCNOT
havebasis = "X"
, whereasControlledPhaseShift
andRZ
havebasis = "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 onndim_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 isNone
. 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 length0
. Returns
The control wires of the operation.
 Return type

estimation_wires
¶ The estimation wires of the QPE

grad_method
= None¶

grad_recipe
= None¶ Gradient recipe for the parametershift 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 nontrainable 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 nonnegative 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.

target_wires
¶ The target wires of the QPE
Methods
adjoint
()Create an operation that is the adjoint of this one.
compute_decomposition
(wires, unitary, …)Representation of the QPE circuit 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).
Representation of the operator as a product of other operators.
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 of an operator that is in singleparameterform.
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.
simplify
()Reduce the depth of nested operators to the minimum.
The parameters required to implement a singlequbit 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
(wires, unitary, target_wires, estimation_wires)[source]¶ Representation of the QPE circuit as a product of other operators.
\[O = O_1 O_2 \dots O_n.\]See also
 Parameters
wires (Any or Iterable[Any]) – wires that the QPE circuit acts on
unitary (Operator) – the phase estimation unitary, specified as an operator
target_wires (Any or Iterable[Any]) – the target wires to apply the unitary
estimation_wires (Any or Iterable[Any]) – the wires to be used for phase estimation
 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
 Parameters
params (list) – trainable parameters of the operator, as stored in the
parameters
attributewires (Iterable[Any], Wires) – wires that the operator acts on
hyperparams (dict) – nontrainable 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.
See also
 Parameters
*params (list) – trainable parameters of the operator, as stored in the
parameters
attribute**hyperparams (dict) – nontrainable 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
 Parameters
*params (list) – trainable parameters of the operator, as stored in the
parameters
attribute**hyperparams (dict) – nontrainable 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
 Parameters
*params (list) – trainable parameters of the operator, as stored in the
parameters
attribute**hyperparams (dict) – nontrainable 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
 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
 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
 Returns
eigenvalues
 Return type
tensor_like

expand
()¶ Returns a tape that contains the decomposition of the operator.
 Returns
quantum tape
 Return type

generator
()¶ Generator of an operator that is in singleparameterform.
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
andSparseHamiltonian
respectively).The default value to return is
None
, indicating that the operation has no defined generator.

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 nonparameter 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(decimals=2) "RX\n(1.23)" >>> op.label(base_label="my_label") "my_label" >>> op.label(decimals=2, base_label="my_label") "my_label\n(1.23)"
If the operation has a matrixvalued 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)[source]¶ 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

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
 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'>)[source]¶ Append the operator to the Operator queue.

simplify
()¶ Reduce the depth of nested operators to the minimum.
 Returns
simplified operator
 Return type

single_qubit_rot_angles
()¶ The parameters required to implement a singlequbit 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.See also
 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