qml.UCCSD¶

class
UCCSD
(weights, wires, s_wires=None, d_wires=None, init_state=None, do_queue=True, id=None)[source]¶ Bases:
pennylane.operation.Operation
Implements the Unitary CoupledCluster Singles and Doubles (UCCSD) ansatz.
The UCCSD ansatz calls the
FermionicSingleExcitation()
andFermionicDoubleExcitation()
templates to exponentiate the coupledcluster excitation operator. UCCSD is a VQE ansatz commonly used to run quantum chemistry simulations.The UCCSD unitary, within the firstorder Trotter approximation, is given by:
\[\hat{U}(\vec{\theta}) = \prod_{p > r} \mathrm{exp} \Big\{\theta_{pr} (\hat{c}_p^\dagger \hat{c}_r\mathrm{H.c.}) \Big\} \prod_{p > q > r > s} \mathrm{exp} \Big\{\theta_{pqrs} (\hat{c}_p^\dagger \hat{c}_q^\dagger \hat{c}_r \hat{c}_s\mathrm{H.c.}) \Big\}\]where \(\hat{c}\) and \(\hat{c}^\dagger\) are the fermionic annihilation and creation operators and the indices \(r, s\) and \(p, q\) run over the occupied and unoccupied molecular orbitals, respectively. Using the JordanWigner transformation the UCCSD unitary defined above can be written in terms of Pauli matrices as follows (for more details see arXiv:1805.04340):
\[\begin{split}\hat{U}(\vec{\theta}) = && \prod_{p > r} \mathrm{exp} \Big\{ \frac{i\theta_{pr}}{2} \bigotimes_{a=r+1}^{p1} \hat{Z}_a (\hat{Y}_r \hat{X}_p  \mathrm{H.c.}) \Big\} \\ && \times \prod_{p > q > r > s} \mathrm{exp} \Big\{ \frac{i\theta_{pqrs}}{8} \bigotimes_{b=s+1}^{r1} \hat{Z}_b \bigotimes_{a=q+1}^{p1} \hat{Z}_a (\hat{X}_s \hat{X}_r \hat{Y}_q \hat{X}_p + \hat{Y}_s \hat{X}_r \hat{Y}_q \hat{Y}_p + \hat{X}_s \hat{Y}_r \hat{Y}_q \hat{Y}_p + \hat{X}_s \hat{X}_r \hat{X}_q \hat{Y}_p  \{\mathrm{H.c.}\}) \Big\}.\end{split}\] Parameters
weights (tensor_like) – Size
(len(s_wires) + len(d_wires),)
tensor containing the parameters \(\theta_{pr}\) and \(\theta_{pqrs}\) entering the Z rotation inFermionicSingleExcitation()
andFermionicDoubleExcitation()
. These parameters are the coupledcluster amplitudes that need to be optimized for each single and double excitation generated with theexcitations()
function.wires (Iterable) – wires that the template acts on
s_wires (Sequence[Sequence]) – Sequence of lists containing the wires
[r,...,p]
resulting from the single excitation \(\vert r, p \rangle = \hat{c}_p^\dagger \hat{c}_r \vert \mathrm{HF} \rangle\), where \(\vert \mathrm{HF} \rangle\) denotes the HarteeFock reference state. The first (last) entryr
(p
) is considered the wire representing the occupied (unoccupied) orbital where the electron is annihilated (created).d_wires (Sequence[Sequence[Sequence]]) – Sequence of lists, each containing two lists that specify the indices
[s, ...,r]
and[q,..., p]
defining the double excitation \(\vert s, r, q, p \rangle = \hat{c}_p^\dagger \hat{c}_q^\dagger \hat{c}_r \hat{c}_s \vert \mathrm{HF} \rangle\). The entriess
andr
are wires representing two occupied orbitals where the two electrons are annihilated while the entriesq
andp
correspond to the wires representing two unoccupied orbitals where the electrons are created. Wires inbetween represent the occupied and unoccupied orbitals in the intervals[s, r]
and[q, p]
, respectively.init_state (array[int]) – Length
len(wires)
occupationnumber vector representing the HF state.init_state
is used to initialize the wires.
Usage Details
Notice that:
The number of wires has to be equal to the number of spin orbitals included in the active space.
The single and double excitations can be generated with the function
excitations()
. See example below.The vector of parameters
weights
is a onedimensional array of sizelen(s_wires)+len(d_wires)
An example of how to use this template is shown below:
import pennylane as qml from pennylane import numpy as np # Define the molecule symbols = ['H', 'H', 'H'] geometry = np.array([[0.01076341, 0.04449877, 0.0], [0.98729513, 1.63059094, 0.0], [1.87262415, 0.00815842, 0.0]], requires_grad = False) electrons = 2 charge = 1 # Build the electronic Hamiltonian H, qubits = qml.qchem.molecular_hamiltonian(symbols, geometry, charge=charge) # Define the HF state hf_state = qml.qchem.hf_state(electrons, qubits) # Generate single and double excitations singles, doubles = qml.qchem.excitations(electrons, qubits) # Map excitations to the wires the UCCSD circuit will act on s_wires, d_wires = qml.qchem.excitations_to_wires(singles, doubles) # Define the device dev = qml.device("default.qubit", wires=qubits) # Define the qnode @qml.qnode(dev) def circuit(params, wires, s_wires, d_wires, hf_state): qml.UCCSD(params, wires, s_wires, d_wires, hf_state) return qml.expval(H) # Define the initial values of the circuit parameters params = np.zeros(len(singles) + len(doubles)) # Define the optimizer optimizer = qml.GradientDescentOptimizer(stepsize=0.5) # Optimize the circuit parameters and compute the energy for n in range(21): params, energy = optimizer.step_and_cost(circuit, params, wires=range(qubits), s_wires=s_wires, d_wires=d_wires, hf_state=hf_state) if n % 2 == 0: print("step = {:}, E = {:.8f} Ha".format(n, energy))
step = 0, E = 1.24654994 Ha step = 2, E = 1.27016844 Ha step = 4, E = 1.27379541 Ha step = 6, E = 1.27434106 Ha step = 8, E = 1.27442311 Ha step = 10, E = 1.27443547 Ha step = 12, E = 1.27443733 Ha step = 14, E = 1.27443761 Ha step = 16, E = 1.27443765 Ha step = 18, E = 1.27443766 Ha step = 20, E = 1.27443766 Ha
Attributes
Arithmetic depth of the operator.
If inverse is requested, this is the name of the original operator to be inverted.
The target operation for controlled gates.
Batch size of the operator if it is used with broadcasted parameters.
Control wires of the operator.
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.
Boolean determining if the inverse of the operation was requested.
This property determines if an operator is hermitian.
Name of the operator.
Number of dimensions per trainable parameter of the operator.
Number of trainable parameters that the operator depends 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.
Wires that the operator acts on.

arithmetic_depth
¶ Arithmetic depth of the operator.

base_name
¶ If inverse is requested, this is the name of the original operator to be inverted.

basis
= None¶ The target operation for controlled gates. 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

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_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.

inverse
¶ Boolean determining if the inverse of the operation was requested.

is_hermitian
¶ This property determines if an operator is hermitian.

name
¶ 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
¶

num_wires
= 1¶

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.
Methods
adjoint
()Create an operation that is the adjoint of this one.
compute_decomposition
(weights, wires, …)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).
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 has recorded the decomposition of the operator.
Generator of an operator that is in singleparameterform.
inv
()Inverts the operator.
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
() → pennylane.operation.Operator¶ 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
(weights, wires, s_wires, d_wires, init_state)[source]¶ Representation of the operator as a product of other operators.
\[O = O_1 O_2 \dots O_n.\]See also
 Parameters
weights (tensor_like) – Size
(len(s_wires) + len(d_wires),)
tensor containing the parameters entering the Z rotation inFermionicSingleExcitation()
andFermionicDoubleExcitation()
.wires (Any or Iterable[Any]) – wires that the operator acts on
s_wires (Sequence[Sequence]) – Sequence of lists containing the wires
[r,...,p]
resulting from the single excitation.d_wires (Sequence[Sequence[Sequence]]) – Sequence of lists, each containing two lists that specify the indices
[s, ...,r]
and[q,..., p]
defining the double excitation.init_state (array[int]) – Length
len(wires)
occupationnumber vector representing the HF state.init_state
is used to initialize the wires.
 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.
 Parameters
params (list) – trainable parameters of the operator, as stored in the
parameters
attributehyperparams (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.
 Parameters
params (list) – trainable parameters of the operator, as stored in the
parameters
attributehyperparams (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
attributehyperparams (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 has recorded 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.

inv
()¶ Inverts the operator.
This method concatenates a string to the name of the operation, to indicate that the inverse will be used for computations.
Any subsequent call of this method will toggle between the original operation and the inverse of the operation.
 Returns
operation to be inverted
 Return type
Operator

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)" >>> op.inv() >>> op.label() "RX⁻¹"
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: dict)¶ 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) → List[pennylane.operation.Operator]¶ 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
() → pennylane.operation.Operator¶ 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