qml.Hamiltonian¶

class
Hamiltonian
(coeffs, observables, simplify=False, grouping_type=None, method='rlf', id=None)[source]¶ Bases:
pennylane.operation.Observable
Operator representing a Hamiltonian.
The Hamiltonian is represented as a linear combination of other operators, e.g., \(\sum_{k=0}^{N1} c_k O_k\), where the \(c_k\) are trainable parameters.
 Parameters
coeffs (tensor_like) – coefficients of the Hamiltonian expression
observables (Iterable[Observable]) – observables in the Hamiltonian expression, of same length as coeffs
simplify (bool) – Specifies whether the Hamiltonian is simplified upon initialization (liketerms are combined). The default value is False.
grouping_type (str) – If not None, compute and store information on how to group commuting observables upon initialization. This information may be accessed when QNodes containing this Hamiltonian are executed on devices. The string refers to the type of binary relation between Pauli words. Can be
'qwc'
(qubitwise commuting),'commuting'
, or'anticommuting'
.method (str) – The graph coloring heuristic to use in solving minimum clique cover for grouping, which can be
'lf'
(Largest First) or'rlf'
(Recursive Largest First). Ignored ifgrouping_type=None
.id (str) – name to be assigned to this Hamiltonian instance
Example:
A Hamiltonian can be created by simply passing the list of coefficients as well as the list of observables:
>>> coeffs = [0.2, 0.543] >>> obs = [qml.PauliX(0) @ qml.PauliZ(1), qml.PauliZ(0) @ qml.Hadamard(2)] >>> H = qml.Hamiltonian(coeffs, obs) >>> print(H) (0.543) [Z0 H2] + (0.2) [X0 Z1]
The coefficients can be a trainable tensor, for example:
>>> coeffs = tf.Variable([0.2, 0.543], dtype=tf.double) >>> obs = [qml.PauliX(0) @ qml.PauliZ(1), qml.PauliZ(0) @ qml.Hadamard(2)] >>> H = qml.Hamiltonian(coeffs, obs) >>> print(H) (0.543) [Z0 H2] + (0.2) [X0 Z1]
The user can also provide custom observables:
>>> obs_matrix = np.array([[0.5, 1.0j, 0.0, 3j], [1.0j, 1.1, 0.0, 0.1], [0.0, 0.0, 0.9, 12.0], [3j, 0.1, 12.0, 0.0]]) >>> obs = qml.Hermitian(obs_matrix, wires=[0, 1]) >>> H = qml.Hamiltonian((0.8, ), (obs, )) >>> print(H) (0.8) [Hermitian0,1]
Alternatively, the
molecular_hamiltonian()
function from the Quantum Chemistry module can be used to generate a molecular Hamiltonian.In many cases, Hamiltonians can be constructed using Pythonic arithmetic operations. For example:
>>> qml.Hamiltonian([1.], [qml.PauliX(0)]) + 2 * qml.PauliZ(0) @ qml.PauliZ(1)
is equivalent to the following Hamiltonian:
>>> qml.Hamiltonian([1, 2], [qml.PauliX(0), qml.PauliZ(0) @ qml.PauliZ(1)])
While scalar multiplication requires native python floats or integer types, addition, subtraction, and tensor multiplication of Hamiltonians with Hamiltonians or other observables is possible with tensorvalued coefficients, i.e.,
>>> H1 = qml.Hamiltonian(torch.tensor([1.]), [qml.PauliX(0)]) >>> H2 = qml.Hamiltonian(torch.tensor([2., 3.]), [qml.PauliY(0), qml.PauliX(1)]) >>> obs3 = [qml.PauliX(0), qml.PauliY(0), qml.PauliX(1)] >>> H3 = qml.Hamiltonian(torch.tensor([1., 2., 3.]), obs3) >>> H3.compare(H1 + H2) True
A Hamiltonian can store information on which commuting observables should be measured together in a circuit:
>>> obs = [qml.PauliX(0), qml.PauliX(1), qml.PauliZ(0)] >>> coeffs = np.array([1., 2., 3.]) >>> H = qml.Hamiltonian(coeffs, obs, grouping_type='qwc') >>> H.grouping_indices [[0, 1], [2]]
This attribute can be used to compute groups of coefficients and observables:
>>> grouped_coeffs = [coeffs[indices] for indices in H.grouping_indices] >>> grouped_obs = [[H.ops[i] for i in indices] for indices in H.grouping_indices] >>> grouped_coeffs [tensor([1., 2.], requires_grad=True), tensor([3.], requires_grad=True)] >>> grouped_obs [[qml.PauliX(0), qml.PauliX(1)], [qml.PauliZ(0)]]
Devices that evaluate a Hamiltonian expectation by splitting it into its local observables can use this information to reduce the number of circuits evaluated.
Note that one can compute the
grouping_indices
for an already initialized Hamiltonian by using thecompute_grouping
method.Attributes
Arithmetic depth of the operator.
Return the coefficients defining the Hamiltonian.
Return the grouping indices attribute.
Integer hash that uniquely represents the operator.
Dictionary of nontrainable variables that this operation depends on.
Custom string to label a specific operator instance.
All observables must be hermitian
String for the name of the operator.
Number of trainable parameters that the operator depends on.
Number of wires the operator acts on.
Return the operators defining the Hamiltonian.
Trainable parameters that the operator depends on.
Measurement type that this observable is called with.
The sorted union of wires from all operators.

arithmetic_depth
¶ Arithmetic depth of the operator.

batch_size
= None¶

coeffs
¶ Return the coefficients defining the Hamiltonian.
 Returns
coefficients in the Hamiltonian expression
 Return type
Iterable[float])

grad_method
= 'A'¶

grouping_indices
¶ Return the grouping indices attribute.
 Returns
indices needed to form groups of commuting observables
 Return type
list[list[int]]

has_adjoint
= False¶

has_decomposition
= False¶

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
¶ All observables must be hermitian

name
¶

ndim_params
= None¶

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.

ops
¶ Return the operators defining the Hamiltonian.
 Returns
observables in the Hamiltonian expression
 Return type
Iterable[Observable])

parameters
¶ Trainable parameters that the operator depends on.

return_type
= None¶ Measurement type that this observable is called with.
 Type
None or ObservableReturnTypes
Methods
adjoint
()Create an operation that is the adjoint of this one.
compare
(other)Determines whether the operator is equivalent to another.
compute_decomposition
(*params[, wires])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_grouping
([grouping_type, method])Compute groups of indices corresponding to commuting observables of this Hamiltonian, and store it in the
grouping_indices
attribute.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 hamiltonian 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])Queues a qml.Hamiltonian instance
simplify
()Simplifies the Hamiltonian by combining liketerms.
sparse_matrix
([wire_order])Computes the sparse matrix representation of a Hamiltonian 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.

compare
(other)[source]¶ Determines whether the operator is equivalent to another.
Currently only supported for
Hamiltonian
,Observable
, orTensor
. Hamiltonians/observables are equivalent if they represent the same operator (their matrix representations are equal), and they are defined on the same wires.Warning
The compare method does not check if the matrix representation of a
Hermitian
observable is equal to an equivalent observable expressed in terms of Pauli matrices, or as a linear combination of Hermitians. To do so would require the matrix form of Hamiltonians and Tensors be calculated, which would drastically increase runtime. Returns
True if equivalent.
 Return type
(bool)
Examples
>>> H = qml.Hamiltonian( ... [0.5, 0.5], ... [qml.PauliZ(0) @ qml.PauliY(1), qml.PauliY(1) @ qml.PauliZ(0) @ qml.Identity("a")] ... ) >>> obs = qml.PauliZ(0) @ qml.PauliY(1) >>> print(H.compare(obs)) True
>>> H1 = qml.Hamiltonian([1, 1], [qml.PauliX(0), qml.PauliZ(1)]) >>> H2 = qml.Hamiltonian([1, 1], [qml.PauliZ(0), qml.PauliX(1)]) >>> H1.compare(H2) False
>>> ob1 = qml.Hamiltonian([1], [qml.PauliX(0)]) >>> ob2 = qml.Hermitian(np.array([[0, 1], [1, 0]]), 0) >>> ob1.compare(ob2) False

static
compute_decomposition
(*params, wires=None, **hyperparameters)¶ Representation of the operator as a product of other operators (static method).
\[O = O_1 O_2 \dots O_n.\]Note
Operations making up the decomposition should be queued within the
compute_decomposition
method.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
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

compute_grouping
(grouping_type='qwc', method='rlf')[source]¶ Compute groups of indices corresponding to commuting observables of this Hamiltonian, and store it in the
grouping_indices
attribute. Parameters
grouping_type (str) – The type of binary relation between Pauli words used to compute the grouping. Can be
'qwc'
,'commuting'
, or'anticommuting'
.method (str) – The graph coloring heuristic to use in solving minimum clique cover for grouping, which can be
'lf'
(Largest First) or'rlf'
(Recursive Largest First).

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)[source]¶ 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 hamiltonian 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 hamiltonian
 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]¶ Queues a qml.Hamiltonian instance

simplify
()[source]¶ Simplifies the Hamiltonian by combining liketerms.
Example
>>> ops = [qml.PauliY(2), qml.PauliX(0) @ qml.Identity(1), qml.PauliX(0)] >>> H = qml.Hamiltonian([1, 1, 2], ops) >>> H.simplify() >>> print(H) (1) [X0] + (1) [Y2]
Warning
Calling this method will reset
grouping_indices
to None, since the observables it refers to are updated.

sparse_matrix
(wire_order=None)[source]¶ Computes the sparse matrix representation of a Hamiltonian in the computational basis.
 Parameters
wire_order (Iterable) – global wire order, must contain all wire labels from the operator’s wires. If not provided, the default order of the wires (self.wires) of the Hamiltonian is used.
 Returns
a sparse matrix in scipy Compressed Sparse Row (CSR) format with dimension \((2^n, 2^n)\), where \(n\) is the number of wires
 Return type
csr_matrix
Example:
>>> coeffs = [1, 0.45] >>> obs = [qml.PauliZ(0) @ qml.PauliZ(1), qml.PauliY(0) @ qml.PauliZ(1)] >>> H = qml.Hamiltonian(coeffs, obs) >>> H_sparse = H.sparse_matrix() >>> H_sparse <4x4 sparse matrix of type '<class 'numpy.complex128'>' with 8 stored elements in Compressed Sparse Row format>
The resulting sparse matrix can be either used directly or transformed into a numpy array:
>>> H_sparse.toarray() array([[ 1.+0.j , 0.+0.j , 0.+0.45j, 0.+0.j ], [ 0.+0.j , 1.+0.j , 0.+0.j , 0.0.45j], [ 0.0.45j, 0.+0.j , 1.+0.j , 0.+0.j ], [ 0.+0.j , 0.+0.45j, 0.+0.j , 1.+0.j ]])

terms
()[source]¶ Representation of the operator as a linear combination of other operators.
\[O = \sum_i c_i O_i\]See also
 Returns
coefficients and operations
 Return type
tuple[Iterable[tensor_like or float], list[Operator]]
Example >>> coeffs = [1., 2.] >>> ops = [qml.PauliX(0), qml.PauliZ(0)] >>> H = qml.Hamiltonian(coeffs, ops)
>>> H.terms() [1., 2.], [qml.PauliX(0), qml.PauliZ(0)]
The coefficients are differentiable and can be stored as tensors: >>> import tensorflow as tf >>> H = qml.Hamiltonian([tf.Variable(1.), tf.Variable(2.)], [qml.PauliX(0), qml.PauliZ(0)]) >>> t = H.terms()
>>> t[0] [<tf.Tensor: shape=(), dtype=float32, numpy=1.0>, <tf.Tensor: shape=(), dtype=float32, numpy=2.0>]

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