# qml.Hamiltonian¶

class Hamiltonian(coeffs, observables, simplify=False, grouping_type=None, method='rlf', id=None)[source]

Operator representing a Hamiltonian.

The Hamiltonian is represented as a linear combination of other operators, e.g., $$\sum_{k=0}^{N-1} c_k O_k$$, 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 (like-terms 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' (qubit-wise 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 if grouping_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 tensor-valued 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], ]


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
>>> 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 the compute_grouping method.

 arithmetic_depth Arithmetic depth of the operator. batch_size coeffs Return the coefficients defining the Hamiltonian. grad_method grouping_indices Return the grouping indices attribute. 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 All observables must be hermitian name String for the name of the operator. ndim_params num_params Number of trainable parameters that the operator depends on. num_wires Number of wires the operator acts on. ops Return the operators defining the Hamiltonian. parameters Trainable parameters that the operator depends on. return_type Measurement type that this observable is called with. wires 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 non-trainable 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

wires

The sorted union of wires from all operators.

Returns

Combined wires present in all terms, sorted.

Return type

(Wires)

 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. Eigenvalues of the operator in the computational basis. Returns a tape that contains the decomposition of the operator. Generator of an operator that is in single-parameter-form. 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. A list of new operators equal to this one raised to the given power. queue([context]) Queues a qml.Hamiltonian instance Simplifies the Hamiltonian by combining like-terms. sparse_matrix([wire_order]) Computes the sparse matrix representation of a Hamiltonian in the computational basis. 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

compare(other)[source]

Determines whether the operator is equivalent to another.

Currently only supported for Hamiltonian, Observable, or Tensor. 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(, [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.

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

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

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.

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.

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.

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 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(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 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.],
>>> op3 = qml.QubitUnitary(np.eye(4), wires=(0,1))
>>> op3.label(cache=cache)
'U(M1)'
>>> cache['matrices']
[tensor([[1., 0.],
tensor([[1., 0., 0., 0.],
[0., 1., 0., 0.],
[0., 0., 1., 0.],

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

Hamiltonian

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.

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 like-terms.

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.

$O = \sum_i c_i O_i$
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
[<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