qml.Hamiltonian

class Hamiltonian(coeffs, observables, simplify=False)

Bases: object

Lightweight class for representing Hamiltonians for Variational Quantum Eigensolver problems.

Hamiltonians can be expressed as linear combinations of observables, e.g., \(\sum_{k=0}^{N-1} c_k O_k\).

This class keeps track of the terms (coefficients and observables) separately.

Parameters

• coeffs (Iterable[float]) – coefficients of the Hamiltonian expression

• observables (Iterable[Observable]) – observables in the Hamiltonian expression

• simplify (bool) – Specifies whether the Hamiltonian is simplified upon initialization (like-terms are combined). The default value is False.

See also

ExpvalCost, molecular_hamiltonian()

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 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.

Warning

Hamiltonians can be constructed using Pythonic arithmetic operations. For example:

>>> 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)])

When Hamiltonians are defined using arithmetic operations inside of QNodes, constituent observables may be queued as operations/an error may be thrown. Thus, Hamiltonians must be defined either outside of QNodes, or inside of QNodes using the conventional method.

Note that this issue also arises when calling the simplify() method.

Attributes

coeffs	Return the coefficients defining the Hamiltonian.
name
ops	Return the operators defining the Hamiltonian.
terms	The terms of the Hamiltonian expression \(\sum_{k=0}^{N-1} c_k O_k\)
wires	The sorted union of wires from all operators.

coeffs

Return the coefficients defining the Hamiltonian.

Returns

coefficients in the Hamiltonian expression

Return type

Iterable[float])

name

ops

Return the operators defining the Hamiltonian.

Returns

observables in the Hamiltonian expression

Return type

Iterable[Observable])

terms

The terms of the Hamiltonian expression \(\sum_{k=0}^{N-1} c_k O_k\)

Returns

tuples of coefficients and operations, each of length N

Return type

(tuple, tuple)

wires

The sorted union of wires from all operators.

Returns

Combined wires present in all terms, sorted.

Return type

(Wires)

Methods

compare(H)	Compares with another Hamiltonian, Observable, or Tensor, to determine if they are equivalent.
queue()	Queues a qml.Hamiltonian instance
simplify()	Simplifies the Hamiltonian by combining like-terms.

compare(H)

Compares with another Hamiltonian, Observable, or Tensor, to determine if they are equivalent.

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

>>> A = np.array([[1, 0], [0, -1]])
>>> H = qml.Hamiltonian(
...     [0.5, 0.5],
...     [qml.Hermitian(A, 0) @ qml.PauliY(1), qml.PauliY(1) @ qml.Hermitian(A, 0) @ qml.Identity("a")]
... )
>>> obs = qml.Hermitian(A, 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

queue()

Queues a qml.Hamiltonian instance

simplify()

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]