Hamiltonian

The Hamiltonian class represents a symbolic Hamiltonian as a sum of weighted Pauli operators. You can create Hamiltonians using the built-in Pauli operators and combine them with standard arithmetic operations.

Constructing

To constuct a Hamiltonian with a single Pauli, you can use the constructors X(i), Y(i), Z(i), I(i). From these single-qubit operators, you can build multi-qubit Hamiltonians using arithmetic operations. The operations follow Python syntax, for example: 2 * Z(0) + Z(1) and Z(0) * Z(1) build multi-qubit Hamiltonian.

List of Operations

Arithmetic operations:

• Addition: H1 + H2

• Scalar multiplication: 5 * H

• multiplication: H0 * H1

• Subtraction: H1 - H2

• Division by scalar: H / 5

• Negation: -H

Extra Symbolic Operators:

• commutator: H1.commutator(H2)

• anticommutator: H1.anticommutator(H2)

• vector_norm: H.vector_norm()

• frobenius_norm: H.frobenius_norm()

• trace: H.trace()

Exporting Hamiltonians:

• to matrix: H.to_matrix(nqubits)

• to qtensor: H.to_qtensor(nqubits)

Importing Hamiltonians:

• from qtensor: Hamiltonian.from_qtensor(qtensor)

• from string: Hamiltonian.parse(hamiltonian_string)

Example: Ising Hamiltonian

To define an Ising Hamiltonian of the form:

\[H_{\text{Ising}} = - \sum_{\langle i, j \rangle} J_{ij} \sigma^Z_i \sigma^Z_j - \sum_j h_j \sigma^Z_j\]

you can use the Pauli Z operators from the library:

from qilisdk.analog import Z

nqubits = 3
J = {(0, 1): 1, (0, 2): 2, (1, 2): 4}
h = {0: 1, 1: 2, 2: 3}

coupling = sum(weight * Z(i) * Z(j) for (i, j), weight in J.items())
fields = sum(weight * Z(i) for i, weight in h.items())

H = -(coupling + fields)
print(H)

Output:

- Z(0) Z(1) - 2 Z(0) Z(2) - 4 Z(1) Z(2) - Z(0) - 2 Z(1) - 3 Z(2)