Ising and QUBO Formulations

To run a problem on an annealer you must write it as the ground state of a diagonal Hamiltonian. Two equivalent classical encodings dominate: the Ising model (spins $\pm 1$ ) and QUBO (binary variables $0/1$ ). This lesson defines both, shows they are interchangeable, and computes a tiny ground-state energy by brute force.

The Ising energy

The Ising model assigns each variable a spin $s_i \in \{-1, +1\}$ and an energy

E_{\text{Ising}}(\mathbf{s}) = \sum_{i<j} J_{ij}\,s_i s_j + \sum_i h_i\,s_i ,

with couplings $J_{ij}$ and local fields $h_i$ . Quantizing means replacing each $s_i$ by the Pauli $Z_i$ (eigenvalues $\pm 1$ ), giving the diagonal problem Hamiltonian

H_P = \sum_{i<j} J_{ij}\,Z_i Z_j + \sum_i h_i\,Z_i .

A negative coupling $J_{ij} < 0$ is ferromagnetic — it rewards aligned spins; a positive coupling is antiferromagnetic and rewards opposite spins.

QUBO

A Quadratic Unconstrained Binary Optimization problem uses binary variables $x_i \in \{0, 1\}$ and minimizes

f(\mathbf{x}) = \sum_{i \le j} Q_{ij}\,x_i x_j .

The two forms are related by the linear change of variable

s_i = 1 - 2 x_i \quad\Longleftrightarrow\quad x_i = \frac{1 - s_i}{2},

so $x_i = 0 \leftrightarrow s_i = +1$ and $x_i = 1 \leftrightarrow s_i = -1$ . Substituting maps any QUBO to an Ising model and back (up to an additive constant), which is why hardware vendors accept either form. Lucas's survey shows how graph coloring, max-cut, partitioning, and many other NP problems become Ising/QUBO instances.

A two-spin example

Take a single ferromagnetic coupling $J_{01} = -1$ and no fields:

E(s_0, s_1) = -\,s_0 s_1 .

The four configurations give energies

E(+,+) = -1,\quad E(-,-) = -1,\quad E(+,-) = +1,\quad E(-,+) = +1.

The ground-state energy is $-1$ , doubly degenerate: both spins up ( $|00\rangle$ ) or both down ( $|11\rangle$ ). Ferromagnetic coupling, as expected, prefers alignment.

Try it

Brute-force the two-spin ferromagnet $E(s_0, s_1) = -s_0 s_1$ over all four spin configurations and return the minimum energy. The grader checks that you return $-1$ .

Run your code to see the quantum state.

The loop returns the ground-state energy $-1$ , realized by the aligned states $|00\rangle$ and $|11\rangle$ .