Energy Models

An energy model defines a classical energy function over binary configurations which is the target distribution the QeMCMC sampler explores. The energy can be any function of the variables, but two common formulations are the Ising model and QUBO (Quadratic Unconstrained Binary Optimization). Both are binary quadratic models, and conversion between them is trivial (via \(s = 2x - 1\)).

In QeMCMC you describe a model by its coefficients, passed to EnergyModel as a list of coupling tensors: a 1D array of linear terms and a 2D matrix of quadratic terms. Higher-order tensors (3D, 4D, …) are also accepted for interactions beyond pairwise.

Ising model

The Ising model is normally used in statistical mechanics and physics based problems. The \(N\) variables \(s = [s_1, \dots, s_N]\) are spins taking values \(s_i \in \{-1, +1\}\). The objective function is given by the energy:

\[E(s) = \sum_{i=1}^{N} h_i s_i + \sum_{i<j} J_{ij}\, s_i s_j, \qquad s_i \in \{-1, +1\}.\]

The linear coefficients \(h_i\) are local fields and the quadratic coefficients \(J_{ij}\) are the coupling strengths between spins.

QUBO

QUBO is generally used in computer science problems. The variables \(x = [x_1, \dots, x_N]\) are binary, taking values \(x_i \in \{0, 1\}\). The objective function is given by the energy:

\[E(x) = \sum_{i} a_i x_i + \sum_{i<j} b_{ij}\, x_i x_j, \qquad x_i \in \{0, 1\},\]

where the \(a_i\) are the linear coefficients and the \(b_{ij}\) the quadratic coefficients.

Note

Provide the quadratic term as a full symmetric matrix with a zero diagonal (not upper-triangular): QeMCMC sums over all \(i, j\) and halves the result, giving the \(\sum_{i<j}\) above. Linear coefficients always go in the separate 1D array, never on the diagonal.

Sign convention

By default QeMCMC negates each term (cost_function_signs=[-1, -1]), i.e. it evaluates

\[E(s) = -\sum_i h_i s_i - \sum_{i<j} J_{ij} s_i s_j\]

This is the convention for Boltzmann sampling \(p(s) \propto e^{-E(s)/T}\).

Quick models for testing

ModelMaker builds a small random Ising or QUBO instance so you don't have to write couplings by hand while experimenting:

from qemcmc.model import ModelMaker

model = ModelMaker(n, "Fully Connected Ising", name="test").model   # or "Fully Connected QUBO"

For constrained models, see Constraints.