A Simple Heuristic for Expressing a Truth Table as a Quadratic Pseudo-Boolean Function

Quadratic pseudo-Boolean functions (i.e., quadratic functions with real-valued coefficients and two-valued variables) are the native input to quantum annealers and also a common problem format to optimize with the QAOA algorithm on circuitmodel quantum computers. In both cases, large, complex optimization problems can be expressed in terms of combinations of simpler problems. A key challenge in problem set-up lies in finding the coefficients for the constituent sub-problems. These need to be chosen such that the function is minimized on valid inputs and is higher elsewhere. One difficulty is that it is not possible in general to solve for a sub-problem’s coefficients without introducing ancillary variables, the number of which and their corresponding per-row values being unknown a priori. These unknowns lead to solving for a sub-problem’s coefficients being an NP-complete problem.In this paper we present a simple heuristic that considers a prioritized subset of the exponential number of possibilities for each per-row ancillary variable. Using this heuristic, coefficients can be found substantially faster than would be possible using a brute-force search and with less software complexity than is required by prior approaches.