Hardware-Aware QUBO Reformulation of Constrained Binary Optimization via the Walsh-Fourier Transform

We present a novel slack-free, penalty-based framework for reformulating constrained binary optimization as Quadratic Unconstrained Binary Optimization (QUBO) on near-term quantum annealing hardware.

Given a user-chosen penalty function that most naturally captures a constraint---typically non-quadratic, such as a Heaviside-function surrogate---and a target probability measure over the Boolean hypercube, our method returns the weighted least-squares projection of the chosen penalty function onto the subspace spanned by linear and quadratic Walsh--Fourier characters that correspond to physically realizable couplings on the target hardware graph.

Within this restricted family, the resulting quadratic surrogate is uniquely and optimally determined by the normal equations: unlike state-of-the-art approaches, it introduces no per-constraint penalty coefficients to tune and avoids dense all-pairs couplings by construction.

Two practical consequences follow.

First, the projected penalty respects device connectivity, reducing chain lengths and physical-qubit overhead after minor embedding.

Second, we show empirically that this hardware-native surrogate can outperform denser full-pairwise projections, despite being drawn from a strictly smaller approximation space.

This advantage widens once the QUBO is embedded and sampled on quantum annealers, yielding samples with the lowest worst-case and mean objective gaps compared to unbalanced penalization and a hardware-blind projection onto all quadratic terms.