In this paper, we study the discrete complex quadratic optimization problem, where the goal is to maximize a multivariate Hermitian quadratic objective function over complex m’th roots of unity. This problem is a generalization of the well-studied quadratic unconstrained binary optimization problem (QUBO), and has applications in signal processing, including in maximal likelihood estimation and multiple-input multiple-output detection, in addition to applications in statistical physics, in particular the computation of ground states of the Potts model. We study the sum-of-squares hierarchy for this problem, which is a sequence of semidefinite relaxations that is known to converge to the optimal solution after at most n steps. A previous result of Fawzi, Saunderson and Parrilo showed that in the the binary case (
m=2 ), at most
⌈n/2⌉ levels of the hierarchy are actually needed. In this paper we generalize this result and establish that the hierarchy converges after at most
⌊n/2⌋+1 level for any
m≥2 . Our proof generalizes the ideas used in the
m=2 case, and is based on constructing chordal covers of appropriate Cayley graphs.
We study the rank of the Sum of Squares (SoS) hierarchy over the Boolean hypercube for Symmetric Quadratic Functions (SQFs) in n variables with roots placed in points k - 1 and k. Functions of this type have played a central role in deepening the understanding of the performance of the SoS method for various unconstrained Boolean hypercube optimization problems, including the Max Cut problem. Recently, Lee, Prakash, de Wolf, and Yuen proved a lower bound on the SoS rank for SQFs of Ω(√k(n - k)) and conjectured the lower bound of Ω(n) by similarity to a polynomial representation of the n-bit OR function. Leveraging recent developments on Chebyshev polynomials, we refute the Lee-Prakash-de Wolf- Yuen conjecture and prove that the SoS rank for SQFs is at most O(√nk log(n)). We connect this result to two constrained Boolean hypercube optimization problems. First, we provide a degree O(√n) SoS certificate that matches the known SoS rank lower bound for an instance of Min Knapsack, a problem that was intensively studied in the literature. Second, we study an instance of the Set Cover problem for which Bienstock and Zuckerberg conjectured an SoS rank lower bound of n/4. We refute the Bienstock-Zuckerberg conjecture and provide a degree O(√n log(n)) SoS certificate for this problem.
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.