Quantum processing units (QPUs) executing annealing algorithms have shown promise in optimization and simulation applications. Hybrid algorithms are a natural bridge to larger applications. We present a simple greedy method for solving larger-than-QPU lattice-structured Ising optimization problems. The method, implemented in the open source D-Wave Hybrid framework, uses a QPU coprocessor operating with generic parameters. Performance is evaluated for standard spin-glass problems on two lattice types with up to 11,616 spin variables, double the size that is directly programmable on any available QPU. The proposed method is shown to converge to low-energy solutions faster than an open source simulated annealing method that is either directly employed or substituted as a coprocessor in the hybrid method. Using newer Advantage QPUs in place of D-Wave 2000Q QPUs is shown to enhance convergence of the hybrid method to low energies and to achieve a lower final energy.

Pairs of binary sequences formed using linear combinations of multiplicative characters of finite fields are exhibited that, when compared with a random sequence pairs, simultaneously achieve significantly lower mean square autocorrelation values (for each sequence in the pair) and significantly lower mean square crosscorrelation values. If we define crosscorrelation merit factor analogously to the usual merit factor for autocorrelation, and if we define demerit factor as the reciprocal of merit factor, then randomly selected binary sequence pairs are known to have an average crosscorrelation demerit factor of 1. Our constructions provide sequence pairs with a crosscorrelation demerit factor significantly less than 1, and at the same time, the autocorrelation demerit factors of the individual sequences can also be made significantly less than 1 (which also indicates better than average performance). The sequence pairs studied here provide combinations of autocorrelation and crosscorrelation performance that are not achievable using sequences formed from single characters, such as maximal linear recursive sequences (m-sequences) and Legendre sequences. In this paper, exact asymptotic formulae are proved for the autocorrelation and crosscorrelation merit factors of sequence pairs formed using linear combinations of multiplicative characters. Data is presented that shows that the asymptotic behavior is closely approximated by sequences of modest length.