On the hardness of quadratic unconstrained binary optimization problems

TL;DR

The paper addresses the hardness of quadratic unconstrained binary optimization problems (QUBOs) by linking small-N landscape statistics, specifically Hamming-distance and level-spacing distributions, to large-N performance of quantum annealing on the D-Wave device. It classifies QUBOs into three families (2SAT-derived, fully-connected random spin-glass, and fully-connected regular spin-glass) and uses exact enumeration, Hamming-distance analysis, and D-Wave 5.1 experiments (up to N=170) to reveal landscape-driven differences in solvability. The key finding is that exponents characterizing the D-Wave success probability correlate well with the small-N landscape metrics, with REG problems generally easiest and 2SAT hardest, while RAN sit in between; this provides a predictive lens for annealing hardness and benchmarking. Overall, the work links microscopic energy landscapes to macroscopic quantum-annealing performance, offering guidance for evaluating and selecting QUBO instances and motivating further exploration of landscape-informed predictions for both quantum and classical solvers.

Abstract

We use exact enumeration to characterize the solutions of quadratic unconstrained binary optimization problems of less than 21 variables in terms of their distributions of Hamming distances to close-by solutions. We also perform experiments with the D-Wave Advantage 5.1 quantum annealer, solving many instances of up to 170-variable, quadratic unconstrained binary optimization problems. Our results demonstrate that the exponents characterizing the success probability of a D-Wave annealer to solve a QUBO correlate very well with the predictions based on the Hamming distance distributions computed for small problem instances.

Figures (6)

• Figure 1: (color online) (a) Frequencies of Hamming distances between the ground state configuration and the first 6036 excited states, obtained by analyzing a $N=20$ 2SAT problem. (b) Level spacing distribution, that is the distribution of the energy difference $\Delta$ between the first excited state and the ground state, the second excited state and the first excited state, etc. For the 2SAT problem considered, the first 6036 excited state are degenerate, yielding one peak at $\Delta=4$.
• Figure 2: (color online) Same as Fig. \ref{['HAMM0']} except that the problem instance belongs to the class of $N=20$ RAN problems and that we only show the results for the 4000 states which are closest to the ground state in energy. In this case, the first 4000 excited energy levels differ by approximately 17 units whereas for 2SAT, the energy of 6036 of the lowest excited states are only 4 units of energy higher than the ground state.
• Figure 3: (color online) Same as Fig. \ref{['HAMM0']} except that the problem instance belongs to the class of $N=20$ REG problems. There are only five distinct energy differences in this case. The energies of the 4000 lowest energy levels differ by approximately 24 units.
• Figure 4: (color online) (a) Mean success probability and its variance as a function of the problem size $N$, obtained by solving all problem instances of the REG class on a D-Wave Advantage 5.1 quantum annealer. Solid lines are least square fits to data for $N<80$ and $N>80$, respectively. (b) Average chain length as a function of the problem size $N$, a measure for the average number of physical qubits that is required to represent one variable in the QUBO problem.
• Figure 5: (color online) Same as Fig. \ref{['RQA2']} except that the problem instances belongs to the RAN class.