Quantum Computational Phase Transition in Combinatorial Problems

Quantum Approximate Optimization algorithm (QAOA) is one of the candidates to achieve a near-term quantum advantage. To search for such a quantum advantage in solving any problem, it is crucial to first understand the difference between problem instances' empirical hardness for QAOA and classical algorithms. We identify a computational phase transition of QAOA when solving hard problems such as 3-SAT -- the performance is worst at the well-known SAT-UNSAT phase transition, where the hardest instances lie. We connect the transition to the controllability and the complexity of QAOA circuits. Such a transition is absent for 2-SAT and QAOA achieves close to perfect performance at the problem size we studied. Then, we show that the high problem density region, which limits QAOA's performance in hard optimization problems (reachability deficits), is actually a good place to utilize QAOA: its approximation ratio has a much slower decay with the problem density, compared to classical approximate algorithms. Indeed, it is exactly in this region that quantum advantages of QAOA can be identified. The computational phase transition generalizes to other Hamiltonian-based algorithms, such as the quantum adiabatic algorithm.