Quantum computational phase transition in combinatorial problems

Abstract

Quantum Approximate Optimization algorithm (QAOA) aims to search for approximate solutions to discrete optimization problems with near-term quantum computers. As there are no algorithmic guarantee possible for QAOA to outperform classical computers, without a proof that bounded-error quantum polynomial time (BQP) ≠ nondeterministic polynomial time (NP), it is necessary to investigate the empirical advantages of QAOA. We identify a computational phase transition of QAOA when solving hard problems such as SAT—random instances are most difficult to train at a critical problem density. We connect the transition to the controllability and the complexity of QAOA circuits. Moreover, we find that the critical problem density in general deviates from the SAT-UNSAT phase transition, where the hardest instances for classical algorithms lies. 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 over classical approximate algorithms can be identified.

Solution of SAT Problems with the Adaptive-Bias Quantum Approximate Optimization Algorithm

“…Note that the QAOA and ab-QAOA are able to solve both Max-1-3-SAT + and 1-3-SAT + problems with the same cost Hamiltonian. In the former problem, we need to find the exact ground state, while in the latter problem, we just need to know whether the optimized energy is smaller than a threshold E th (SAT) or not (UNSAT) [24], where the threshold E th is E th = 0.5 in this paper. The penalty terms for 1-3-SAT + and Max-1-3-SAT + problems are,…”

“…The classical computational cost suffers from an easyhard-easy pattern, where the problems near α c are known to be the hardest [26,29]. It was found in [24] that the QAOA also follows an easy-hard-easy pattern in the same region.…”

“…It displays an easy-hard-easy pattern in the random-3-SAT problem and an easy-hard pattern in the random Max-3-SAT problem [27][28][29]. The same pattern is evident in the QAOA [24,25], where in the hard regions, a very large number of levels is required to obtain an accurate ground state. This means that such problems are out of reach for current NISQ devices [20], and prospects are dim for the foreseeable future.…”

“…However, although the QAOA does well on the Max-Cut problem, it encounters difficulties when applied to the random 3-SAT problems [24] and random Max-3-SAT problems [25]. These problems have been intensely studied in the framework of classical algorithms [26][27][28][29][30].…”

“…Reachability deficits can occur even in the absence of the barren plateaus, e.g. for large clause density α with a fixed n. Nevertheless, the hard-region SAT problems does seem to exhibit small variance of the energy gradients [24].…”