Quantum Approximate Optimization algorithm (QAOA) is one of the candidates to achieve a near-term quantum advantage. As QAOA seems only capable of solving optimization problems, there is a folklore that QAOA cannot see the difference between easy problems such as 2-SAT and hard problems such as 3-SAT -- although 2-SAT is in the polynomial-time (P) class, its optimization version is also nondeterministic polynomial-time (NP)-hard. In this paper, we show that the folklore is not true. We find a computational phase transition in QAOA when solving a variant of 3-SAT -- the amplitude of gradient and the success probability achieve their minimum at the well-known SAT-UNSAT phase transition. On the contrary, for 2-SAT, such a phenomena is absent at SAT-UNSAT phase transition and the success probability is unity for a reasonable circuit depth. In solving the NP-hard optimization versions of SAT, we identify quantum advantages over a classical approximate algorithm at quite a shallow depth of p=4 for the problem size of $n=10$.
翻译:QAOA似乎只能够解决优化问题,因此,QAOA不能看到2SAT等简单问题与3SAT等棘手问题之间的差别 -- -- 虽然2SAT是多米时间(P)级的,但其优化版也是非决定性的多球时间(NP)硬(NP)。在本文中,我们显示民俗是不真实的。当解决3SAT的变种时,我们在QAOA中发现一个计算阶段的过渡 -- -- 梯度和成功概率在众所周知的SAT-UNSAT阶段过渡阶段达到最低。相反,对于2SAT来说,这种现象在SAT-UNSAT阶段的过渡中并不存在,成功概率是合理电路深度的统一。在解决NP-硬化的SAT版本时,我们发现在QAAAA中,在相当浅的P=4的质深P=10美元(美元)的典型算法上,有定量优势。