2020/12/01 by Mudigonda, Abhijit S., Williams, R. Ryan
#68Q15 #Computational Complexity (cs.CC) #F.0 #FOS: Computer and information sciences
paper · doi:10.48550/arxiv.2012.00330
A line of work initiated by Fortnow in 1997 has proven model-independent time-space lower bounds for the SAT problem and related problems within the polynomial-time hierarchy. For example, for the SAT problem, the state-of-the-art is that the problem cannot be solved by random-access machines in nc time and no(1) space simultaneously for c < 2cos(\fracπ7) ≈ 1.801. We extend this lower bound approach to the quantum and randomized domains. Combining Grover's algorithm with components from SAT time-space lower bounds, we show that there are problems verifiable in O(n) time with quantum Merlin-Arthur protocols that cannot be solved in nc time and no(1) space simultaneously for c < (3+√(3))/(2) ≈ 2.366, a super-quadratic time lower bound. This result and the prior work on SAT can both be viewed as consequences of a more general formula for time lower bounds against small-space algorithms, whose asymptotics we study in full. We also show lower bounds against randomized algorithms: there are problems verifiable in O(n) time with (classical) Merlin-Arthur protocols that cannot be solved in nc randomized time and no(1) space simultaneously for c < 1.465, improving a result of Diehl. For quantum Merlin-Arthur protocols, the lower bound in this setting can be improved to c < 1.5.