Relational problems (those with many possible valid outputs) are different from decision problems, but it is easy to forget just how different. This paper initiates the study of FBQP/qpoly, the class of relational problems solvable in quantum polynomial-time with the help of polynomial-sized quantum advice, along with its analogues for deterministic and randomized computation (FP, FBPP) and advice (/poly, /rpoly). Our first result is that FBQP/qpoly != FBQP/poly, unconditionally, with no oracle -- a striking contrast with what we know about the analogous decision classes. The proof repurposes the separation between quantum and classical one-way communication complexities due to Bar-Yossef, Jayram, and Kerenidis. We discuss how this separation raises the prospect of near-term experiments to demonstrate "quantum information supremacy," a form of quantum supremacy that would not depend on unproved complexity assumptions. Our second result is that FBPP is not contained in FP/poly -- that is, Adleman's Theorem fails for relational problems -- unless PSPACE is contained in NP/poly. Our proof uses IP=PSPACE and time-bounded Kolmogorov complexity. On the other hand, we show that proving FBPP not in FP/poly will be hard, as it implies a superpolynomial circuit lower bound for PromiseBPEXP. We prove the following further results: * Unconditionally, FP != FBPP and FP/poly != FBPP/poly (even when these classes are carefully defined). * FBPP/poly = FBPP/rpoly (and likewise for FBQP). For sampling problems, by contrast, SampBPP/poly != SampBPP/rpoly (and likewise for SampBQP).
翻译:关系问题(具有许多可能的有效产出)与决策问题不同,但很容易忘记差异。本文启动FBQP/qpoly研究FBQP/poly, 无条件且无任何奥秘, 与我们所了解的类似决定类别形成鲜明对比。 证据重新定义了量与典型的单向通信问题之间的分离, 原因是巴-约瑟夫、 贾拉姆和凯雷尼蒂斯。 我们讨论这种分离如何增加近期实验的前景, 以显示“ QPPPP(FB, FBP, /ramply ) 和咨询意见(/poly, /ramply ) 。 我们的第一个结果是FBPP/qpoly, 无条件且不附带任何奥秘( FBPP/poly) 。 我们的第二个结果是, FBPP/poly 无法在SAFP/PLM 上显示时间关系, 除非我们FP/PLI 显示我们FP( ) 的FP( ) 和 SLI) 。