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arXiv quant-ph/9802040quant-ph
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量子与经典通信和计算

Quantum vs. Classical Communication and Computation

Harry Buhrman, Richard Cleve, Avi Wigderson

摘要:

我们提出了一个简单而通用的模拟技术,将任何黑盒量子算法(一个la格罗弗的数据库搜索算法)的量子通信协议的相关问题,在某种程度上,充分利用量子并行性。这使我们能够获得新的积极和消极的结果。积极的结果是新的量子通信协议,通过这种模拟从非平凡的量子算法建立。这些协议,结合(旧的和新的)经典的下限,提供量子和经典(概率)两方通信复杂性模型之间的第一个渐近分离结果。特别地,我们得到了有界误差模型的二次分离,零误差模型的指数分离。否定的结果将已知的量子通信下限转换为黑箱模型中的计算下限。特别是,我们证明了Grover为OR函数实现的二次加速对于有界误差模型中的PARITY函数或MAJORITY函数是不可能的,对于OR函数本身也是不可能的。这种二分法自然意味着研究OR和MAJORITY之间的有界深度谓词(即多项式层次中的谓词)。我们提出了黑盒算法,实现近二次加速所有这样的谓词。

英文摘要:

We present a simple and general simulation technique that transforms any black-box quantum algorithm (a la Grover's database search algorithm) to a quantum communication protocol for a related problem, in a way that fully exploits the quantum parallelism. This allows us to obtain new positive and negative results. The positive results are novel quantum communication protocols that are built from nontrivial quantum algorithms via this simulation. These protocols, combined with (old and new) classical lower bounds, are shown to provide the first asymptotic separation results between the quantum and classical (probabilistic) two-party communication complexity models. In particular, we obtain a quadratic separation for the bounded-error model, and an exponential separation for the zero-error model. The negative results transform known quantum communication lower bounds to computational lower bounds in the black-box model. In particular, we show that the quadratic speed-up achieved by Grover for the OR function is impossible for the PARITY function or the MAJORITY function in the bounded-error model, nor is it possible for the OR function itself in the exact case. This dichotomy naturally suggests a study of bounded-depth predicates (i.e. those in the polynomial hierarchy) between OR and MAJORITY. We present black-box algorithms that achieve near quadratic speed up for all such predicates.

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