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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Comparative power of quantum and classical computation models

Farid Mansurovich Ablayev

Published Jun 1, 2005• 0 citations
Publisher

Abstract

In the talk we present results on comparitve power of classical and quantum computational models. We focus on two well known in Computer Science models: finite automata which is known as uniform computational model and branching programs which is known as nonuniform computational model.
DOI
10.1117/12.620484

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We present a classical stochastic simulation technique of quantum Branching programs. This technique allows to prove the following relations among complexity classes: PrQP-BP &subuline; PP-BP and BQP-BP &subuline; PP-BP. Here BPP-BP and PP-BP stands for the classes of functions computable with bounded error and unbounded error respectively by stochastic branching program of polynomial size. BQP-BP and PrQP-BP stands the classes of functions computable with bounded error and unbounded error respectively by quantum branching program of polynomial size. Second. We present two different types, of complexity lower bounds for quantum nonuniform automata (OBDDs). We call them "metric" and "entropic" lower bounds in according to proof technique used. We present explicit Boolean functions that show that these lower bounds are tight enough. We show that when considering "almost all Boolean functions" on n variables our entropic lower bounds gives exponential (2c(δ)(n-log n)) lower bound for the width of quantum OBDDs depending on the error δ allowed.
Quantum automata have been studied as simple quantum computation models. They can be considered models of small (or restricted)quantum computers. In this paper, we give descriptions of several kinds of quantum automata and show their power in comparison to their classical counterparts. We also give descriptions of quantum automata that have additional classical computational resources. Introducing classical computational resources can enhance the power of quantum automata, since this approach relaxes such restrictions as reversible state transitions.
Even though quantum computation is useful for solving certain problems, classical computation is more powerful in some cases. Thus, it is significant to compare the abilities of quantum computation and its classical counterpart, based on such a simple computation model as automata. In this paper we focus on the quantum pushdown automata which were defined by Golovkins in 2000, who showed that the class of languages recognized by quantum pushdown automata properly contains the class of languages recognized by finite automata. However, no one knows the entire relationship between the recognitive abilities of quantum and classical pushdown automata. As a part, we show a proposition that quantum pushdown automata can deterministically solve a certain problem that cannot be solved by any deterministic pushdown automata.
In automata theory, the quantum computation has been widely examined for finite state machines, known as quantum finite automata (QFAs), and less attention has been given to the QFAs augmented with counters or stacks. Moreover, to our knowledge, there is no result related to QFAs having more than one input head. In this paper, we focus on such generalizations of QFAs whose input head(s) operate(s) in one-way or realtime mode and present many superiority of them to their classical counterparts. Furthermore, we propose some open problems and conjectures in order to investigate the power of quantumness better. We also give some new results on classical computation.
Since some quantum algorithms were introduced by Shor [1] or by Grover [2], much attention has been given to the field of quantum computation.The quantum computation is so useful for solving certain problems, however, the classical computation is more powerful in some cases.Thus, it is significant to compare the ability of quantum computation with the ability of classical counterpart, based on simple computation models, like automata.In this paper, we focus on the quantum pushdown automata, QPAs, which was defined by Golovkins in 2000 [4].It is shown that the class of languages recognized by QPAs contains the class of languages recognized by finite automata.However, no one knows the relationships between the cognitive ablity of QPAs and the classical counterparts.We make a proposition in this paper, the QPAs that deterministically solve a certain problem, which cannot be solved by deterministic pushdown automata.