Strengths and Weaknesses of Quantum Computing
Charles H. Bennett, Ethan Bernstein, Gilles Brassard, Umesh Vazirani
Published January 1, 1997
Abstract
Recently a great deal of attention has focused on quantum computation following a sequence of results suggesting that quantum computers are more powerful than classical probabilistic computers. Following Shor's result that factoring and the extraction of discrete logarithms are both solvable in quantum polynomial time, it is natural to ask whether all of NP can be efficiently solved in quantum polynomial time. In this paper, we address this question by proving that relative to an oracle chosen uniformly at random, with probability 1, the class NP cannot be solved on a quantum Turing machine in time $o(2^{n/2})$. We also show that relative to a permutation oracle chosen uniformly at random, with probability 1, the class $NP \cap coNP$ cannot be solved on a quantum Turing machine in time $o(2^{n/3})$. The former bound is tight since recent work of Grover shows how to accept the class NP relative to any oracle on a quantum computer in time $O(2^{n/2})$.
Author comment
18 pages, latex, no figures, to appear in SIAM Journal on Computing (special issue on quantum computing)
More in quant-ph
- Tianyan: Cloud services with quantum advantageTianyan Quantum Group
- Unifying communication paradigms in measurement-based delegated quantum computingFabian Wiesner, Jens Eisert, Anna Pappa
- Entangling Disciplines: Causality, Entropy and Time-Travel Paradoxes on a Quantum ComputerMaria Violaris
- Quantum Galilei group as quantum reference frame transformationsAngel Ballesteros, Diego Fernandez-Silvestre, Flaminia Giacomini +1
- Classical (ontological) dual states in quantum theory and the minimal group representation Hilbert spaceDiego J. Cirilo-Lombardo, Norma G. Sanchez