Strengths and Weaknesses of Quantum Computing
Charles H. Bennett, Ethan Bernstein, Gilles Brassard, Umesh Vazirani
TL;DR
The paper investigates whether NP problems can be solved efficiently by quantum computers using a relativized (oracle) framework. It introduces oracle quantum Turing machines and proves strong lower bounds: relative to a random oracle, NP cannot be solved in time $o(2^{n/2})$ by a quantum Turing machine, and relative to a random permutation oracle, $NP\cap coNP$ cannot be solved in time $o(2^{n/3})$, with Grover’s bound showing tightness in the oracle setting. It also develops methods for using bounded-error quantum subroutines safely, proving that $\mathbf{BQP}^{\mathbf{BQP}} = \mathbf{BQP}$ via tidy quantum computations and boosting techniques, thereby clarifying the black-box limits of quantum speedups for NP-type problems. Overall, the results illustrate inherent relativized barriers to exponential quantum speedups for nondeterministic problems and establish a formal framework for analyzing quantum subroutines and programming primitives in QTMs.
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 $O(2^{n/2})$.