Strengths and Weaknesses of Quantum Computing

TL;DR

This paper investigates the limitations of quantum computers in solving NP problems, demonstrating that, relative to random oracles, quantum algorithms cannot efficiently solve NP in sub-exponential time, highlighting inherent computational boundaries.

Contribution

It proves that, relative to certain oracles, quantum computers cannot solve NP problems faster than specific exponential time bounds, revealing fundamental limits of quantum computational power.

Findings

Quantum computers cannot solve NP in sub-$2^{n/2}$ time relative to a random oracle.

Quantum computers cannot solve $NP igcap coNP$ in sub-$2^{n/3}$ time relative to a permutation oracle.

The bounds are tight, aligning with known quantum algorithms like Grover's search.

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 $O(2^{n/2})$.