Strengths and Weaknesses of Quantum Computing

Abstract

Recently a great deal of attention has focused on quantum computation following a sequence of results [4,16,15] 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 ∩ co-NP cannot be solved on a quantum Turing machine in time o(2 n/3 ). The former bound is tight since recent work of Grover [13] shows how to accept the class NP relative to any oracle on a quantum computer in time O(2 n/2 ).

Quantum versus classical learnability

“…Our results draw on lower bound techniques from both quantum computation and computational learning theory [2,5,6,8,12,24]. A detailed description of the relationship between our results and previous work on quantum versus classical black-box query complexity is given in Section 3.4.…”

“…In recent years many researchers have investigated the power of quantum computers which can query a black-box oracle for an unknown function [1,5,6,9,14,10,11,15,17,20,21,23,32,37]. The broad goal of research in this area is to understand the relationship between the number of quantum versus classical oracle queries which are required to answer various questions about the function computed by the oracle.…”

“…The broad goal of research in this area is to understand the relationship between the number of quantum versus classical oracle queries which are required to answer various questions about the function computed by the oracle. For example, a well-known result due to Deutsch and Jozsa [17] shows that exponentially fewer queries are required in the quantum model in order to determine with certainty whether a black-box oracle computes a constant Boolean function or a function which is balanced between outputs and More recently, several researchers have studied the number of quantum oracle queries which are required to determine whether the function computed by a black-box oracle is identically zero [5,6,9,15,23,37].…”

“…More formally, a gate maps the basis state (where and ) to the state If is a quantum network which has gates as its oracle gates, then each is the unitary transformation which maps (where and ) to . 2 Our oracle is identical to the well-studied notion of a quantum black-box oracle for [5,6,7,9,10,11,15,17,23,37].…”

“…The difference matrix is the zero/one matrix where rows are indexed by concepts in columns are indexed by strings in and iff By our choice of and the definition of each column of has at most ones, so the matrix norm of is Our quantum lower bound proof uses ideas which were first introduced by Bennett et al [6]. Let be a fixed quantum network architecture and let be the corresponding sequence of transformations.…”