Quantum computing without entanglement*1

Abstract

It is generally believed that entanglement is essential for quantum computing. We present here a few simple examples in which quantum computing without entanglement is better than anything classically achievable, in terms of the reliability of the outcome after a fixed number of oracle calls. Using a separable (that is, unentangled) state, we show that the Deutsch–Jozsa problem and the Simon problem can be solved more reliably by a quantum computer than by the best possible classical algorithm, even probabilistic. We conclude that: (a)~entanglement is not essential for quantum computing; and (b)~some advantage of quantum algorithms over classical algorithms persists even when the quantum state contains an arbitrarily small amount of information—that is, even when the state is arbitrarily close to being totally mixed.