Power of One Bit of Quantum Information

Abstract

In standard quantum computation, the initial state is pure and the answer is determined by making a measurement of some of the bits in the computational basis. What can be accomplished if the initial state is a highly mixed state and the answer is determined by measuring the expectation of σz on the first bit with bounded sensitivity? This is the situation in high temperature ensemble quantum computation. We show that in this model it is possible to perform interesting physics simulations which have no known efficient classical algorithms, even though the model is less powerful then standard quantum computing in the presence of oracles.Recent discoveries show that quantum computers can solve problems of practical interest much faster than known algorithms for classical computers [1,2]. This has lead to widespread recognition of the potential benefits of quantum computation. Where does the apparent power of quantum computers come from? This power is frequently attributed to "quantum parallelism" , interference phenomena derived from the superposition principle, and the ability to prepare and control pure states according to the Schrödinger equation. Real quantum computers are rarely in pure states and interact with their environments, which leads to non-unitary evolution. Furthermore, recent proposals and experiments using NMR at high temperature to study quantum computation involve manipulations of extremely mixed states. Recent research in error-correction and fault-tolerant computation has shown that non-unitary evolution due to weak interactions with the environment results in no loss of computational power, if sufficiently pure states can be prepared [3][4][5][6]. Here we consider the situation where there are no errors or interactions with the environment, but the initial state is highly mixed. We investigate the power of one bit of quantum information available for computing, by which we mean that the input state is equivalent to having one bit in a pure state and arbitrarily many additional bits in a completely random state. The model of computation which consists of a classical computer with access to a state of this form is called deterministic quantum computation with one quantum bit (DQC1). We demonstrate that in the presence of oracles, such a computer is less powerful than one with access to pure state bits. However, it can solve problems related to physics simulations for which no efficient classical algorithms are known. DQC1 is the first non-trivial entry in the class of models of computations which are between classical computation and standard quantum computation. Investigations of such models are expected to lead to a better understanding of the reasons for the power of quantum computation.There are many kinds of problems that one might like to solve using a computational device. The three main problems not involving communication are function evaluation, non-deterministic function evaluation and distribution sampling. Let S be the set of all binary strings and S n the set of binary strings of...

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