One‐way Quantum Computation

Abstract

The circuit model of quantum computation [1, 2, 3] has been a powerful tool for the development of quantum computation, acting both as a framework for theoretical investigations and as a guide for experiment. In the circuit model (also called the network model), unitary operations are represented by a network of elementary quantum gates such as the CNOT gate and single-qubit rotations. Many proposals for the implementation of quantum computation are designed around this model, including physical prescriptions for implementing the elementary gates. By formulating quantum computation in a different way, one can gain both a new framework for experiments and new theoretical insights. One-way quantum computation [4] has achieved both of these. Measurements on entangled states play a key role in many quantum information protocols, such as quantum teleportation and entanglement-based quantum key distribution. In these applications an entangled state is required, which must be generated beforehand. Then, during the protocol, measurements are made which convert the quantum correlations into, for example, a secret key. To repeat the protocol a fresh entangled state must be prepared. In this sense, the entangled state, or the quantum correlations embodied by the state, can be considered a resource which is “used up” in the protocol. In one-way quantum computation, the quantum correlations in an entangled state called a cluster state [6] or graph state [7] are exploited to allow universal quantum computation through single-qubit measurements alone. The quantum algorithm is specified in the choice of bases for these measurements and the “structure” of the entanglement (as explained below) of the resource state. The name “one-way” reflects the resource nature of the graph state. The state can be used only once, and (irreversible) projective measurements drive the computation forward, in contrast to the reversibility of every gate in the standard network model. In this chapter, we will provide an introduction to one-way quantum computation, and several of the techniques one can use to describe it. In this section we will introduce graph and cluster states and develop a notation for general single-qubit measurements. In section 2 we will introduce the key concepts of one-way quantum computation with some simple examples. After this, in section 3, we shall investigate how one-way quantum computation can