Classically controlled quantum computation

Abstract

It is reasonable to assume that quantum computations take place under the control of the classical world. For modelling this standard situation, we introduce a Classically controlled Quantum Turing Machine (CQTM), which is a Turing machine with a quantum tape for acting on quantum data, and a classical transition function for formalised classical control. In a CQTM, unitary transformations and quantum measurements are allowed. We show that any classical Turing machine can be simulated by a CQTM without loss of efficiency. Furthermore, we show that any $k$-tape CQTM can be simulated by a 2-tape CQTM with a quadratic loss of efficiency. In order to compare CQTMs with existing models of quantum computation, we prove that any uniform family of quantum circuits (Yao 1993) is efficiently approximated by a CQTM. Moreover, we prove that any semi-uniform family of quantum circuits (Nishimura and Ozawa 2002), and any measurement calculus pattern (Danos et al. 2004) are efficiently simulated by a CQTM. Finally, we introduce a Measurement-based Quantum Turing Machine (MQTM), which is a restriction of CQTMs in which only projective measurements are allowed. We prove that any CQTM is efficiently simulated by a MQTM. In order to appreciate the similarity between programming classical Turing machines and programming CQTMs, some examples of CQTMs are given.