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Electronic Notes in Theoretical Computer Science

Classically-controlled Quantum Computation

Simon Perdrix, Philippe Jorrand

Published Feb 24, 2006• 12 citations
Publisher

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 a formalized classical control. In CQTM, unitary transformations and quantum measurements are allowed. We show that any classical Turing machine is simulated by a CQTM without loss of efficiency. Furthermore, we show that any k -tape CQTM is simulated by a 2-tape CQTM with a quadratic loss of efficiency. The gap between classical and quantum computations which was already pointed out in the framework of measurement-based quantum computation (see [S. Perdrix, Ph. Jorrand, Measurement-Based Quantum Turing Machines and their Universality, arXiv, quant-ph/0404146 , 2004]) is confirmed in the general case of classically-controlled quantum computation. In order to appreciate the similarity between programming classical Turing machines and programming CQTM, some examples of CQTM will be given in the full version of the paper. Proofs of lemmas and theorems are omitted in this extended abstract.

DOI
10.1016/j.entcs.2005.09.026

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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 a formalized classical control. In CQTM, unitary transformations and quantum measurements are allowed. We show that any classical Turing machine is simulated by a CQTM without loss of efficiency. Furthermore, we show that any k -tape CQTM is simulated by a 2-tape CQTM with a quadratic loss of efficiency. The gap between classical and quantum computations which was already pointed out in the framework of measurement-based quantum computation (see [S. Perdrix, Ph. Jorrand, Measurement-Based Quantum Turing Machines and their Universality, arXiv, quant-ph/0404146 , 2004]) is confirmed in the general case of classically-controlled quantum computation. In order to appreciate the similarity between programming classical Turing machines and programming CQTM, some examples of CQTM will be given in the full version of the paper. Proofs of lemmas and theorems are omitted in this extended 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 -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.
Quantum measurement is universal for quantum computation. This universality allows alternative schemes to the traditional three-step organisation of quantum computation: initial state preparation, unitary transformation, measurement. In order to formalize these other forms of computation, while pointing out the role and the necessity of classical control in measurement-based computation, and for establishing a new upper bound of the minimal resources needed to quantum universality, a formal model is introduced by means of Measurement-based Quantum Turing Machines.
Quantum measurement is universal for quantum computation. This universality allows alternative schemes to the traditional three-step organisation of quantum computation: initial state preparation, unitary transformation, measurement. In order to formalize these other forms of computation, while pointing out the role and the necessity of classical control in measurement-based computation, and for establishing a new upper bound of the minimal resources needed to quantum universality, a formal model is introduced by means of Measurement-based Quantum Turing Machines.
We construct a universal quantum computer following Deutsch's original proposal of a universal quantum Turing machine (UQTM). Like Deutsch's UQTM, our machine can emulate any classical Turing machine and can execute any algorithm that can be implemented in the quantum gate array framework but under the control of a quantum program, and hence is universal. We present the architecture of the machine, which consists of a memory tape and a processor and describe the observables that comprise the registers of the processor and the instruction set, which includes a set of operations that can approximate any unitary operation to any desired accuracy and hence is quantum computationally universal. We present the unitary evolution operators that act on the machine to achieve universal computation and discuss each of them in detail and specify and discuss explicit program halting and concatenation schemes. We define and describe a set of primitive programs in order to demonstrate the universal nature of the machine. These primitive programs facilitate the implementation of more complex algorithms and we demonstrate their use by presenting a program that computes the NAND function, thereby also showing that the machine can compute any classically computable function.
We study the observation of quantum Turing machines by allowing interactions between a quantum machine and its environment during the computation, whereas a quantum Turing machine --original model introduced by Deutsch-- remains isolated. We show that the introduction of observations leads to a weakening of the well formedness conditions of quantum Turing machines such that any (reversible or not) classical Turing machine is a special instance of a quantum machine. Moreover, observation of quantum Turing machines provides a formal solution to the halting process problem: the impossibility to know whether a given quantum Turing machine has actually reached its halting state. It also provides a more realistic abstract architecture of a quantum computer while most of the physical proposals of quantum computers are based on an hybrid classical-quantum architecture. However, we show that a natural formalisation of an observable quantum Turing machines leads to an over-powerful model solving undecidable problems. As a consequence, we introduce a restricted version of observable quantum Turing machines and we show that, under this restriction, any observable quantum Turing machine can be efficiently simulated by a quantum Turing machine. Finally, we discuss the potential application of the observable quantum Turing machine in the quest of a universal quantum Turing machine while recent papers have pointed out that a classical control is a key feature of a universal quantum machine.
The quantum Turing machine (QTM) has been introduced by Deutsch as an abstract model of quantum computation. The transition function of a QTM is linear, and has to be unitary to be a well-formed QTM. This well-formedness condition ensures that the evolution of the machine does not violate the postulates of quantum mechanics. However, we claim in this paper that the well-formedness condition is too strong and we introduce a weaker condition, leading to a larger class of Turing machines called Observable Quantum Turing Machines (OQTMs). We prove that the evolution of such OQTM does not violate the postulates of quantum mechanics while offering a more general abstract model for quantum computing. This novel abstract model unifies classical and quantum computations, since every well-formed QTM and every deterministic TM are OQTMs, whereas a deterministic TM has to be reversible to be a well-formed QTM. In this paper we introduce the fundamentals of OQTM like a well-observed lemma and a completion lemma. The introduction of such an abstract machine allowing classical and quantum computations is motivated by the emergence of models of quantum computation like the one-way model. More generally, the OQTM aims to be an abstract framework for the pragmatic paradigm of quantum computing: quantum data, classical control'. Furthermore, this model allows a formal and rigorous treatment of problems requiring classical interactions, like the halting of QTM. Finally, it opens new perspectives for the construction of a universal QTM.