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.
As was well known, in classical computation, Turing machines, circuits, multi-stack machines, and multi-counter machines are equivalent, that is, they can simulate each other in polynomial time. In quantum computation, Yao [11] first proved that for any quantum Turing machines
M, there exists quantum Boolean circuit
(n,t)-simulating
M, where
n denotes the length of input strings, and
t is the number of move steps before machine stopping. However, the simulations of quantum Turing machines by quantum multi-stack machines and quantum multi-counter machines have not been considered, and quantum multi-stack machines have not been established, either. Though quantum counter machines were dealt with by Kravtsev [6] and Yamasaki {\it et al.} [10], in which the machines count with
0,±1 only, we sense that it is difficult to simulate quantum Turing machines in terms of this fashion of quantum computing devices, and we therefore prove that the quantum multi-counter machines allowed to count with
0,±1,±2,...,±n for some
n>1 can efficiently simulate quantum Turing machines. Therefore, our mail goals are to establish quantum multi-stack machines and quantum multi-counter machines with counts
0,±1,±2,...,±n and
n>1, and particularly to simulate quantum Turing machines by these quantum computing devices.