The basic idea of modern computing science is the view of computation as a mechanical, rather than a purely mental process. A method, or procedure for achieving some desired result is called effective or mechanical just in case 
Alan Turing and Alonzo Church both formalized the above definition by introducing the concept of computability by Turing machine and the mathematically equivalent concept of recursive functions with the following conclusions:
Despite its operationalistic approach, the above computability concept doesn't have much in common with the continuous nature of physics, so in order to build a computing machine , we have to introduce a labeling function which maps the analog physical states (e.g. the tension of a capacitor) to digital computational states (e.g. the value of a bit) The digital states have to be strings over some finite alphabet .
Since the above definition of computability requires a finite number of both, symbols and instructions, the labeling function only needs to apply on discrete intermediate machine states so the temporal evolution of the machine state is mapped onto a sequence of computational states where each transition corresponds to one function from a enumerable set of (simple) instructions.2.1The sequence is called program.
The states and are called the input- and the
output-state. The machine
thus implements the
The above definition of a computing machine poses severe restrictions on the interpretation of physical states. If we consider computation as a physical process, rather than a ``mechanical'' manipulation of symbols as defined in 2.1.1, we can drop all restrictions in the above definition which don't have a physical equivalent.
As we have showed in 126.96.36.199, the measurement of an observable with the according operator is only deterministic, if a system is in an eigenstate of . To account for the stochastic nature of quantum measurement, we have to replace the labeling function by a probabilistic operator which randomly chooses a string according to some probability distribution with .
Since it is not possible to non-destructively measure a quantum system and we are only interested in the result of a computation, anyway, it is not necessary that a labeling is defined for the intermediate steps to of a computation i.e. it is not required to ``watch'' the temporal evolution of the system, as long as a labeling for the input- and output-state and is given.
While the transitions still have to correspond to (simple) operations from a enumerable instruction set of quantum transformations, the operators , don't have to directly correspond to functions in .2.2
In 188.8.131.52 we have shown that the temporal evolution of a quantum system is mathematically described by unitary operators, so a quantum program is a composition of elementary unitary transformations.
The Turing Machine avoids this problem by extending the computational states with an integer and using this ``head position'' as an additional parameter to the generated instructions. As (which can get arbitrary large) would be ``stored'' in the physical position of the head and not in the state of the head itself, it can still be claimed that the TM operates ``by finite means''.
Even with our simpler model, we could avoid an infinitely instruction set, e.g. by interpreting a state with as the pair , and define the instructions as .
As we are discussing physical computers, which usually don't have unlimited memory, we can ignore this problem, and use the simpler and more general computer definition given above.