Characterization of quantum computable decision problems by state discrimination

Abstract

Abstract. One advantage of quantum algorithms over classical computation is the possibility to spread out, process, analyse and extract information in multipartite configurations in coherent superpositions of classical states. This will be discussed in terms of quantum state identification problems based on a proper partitioning of mutually orthogonal sets of states. The question arises whether or not it is possible to encode equibalanced decision problems into quantum systems, so that a single invocation of a filter used for state discrimination suffices to obtain the result. OUTLINEThe question as to what might be considered the "essence" of quantum computation, and its possible advantages over classical computation, has been the topic of numerous considerations, both from a physical (e.g., Ref. [1,2,3,4,5,6,7]) as well as from a computer science (e.g., Ref. [8,9,10,11,12,13]) perspective. Contributing to this ongoing research, we will present an analysis of novel propositional structures in quantum mechanics; i.e., on the issue of what kind of propositions about quantum computers exist which do not correspond to any classical statement. We will consider coherent superpositions of states and will make explicit use of the fact that in quantum mechanics information can be coded in or "spread among" entangled multipartite systems in such a way that information about the single quanta is not useful for (and even makes impossible) a decryption of the quantum computation.Alas, it is quite evident that not all decision problems have a proper encoding into some quantum mechanical system such that their resources (computation time, memory usage) is bound by some criterion such as polynomiality or even finiteness. Take, as a concrete example, a particular type of halting problem: Alice presents Bob a black box with input and output interfaces. Bob's task is to find out whether an arbitrary function of n bits encoded in the black box will ever output "0." As this configuration could essentially get as worse as a busy beaver problem [14], the time it takes for Alice's box to ever output a "0" may grow faster than any recursive (i.e., computable [15,16]) function of n.Is it possible to characterize the exact domain of functions and propositions about them which can be "reasonably" (e.g., polynomially) coded into a quantum computation, given an fairly general set of coding strategies, such as unitary transformations? In what follows, an attempt is made to characterize the class of quantum computable functions whose computational complexity grows linearly with the number of bits by considering partitioning of states and the associated propositions and observables [17,18,19,20]. Certain quantum computations such as the Deutsch algorithm will be expressed as state identification problems, resulting in the systematic construction of a great variety of computations corresponding to (incomplete) state identifications based on superposition and interference.The notation of Mermin [21,6,22] will be adopted. Consider at first a sing...

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