Notwithstanding interest and excitement building around quantum computing in the last decades, a concise statement saying where this computing can truly help is still missing. As it is shown in the present paper, equal cost of computation for truth and falsity of experimental quantum propositions (required in order to infer a conclusion from a premise) cannot be achieved with classical computing. On the other hand, this equality might be realized with quantum parallel computing provided that the efficiency of such computing can be greater than 1.
Notwithstanding interest and excitement building around quantum computing in the last decades, a concise statement saying where this computing can truly help is still missing. As it is shown in the present paper, equal cost of computation for truth and falsity of experimental quantum propositions (required in order to infer a conclusion from a premise) cannot be achieved with classical computing. On the other hand, this equality might be realized with quantum parallel computing provided that the efficiency of such computing can be greater than 1.
We briefly review what a quantum computer is, what it promises to do for us and why it is so hard to build one. Among the first applications anticipated to bear fruit is the quantum simulation of quantum systems. While most quantum computation is an extension of classical digital computation, quantum simulation differs fundamentally in how the data are encoded in the quantum computer. To perform a quantum simulation, the Hilbert space of the system to be simulated is mapped directly onto the Hilbert space of the (logical) qubits in the quantum computer. This type of direct correspondence is how data are encoded in a classical analogue computer. There is no binary encoding, and increasing precision becomes exponentially costly: an extra bit of precision doubles the size of the computer. This has important consequences for both the precision and error-correction requirements of quantum simulation, and significant open questions remain about its practicality. It also means that the quantum version of analogue computers, continuous-variable quantum computers, becomes an equally efficient architecture for quantum simulation. Lessons from past use of classical analogue computers can help us to build better quantum simulators in future.
We briefly review what a quantum computer is, what it promises to do for us and why it is so hard to build one. Among the first applications anticipated to bear fruit is the quantum simulation of quantum systems. While most quantum computation is an extension of classical digital computation, quantum simulation differs fundamentally in how the data are encoded in the quantum computer. To perform a quantum simulation, the Hilbert space of the system to be simulated is mapped directly onto the Hilbert space of the (logical) qubits in the quantum computer. This type of direct correspondence is how data are encoded in a classical analogue computer. There is no binary encoding, and increasing precision becomes exponentially costly: an extra bit of precision doubles the size of the computer. This has important consequences for both the precision and errorcorrection requirements of quantum simulation, and significant open questions remain about its practicality. It also means that the quantum version of analogue computers, continuous-variable quantum computers, becomes an equally efficient architecture for quantum simulation. Lessons from past use of classical analogue computers can help us to build better quantum simulators in future.
Of what are experimental quantum propositions primary bearers? As it is widely accepted in the modern literature, rather than being bearers of truth and falsity, these entities are bearers of probability values. Consequently, their truth values can be regarded as no more than degenerate probabilities (i.e., ones that have only the values 0 and 1). The mathematical motivation for precedence of probabilistic semantics over propositional semantic for the logic of experimental quantum propositions is Gleason's theorem. It proves that the theory of probability measures on closed linear subspaces of a Hilbert space (which represent experimental quantum propositions) does not admit any probability measure having only the values 0 and 1. -- By contrast, in the present paper, it is proclaimed that experimental propositions about quantum systems are primary bearers of truth values. As this paper demonstrates, algebraic properties of separable Hilbert spaces of finite dimension equal or greater than 2 do not allow in valuations (that is, truth assignments) which are dispersion-free, i.e., total functions from the set of atomic propositions to the set of two objects, true and false. Providing a probability function can be interpreted as a measure of the (un)certainty in the assignment of truth values, the fact that valuations cannot be dispersion-free gives rise to probabilistic semantics for the logic of experimental quantum propositions.
I give an informal discussion of the nature of quantum computation, arguing that, in terms of the amount of information manipulated in a given time, quantum and classical computation are equally efficient. Quantum superposition does not permit quantum computers to "perform many computations simultaneously" except in a highly qualified and to some extent misleading sense. Quantum computation is therefore not well described by interpretations of quantum mechanics which invoke the concept of vast numbers of parallel universes. Rather, entanglement makes available types of computation process which, while not exponentially larger than classical ones, are unavailable to classical systems.
In this paper we raise questions about the reality of computational quantum parallelism. Such questions are important because while quantum theory is rigorously established, the hypothesis that it supports a more powerful model of computation remains speculative. More specifically, we suggest the possibility that the seeming computational parallelism offered by quantum superpositions is actually effected by gate-level parallelism in the reversible implementation of the quantum operator. In other words, when the total number of logic operations is analyzed, quantum computing may not be more powerful than classical. This fact has significant public policy implications with regard to the relative levels of effort that are appropriate for the development of quantumparallel algorithms and associated hardware (i.e., qubit-based) versus quantum-scale classical hardware.
In an empirical logic, an experimentally verifiable proposition P relating to a quantum system is assigned the value of either true of false if the system is in the pure state that belongs or, respectively, does not belong to the Hilbert subspace that represents P. Determined in such a way, truth or falsity of P can be termed “a factual truth-value” of P. In this present paper, it is proposed to consider “a counter-factual truth-value” of P, i.e., either of the values, true or false, that might have been taken by P if the system had been in a pure state belonging to a Hilbert subspace that does not represent P. The assumption that it is always possible to speak meaningfully of counter-factual truth-values of experimental propositions can be called “the hypothesis of propositional counter-factual definiteness.” As shown in this paper, this hypothesis lies at the basis of the Einstein-Podolsky-Rosen (known as EPR) paradox, a striking and influential thought experiment intended to defy predictions of quantum mechanics, such as the one where measurements of spin along the different axes are incompatible. The purpose of this paper is to show that this hypothesis can be falsified by declining to paste together invariant-subspace lattices of contexts associated with the system (in other words, Boolean algebras or blocks) into one Hilbert lattice. Without such pasting, the EPR paradoxical inference cannot be reached.