On the physical implementation of logical transformations: Generalized [formula omitted]-machines

Abstract

Any account of computation in a physical system, whether an artificial computing device or a natural system considered from a computational point of view, invokes some notion of the relationship between the abstract-logical and concrete-physical aspects of computation. In a recent paper, James Ladyman explored this relationship using a “hybrid physical–logical entity”–the L -machine–and the general account of computation that it supports [J. Ladyman, What does it mean to say that a physical system implements a computation?, Theoretical Computer Science 410 (2009) 376–383]. The underlying L -machine of Ladyman’s analysis is, however, classical and highly idealized, and cannot capture essential aspects of computation in important classes of physical systems (e.g. emerging nanocomputing devices) where logical states do not have faithful physical representations and where noise and quantum effects prevail. In this work we generalize the L -machine to allow for generally unfaithful and noisy implementations of classical logical transformations in quantum mechanical systems. We provide a formal definition and physical-information-theoretic characterization of generalized quantum L -machines (QLMs), identify important classes of QLMs, and introduce new efficacy measures that quantify the faithfulness and fidelity with which logical transformations are implemented by these machines. Two fundamental issues emphasized by Ladyman–realism about computation and the connection between logical and physical irreversibility–are reconsidered within the more comprehensive account of computation that follows from our generalization of the L -machine.