Perspective on quantum computation from topos and sheaf theory

William David Miller

doi:10.1117/12.277647

1 July 1997 Perspective on quantum computation from topos and sheaf theory

William David Miller

Proceedings Volume 3076, Photonic Quantum Computing; (1997) https://doi.org/10.1117/12.277647
Event: AeroSense '97, 1997, Orlando, FL, United States

Abstract

The forcing process from mathematical logic offers a promising framework for studying the feasibility of quantum computation on a practical scale, since decoherence is a serious concern and, so far, questions of control, communication, and the implementation of operations that are important for a working computational system have received less attention than mathematical research on algorithms and basic physical investigations of creating simple gates and storing mixed states. Using forcing in this way is a new application of areas of model theory in which propositions and predicates take values in a lattice. Takeuti develops set theory for any universe built on a Boolean algebra generated by commutable projection operators on a Hilbert space, each such universe being a elementary topos of set- valued sheaves on the algebra, which thus is the lattice of truth values for the topos. Since it has a natural numbers object, it thus supports the general forcing method of Scedrov, but the idea of forcing is demonstrated by an example in the simpler context of partially ordered sets. Stout's lamination construction assembles toposes into objects with truth-value lattices that are orthomodular, like the lattice of all projection operators on Hilbert space, and clarifies some difficulties identified by Takeuti in the case where truth values are noncommutable operators. A substantial body of existing sheaf and topos theory thus is potentially relevant to quantum computation, and further work may provide guidance for system development.

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William David Miller "Perspective on quantum computation from topos and sheaf theory", Proc. SPIE 3076, Photonic Quantum Computing, (1 July 1997); https://doi.org/10.1117/12.277647

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KEYWORDS

Quantum computing

Statistical modeling

Computing systems

Logic

Mathematical modeling

Space operations

Mathematics

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