Polynomial time quantum computation with advice

Advice is supplementary information that enhances the computational power of an underlying computation. This paper focuses on advice that is given in the form of a pure quantum state and examines the influence of such advice on the behaviors of an underlying polynomial-time quantum computation with bounded-error probability.Comment: 9 page

The Structure of logarithmic advice complexity classes

A nonuniform class called here Full-P/log, due to Ko, is studied. It corresponds to polynomial time with logarithmically long advice. Its importance lies in the structural properties it enjoys, more interesting than those of the alternative class P/log; specifically, its introduction was motivated by the need of a logarithmic advice class closed under polynomial-time deterministic reductions. Several characterizations of Full-P/log are shown, formulated in terms of various sorts of tally sets with very small information content. A study of its inner structure is presented, by considering the most usual reducibilities and looking for the relationships among the corresponding reduction and equivalence classes defined from these special tally sets.Postprint (published version

Characterizations of logarithmic advice complexity classes

The complexity classes P/log and Full-P/log, corresponding to the two standard forms of logarithmic advice for polynomial time, are studied. Characterizations are found in terms of Kolmogorov-simple circuits, bounded query classes, prefix-closed advice, reduction classes of regular or Kolmogorov-regular tally sets, and reduction classes of logarithmically sparse-capturable sets. The proofs are based on the Shannon-Lupanov effect and on the novel technique of “doubly exponential skip”. The techniques can be also applied to polynomial time classes with sublinear advice and to classes based on different uniform counterparts, such as NP, PSPACE, and many others.Postprint (author's final draft

Characterizations of Logarithmic Advice Complexity Classes

The complexity classes P=log and Full-P=log, corresponding to the two standard forms of logarithmic advice for polynomial time, are studied. The novel proof technique of "doubly exponential skip" is introduced, and characterizations for these classes are found in terms of several other concepts, among them easy-to-describe boolean circuits and reduction classes of tally sets with high regularity. Similar results hold for many other complexity classes. In this extended abstract most of the proofs are deferred to the appendix, where they are provided for the interested reader but are intended to be read discretionarily. 1. Introduction The study of nonuniform complexity classes stems from the comparison between uniform models of computation, in which a program is valid for arbitrarily long inputs, and nonuniform models in which each program is valid only for inputs of a fixed length. There are many well-known models for both. Typical examples of uniform models are the Turing machine and..