7,036 research outputs found
Quantum Bounded Query Complexity
We combine the classical notions and techniques for bounded query classes
with those developed in quantum computing. We give strong evidence that quantum
queries to an oracle in the class NP does indeed reduce the query complexity of
decision problems. Under traditional complexity assumptions, we obtain an
exponential speedup between the quantum and the classical query complexity of
function classes.
For decision problems and function classes we obtain the following results: o
P_||^NP[2k] is included in EQP_||^NP[k] o P_||^NP[2^(k+1)-2] is included in
EQP^NP[k] o FP_||^NP[2^(k+1)-2] is included in FEQP^NP[2k] o FP_||^NP is
included in FEQP^NP[O(log n)] For sets A that are many-one complete for PSPACE
or EXP we show that FP^A is included in FEQP^A[1]. Sets A that are many-one
complete for PP have the property that FP_||^A is included in FEQP^A[1]. In
general we prove that for any set A there is a set X such that FP^A is included
in FEQP^X[1], establishing that no set is superterse in the quantum setting.Comment: 11 pages LaTeX2e, no figures, accepted for CoCo'9
Uniform Diagonalization Theorem for Complexity Classes of Promise Problems including Randomized and Quantum Classes
Diagonalization in the spirit of Cantor's diagonal arguments is a widely used
tool in theoretical computer sciences to obtain structural results about
computational problems and complexity classes by indirect proofs. The Uniform
Diagonalization Theorem allows the construction of problems outside complexity
classes while still being reducible to a specific decision problem. This paper
provides a generalization of the Uniform Diagonalization Theorem by extending
it to promise problems and the complexity classes they form, e.g. randomized
and quantum complexity classes. The theorem requires from the underlying
computing model not only the decidability of its acceptance and rejection
behaviour but also of its promise-contradicting indifferent behaviour - a
property that we will introduce as "total decidability" of promise problems.
Implications of the Uniform Diagonalization Theorem are mainly of two kinds:
1. Existence of intermediate problems (e.g. between BQP and QMA) - also known
as Ladner's Theorem - and 2. Undecidability if a problem of a complexity class
is contained in a subclass (e.g. membership of a QMA-problem in BQP). Like the
original Uniform Diagonalization Theorem the extension applies besides BQP and
QMA to a large variety of complexity class pairs, including combinations from
deterministic, randomized and quantum classes.Comment: 15 page
Topological conjugations are not constructable
We construct two computable topologically conjugate functions for which no
conjugacy is computable, or even hyperarithmetic, resolving an open question of
Kennedy and Stockman.Comment: 9 pages. v2 adds a note on folklor
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