Uniform Diagonalization Theorem for Complexity Classes of Promise Problems including Randomized and Quantum Classes

TL;DR

This paper generalizes the Uniform Diagonalization Theorem to promise problems, including randomized and quantum classes, enabling the construction of problems outside these classes and exploring their structural properties.

Contribution

It extends the theorem to promise problems with total decidability, allowing analysis of intermediate and undecidable problems across various complexity class pairs.

Findings

Existence of intermediate problems like between BQP and QMA

Undecidability results for class containment problems

Applicability to a wide range of complexity class pairs

Abstract

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.