On exact counting and quasi-quantum complexity

TL;DR

This paper characterizes exact and zero-error quantum complexity classes using modified quantum models with nonunitary gates and postselection, revealing their equivalence and implications for complexity class collapses.

Contribution

It extends Aaronson's work by providing characterizations of exact quantum classes with nonunitary operations and postselection, highlighting their computational power and relationships.

Findings

Exact quantum classes can be characterized by nonunitary, quantum-like algorithms.

Postselection and nonunitarity are equivalent in power only if certain classes collapse.

The work links classical gap-definable classes with quantum computational models.

Abstract

We present characterisations of "exact" gap-definable classes, in terms of indeterministic models of computation which slightly modify the standard model of quantum computation. This follows on work of Aaronson [arXiv:quant-ph/0412187], who shows that the counting class PP can be characterised in terms of bounded-error "quantum" algorithms which use invertible (and possibly non-unitary) transformations, or postselections on events of non-zero probability. Our work considers similar modifications of the quantum computational model, but in the setting of exact algorithms, and algorithms with zero error and constant success probability. We show that the gap-definable counting classes [J. Comput. Syst. Sci. 48 (1994), p.116] which bound exact and zero-error quantum algorithms can be characterised in terms of "quantum-like" algorithms involving nonunitary gates, and that postselection and nonunitarity have equivalent power for exact quantum computation only if these classes collapse.