Quantum Programming in Polylogarithmic Time

TL;DR

This paper introduces a quantum programming language that characterizes the class of functions computable in polylogarithmic time on quantum models, establishing a formal link between programming and complexity classes.

Contribution

It provides the first programming-language-based characterization of FBQPOLYLOG and a compilation strategy to quantum circuits, connecting complexity theory with practical quantum programming.

Findings

The language characterizes FBQPOLYLOG exactly.

Programs can be compiled into quantum circuits of polylogarithmic depth.

Reveals the separation FBQPOLYLOG $ eq$ QNC.

Abstract

Polylogarithmic time delineates a relevant notion of feasibility on several classical computational models such as Boolean circuits or parallel random access machines. As far as the quantum paradigm is concerned, this notion yields the complexity class FBQPOLYLOG of functions approximable in polylogarithmic time with a quantum random-access Turing machine. We introduce a quantum programming language with first-order recursive procedures, which provides the first programming-language-based characterization of FBQPOLYLOG. Each program computes a function in FBQPOLYLOG (soundness) and, conversely, each function of this complexity class is computed by a program (completeness). We also provide a compilation strategy from programs to uniform families of quantum circuits of polylogarithmic depth and polynomial size, whose set of computed functions is known as QNC, and recover the well-known separation result FBQPOLYLOG $\subsetneq$ QNC.