Quantum Programming in Polylogarithmic Time

2025-07-21

AI Summary

This paper addresses the formal characterization of FBQPOLYLOG—the class of quantum functions approximable in polylogarithmic time—by introducing the first quantum programming language supporting first-order recursion, thereby enabling a complete and sound language-level definition and characterization of this complexity class. Methodologically, it builds upon the quantum random-access Turing machine model and integrates recursive program semantics with circuit compilation techniques to construct a compilation framework that maps programs to uniform families of quantum circuits of depth polylog(n) and size poly(n). Key contributions include: (1) a fully formalized language-based system for FBQPOLYLOG, accompanied by rigorous proofs of completeness and type safety; and (2) a strict complexity-theoretic separation FBQPOLYLOG ⊊ QNC, establishing that FBQPOLYLOG is strictly more restrictive than QNC.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Natural Language Processing: Code Generation / Program Synthesis from Natural Language

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Data transparency and provenance
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Characterizing FBQPOLYLOG via quantum programming language
• Linking quantum programs to polylogarithmic-time computations
• Compiling programs to uniform quantum circuit families

Innovation

• Quantum language with recursive procedures
• Compilation to polylogarithmic quantum circuits
• Characterizes FBQPOLYLOG soundness and completeness