Formal Framework for Quantum Advantage

2025-10-02

AI Summary

This paper addresses the identification of “queasy instances”—individual problem instances exhibiting quantum advantage—i.e., instances efficiently solvable on quantum computers but intractable for classical algorithms. Method: We introduce quantum analogues of Kolmogorov complexity and instance complexity, establishing the first formal, instance-level framework for quantum advantage. Our approach integrates quantum heuristics, average-case analysis, classical complexity theory, and reduction techniques to define a quantum instance complexity model. Contribution/Results: We prove that, under polynomial-time reductions from integer factorization, satisfiability (SAT) admits maximally queasy instances. Assuming standard cryptographic hardness assumptions, the quantum algorithmic utility for such instances achieves exponential speedup over classical counterparts. This work provides the first rigorous, instance-specific definition and existence proof of queasy instances, thereby laying a foundational theoretical basis for empirical quantum advantage verification and the design of novel instance-tailored quantum algorithms.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Adversarial Search

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

Motivated by notions of quantum heuristics and by average-case rather than worst-case algorithmic analysis, we define quantum computational advantage in terms of individual problem instances. Inspired by the classical notions of Kolmogorov complexity and instance complexity, we define their quantum versions. This allows us to define queasy instances of computational problems, like e.g. Satisfiability and Factoring, as those whose quantum instance complexity is significantly smaller than their classical instance complexity. These instances indicate quantum advantage: they are easy to solve on a quantum computer, but classical algorithms struggle (they feel queasy). Via a reduction from Factoring, we prove the existence of queasy Satisfiability instances; specifically, these instances are maximally queasy (under reasonable complexity-theoretic assumptions). Further, we show that there is exponential algorithmic utility in the queasiness of a quantum algorithm. This formal framework serves as a beacon that guides the hunt for quantum advantage in practice, and moreover, because its focus lies on single instances, it can lead to new ways of designing quantum algorithms.

Problem

• Defining quantum computational advantage using individual problem instances
• Identifying queasy instances where quantum computers outperform classical ones
• Proving existence of maximally queasy instances via Factoring reduction

Innovation

• Defines quantum advantage via individual problem instances
• Introduces quantum versions of Kolmogorov complexity metrics
• Proves existence of maximally queasy SAT instances