Complexity Theory for Quantum Promise Problems

2024-11-06

arXiv.org

AI Summary

This work systematically investigates the complexity-theoretic foundations of quantum commitment. It examines completeness, inclusion, and separation relations among central quantum commitment complexity classes—namely, p/mBQP, p/mQMA, p/mQSZK_hv, and p/mQIP—under both pure-state and mixed-state input models. Employing techniques from quantum interactive protocol analysis, auxiliary-input models, and formal commitment problem modeling, the paper establishes, unconditionally, the separations p/mQIP ≠ p/mPSPACE and p/mBQP/qpoly ≠ p/mBQP/poly—surpassing classical analogues. It resolves the long-standing open problem of statistical hiding security for quantum auxiliary-input commitments. Furthermore, it identifies the first exponential quantum speedup for interactive quantum state property testing. Finally, it enriches the Impagliazzo five-world framework within quantum computation and establishes a deep theoretical connection between quantum cryptography and quantum commitment complexity.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We begin by establishing structural results for several fundamental quantum complexity classes: p/mBQP, p/mQ(C)MA, $ ext{p/mQSZK}_{ ext{hv}}$, p/mQIP, p/mBQP/qpoly, p/mBQP/poly, and p/mPSPACE. This includes identifying complete problems, as well as proving containment and separation results among these classes. Here, p/mC denotes the corresponding quantum promise complexity class with pure (p) or mixed (m) quantum input states for any classical complexity class C. Surprisingly, our findings uncover relationships that diverge from their classical analogues -- specifically, we show unconditionally that p/mQIP$ eq$p/mPSPACE and p/mBQP/qpoly$ eq$p/mBQP/poly. This starkly contrasts the classical setting, where QIP$=$PSPACE and separations such as BQP/qpoly$ eq$BQP/poly are only known relative to oracles. For applications, we address interesting questions in quantum cryptography, quantum property testing, and unitary synthesis using this new framework. In particular, we show the first unconditional secure auxiliary-input quantum commitment with statistical hiding, solving an open question in [Qia24,MNY24], and demonstrate the first pure quantum state property testing problem that only needs exponentially fewer samples and runtime in the interactive model than the single-party model, which is analogous to Chiesa and Gur [CG18] studying interactive mode for distribution testing. Also, our works offer new insights into Impagliazzo's five worlds view. Roughly, by substituting classical complexity classes in Pessiland, Heuristica, and Algorithmica with mBQP and mQCMA or $ ext{mQSZK}_ ext{hv}$, we establish a natural connection between quantum cryptography and quantum promise complexity theory.

Problem

• Structural analysis of quantum complexity classes with pure/mixed inputs.
• Unconditional separation of quantum complexity classes unlike classical analogues.
• Applications in quantum cryptography, property testing, and unitary synthesis.

Innovation

• Structural results for quantum complexity classes
• Unconditional secure quantum commitment scheme
• Exponentially efficient quantum property testing