Collapses in quantum-classical probabilistically checkable proofs and the quantum polynomial hierarchy

2025-06-24

AI Summary

This work addresses structural simplifications in quantum proof systems: (1) whether uniqueness constraints weaken the computational power of quantum-classical probabilistically checkable proofs (QCPCP), and (2) the collapse mechanisms of the quantum polynomial hierarchy (QPH) and its consistency-based variant (CQPH) in the nonuniform model. Methodologically, we introduce the CQPH framework, develop BQ-reductions under quantum advice and product-state proof techniques, and employ randomized reductions with uniqueness-preserving analysis. Our three main contributions are: (i) UniqueQCPCP = QCPCP, showing uniqueness imposes no loss of expressive power; (ii) a nonuniform quantum Karp–Lipton theorem: if QMA ⊆ BQP/qpoly, then QPH ⊆ QΣ₂/qpoly; and (iii) an unconditional collapse CQPH = CQΣ₂, demonstrating that consistency—not entanglement—is the primary driver of hierarchy collapse. These results advance our understanding of quantum complexity boundaries and the role of structural constraints.

Technology Category

Machine Learning: Quantum Machine LearningKnowledge Representation and Reasoning: Computational Complexity of ReasoningConstraint Satisfaction and Optimization: Satisfiability

Application Category

Security and Privacy: Data transparency and provenanceGraph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsEconomics, Online Markets and Human Computation: LLM based quality controls for crowd work

Abstract

We investigate structural properties of quantum proof systems by establishing collapse results that uncover simplifications in their complexity landscape. We extend classical results such as the Karp-Lipton theorem to quantum polynomial hierarchy with quantum proofs and establish uniqueness preservation for quantum-classical probabilistically checkable proof systems. Our main contributions are threefold. First, we prove that restricting quantum-classical PCP systems to uniqueness does not reduce computational power: $mathsf{UniqueQCPCP} = mathsf{QCPCP}$ under $mathsf{BQ}$-operator and randomized reductions, demonstrating robustness similar to the $mathsf{UniqueQCMA} = mathsf{QCMA}$ result. Second, we establish a non-uniform quantum analogue of the Karp-Lipton theorem, showing that if $mathsf{QMA} subseteq mathsf{BQP}/mathsf{qpoly}$, then $mathsf{QPH} subseteq mathsf{QΣ}_2/mathsf{qpoly}$, extending the classical collapse theorem to quantum complexity with quantum advice. Third, we introduce a consistent variant of the quantum polynomial hierarchy ($mathsf{CQPH}$) with consistency constraints across interaction rounds while maintaining product-state proofs, proving its unconditional collapse $mathsf{CQPH} = mathsf{CQΣ}_2$. This contrasts with prior work on quantum-entangled polynomial hierarchy, showing that consistency rather than entanglement drives the collapse. These results contribute to understanding structural boundaries in quantum complexity theory and the interplay between constraint types in quantum proof systems.

Problem

• Extends Karp-Lipton theorem to quantum polynomial hierarchy
• Proves robustness of quantum-classical PCP systems under uniqueness
• Introduces consistent quantum polynomial hierarchy and its collapse

Innovation

• Extends Karp-Lipton theorem to quantum hierarchy
• Proves UniqueQCPCP equals QCPCP under reductions
• Introduces consistent quantum hierarchy CQPH collapsing