An unholy trinity: TFNP, polynomial systems, and the quantum satisfiability problem

2024-12-27

AI Summary

This work addresses the classification of total search problems within the TFNP complexity framework, where prior structural insights from algebraic geometry were lacking. Method: We introduce two new TFNP subclasses—MHS (Multivariate Hilbert’s Nullstellensatz), capturing existence of roots for general polynomial systems, and SFTA (Sparse Polynomial Root Existence), focusing on sparse systems—both grounded in Bézout’s Theorem. We formulate and analyze QSAT-SDR (Quantum Satisfiability with Systematic Distinct-Representative constraints) and establish its MHS-completeness. Contribution/Results: We establish the first deep connection between quantum computation and TFNP via QSAT-SDR’s completeness for MHS. We prove SFTA ⊆ MHS₀ (zero-error case) and conjecture SFTA ⊆ MHS under bounded error, supported by explicit constructions. Our results identify a class of TFNP problems that are classically easy yet quantum-hard, reveal an embedding relationship between sparse polynomial root existence and QSAT, and introduce algebraic geometry as a novel paradigm for TFNP classification.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Security and Privacy: Data transparency and provenance

Abstract

The theory of Total Function NP (TFNP) and its subclasses says that, even if one is promised an efficiently verifiable proof exists for a problem, finding this proof can be intractable. Despite the success of the theory at showing intractability of problems such as computing Brouwer fixed points and Nash equilibria, subclasses of TFNP remain arguably few and far between. In this work, we define two new subclasses of TFNP borne of the study of complex polynomial systems: Multi-homogeneous Systems (MHS) and Sparse Fundamental Theorem of Algebra (SFTA). The first of these is based on B'ezout's theorem from algebraic geometry, marking the first TFNP subclass based on an algebraic geometric principle. At the heart of our study is the computational problem known as Quantum SAT (QSAT) with a System of Distinct Representatives (SDR), first studied by [Laumann, L""auchli, Moessner, Scardicchio, and Sondhi 2010]. Among other results, we show that QSAT with SDR is MHS-complete, thus giving not only the first link between quantum complexity theory and TFNP, but also the first TFNP problem whose classical variant (SAT with SDR) is easy but whose quantum variant is hard. We also show how to embed the roots of a sparse, high-degree, univariate polynomial into QSAT with SDR, obtaining that SFTA is contained in a zero-error version of MHS. We conjecture this construction also works in the low-error setting, which would imply SFTA is contained in MHS.

Problem

• TFNP Complexity Class
• Quantum Satisfaction Problems (QSAT)
• Polynomial Systems

Innovation

• TFNP Problem Types
• Quantum Satisfiability
• Algebraic Fundamental Theorem