Problem
Research questions and friction points this paper is trying to address.
Quantum Graph Theory
Computational Complexity
Quantum vs Classical Computing
New Approaches to Complexity via Quantum Graphs
🤖 AI Summary
This study investigates the computational complexity of the clique and independent set problems on quantum graphs. Focusing on the natural quantum model where inputs are quantum circuits implementing quantum channels, we formally define and systematically analyze the quantum clique problem, establishing its precise relationships with complexity classes QMA(2), QMA, MA, and NP. Our contributions are: (1) proving that the quantum clique problem is QMA(2)-complete—even under restricted quantum channels—making it the first natural quantum analogue of a classical problem that lies strictly in QMA(2) but not in QMA; (2) introducing a novel, self-testing–based direct proof of QMA(2)-completeness; (3) providing a generic reduction from QMA(k) to QMA(2); and (4) demonstrating that the quantum independent set problem may be strictly easier than its clique counterpart, thereby enabling a unified, comparable characterization of NP, MA, QMA, and QMA(2) within a single quantum graph framework.
📝 Abstract
Problems based on the structure of graphs -- for example finding cliques, independent sets, or colourings -- are of fundamental importance in classical complexity. Defining well-formulated decision problems for quantum graphs, which are an operator system generalisation of graphs, presents several technical challenges. Consequently, the connections between quantum graphs and complexity have been underexplored. In this work, we introduce and study the clique problem for quantum graphs. Our approach utilizes a well-known connection between quantum graphs and quantum channels. The inputs for our problems are presented as circuits inducing quantum channel, which implicitly determine a corresponding quantum graph. We show that, quantified over all channels, this problem is complete for QMA(2); in fact, it remains QMA(2)-complete when restricted to channels that are probabilistic mixtures of entanglement-breaking and partial trace channels. Quantified over a subset of entanglement-breaking channels, this problem becomes QMA-complete, and restricting further to deterministic or classical noisy channels gives rise to complete problems for NP and MA, respectively. In this way, we exhibit a classical complexity problem whose natural quantisation is QMA(2), rather than QMA, and provide the first problem that allows for a direct comparison of the classes QMA(2), QMA, MA, and NP by quantifying over increasingly larger families of instances. We use methods that are inspired by self-testing to provide a direct proof of QMA(2)-completeness, rather than reducing to a previously-studied complete problem. We also give a new proof of the celebrated reduction of QMA(k) to QMA(2). In parallel, we study a version of the closely-related independent set problem for quantum graphs, and provide preliminary evidence that it may be in general weaker in complexity, contrasting to the classical case.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Quantum Graph Theory
Complexity Classes Distinction
Information Transmission Optimization
🔎 Similar Papers
No similar papers found.
