New Approaches to Complexity via Quantum Graphs

TL;DR

The paper defines and analyzes clique and independent-set decision problems for quantum graphs derived from quantum channels, with inputs provided as circuits that implement the channels. It proves that the 2-clique problem is in $\textsf{QMA}(2)$ and is $\textsf{QMA}(2)$-complete when quantified over all channels, and shows $\textsf{QMA}$-hardness for restricted entanglement-breaking channel classes, giving a pathway to $\textsf{QMA}$-completeness for a subset. It also establishes a hierarchy connecting $\textsf{NP}$, $\textsf{MA}$, $\textsf{QMA}$, and $\textsf{QMA}(2)$ by varying channel families, and demonstrates that classical deterministic and probabilistic variants reduce to $\textsf{NP}$ and $\textsf{MA}$ respectively. The work introduces non-unitary circuit extensions and direct-sum constructions, borrows self-testing techniques to prove hardness, and leverages Brandão’s Bell-measurement results to place certain quantum-clique problems in $\textsf{QMA}$. Overall, it provides a canonical quantum analogue of classical graph problems that yields a natural, provable separation between complexity classes and clarifies the role of quantum resources in graph-theoretic decision problems.

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.

Figures (3)

• Figure 1: Quantum circuit representation of the swap test.
• Figure 2: Representation of the construction of the circuit $C_1\oplus_p C_2$ from the circuits $C_1$ and $C_2$ in canonical form, with $\texttt{out}(C_2)<\texttt{out}(C_1)$. $U_p$ is a unitary that implements the map $\lvert0\rangle\mapsto\sqrt{p}\lvert0\rangle+\sqrt{1-p}\lvert1\rangle$ to good approximation.
• Figure 3: Construction of the channel from a $\texttt{QMA}(2)$-language verification circuit.

Theorems & Definitions (78)

• Definition 1.1
• Definition 1.2
• Theorem 1.3: Quantum Case
• Theorem 1.4: Restricted Quantum Case
• Theorem 1.5: Deterministic case
• Theorem 1.6: Probabilistic case
• Definition 1.7
• Lemma 2.1
• proof
• Definition 3.1