On the Power of Quantum Distributed Proofs

TL;DR

The paper analyzes distributed quantum Merlin-Arthur protocols (dQMA), extending prior work to establish new upper bounds, robust quantum advantages, and the first lower bounds for dQMA costs. By introducing a permutation test and symmetrization, it strengthens EQ protocols on both paths and general graphs, achieving total proof sizes of ~\tilde{O}(r n^{2/3}) on paths and O$(r^2 \log n)$ per node on general graphs, while maintaining perfect completeness in key cases. It also develops efficient dQMA protocols for the GT and ranking-verification tasks, and extends Hamming-distance-type verifications to general graphs with scalable costs. Finally, the authors derive separable-proof reductions from general dQMA protocols and establish lower bounds for proof and communication sizes, including Omega$(r\log n)$ bounds for separable proofs and Omega$((\log n)^{1/4-\epsilon})$ for entangled proofs, along with reductions to QMA-communication lower bounds for several hard problems. These results collectively illuminate the power and limits of quantum proofs in distributed verification and open paths to further optimizing quantum verification on complex network topologies.

Abstract

Quantum nondeterministic distributed computing was recently introduced as dQMA (distributed quantum Merlin-Arthur) protocols by Fraigniaud, Le Gall, Nishimura and Paz (ITCS 2021). In dQMA protocols, with the help of quantum proofs and local communication, nodes on a network verify a global property of the network. Fraigniaud et al. showed that, when the network size is small, there exists an exponential separation in proof size between distributed classical and quantum verification protocols, for the equality problem, where the verifiers check if all the data owned by a subset of them are identical. In this paper, we further investigate and characterize the power of the dQMA protocols for various decision problems. First, we give a more efficient dQMA protocol for the equality problem with a simpler analysis. This is done by adding a symmetrization step on each node and exploiting properties of the permutation test, which is a generalization of the SWAP test. We also show a quantum advantage for the equality problem on path networks still persists even when the network size is large, by considering ``relay points'' between extreme nodes. Second, we show that even in a general network, there exist efficient dQMA protocols for the ranking verification problem, the Hamming distance problem, and more problems that derive from efficient quantum one-way communication protocols. Third, in a line network, we construct an efficient dQMA protocol for a problem that has an efficient two-party QMA communication protocol. Finally, we obtain the first lower bounds on the proof and communication cost of dQMA protocols. To prove a lower bound on the equality problem, we show any dQMA protocol with an entangled proof between nodes can be simulated with a dQMA protocol with a separable proof between nodes by using a QMA communication-complete problem introduced by Raz and Shpilka (CCC 2004).

Theorems & Definitions (107)

• Theorem 1: Theorem \ref{['theorem:eq_tree']}
• Theorem 2: Informal version of Theorem \ref{['theorem:robust_QA']} and Corollary \ref{['corollary:classical_lowerbound_for_EQ']}
• Definition 1: Ranking verification problem, informal version of Definition \ref{['definition:ranking']}
• Theorem 3: Informal version of Theorem \ref{['theorem:ranking_verification']}
• Theorem 4: Theorem \ref{['theorem:greater-than']}
• Theorem 5: Theorem \ref{['proposition:multi-input']}
• Theorem 6: Informal version of Proposition \ref{['proposition:dQMA_for_QMAcc']}
• Theorem 7: Informal version of Theorem \ref{['theorem:dQMAsep_for_dQMA']}
• Theorem 8: Informal version of Theorem \ref{['theorem:quantum_lower_bound_separable']}
• Theorem 9: Informal version of Theorem \ref{['theorem:quantum_lower_bound_entangled2']}