New Limits on Distributed Quantum Advantage: Dequantizing Linear Programs

TL;DR

The paper interrogates the limits of distributed quantum advantage in LOCAL, proving there is no quantum benefit for distributed fractional linear programs: any $\alpha$-approximation in the quantum-LOCAL model implies a matching $\alpha$-approximation in the classical deterministic LOCAL model, and this extends to non-signaling models. This dequantization leads to the corollary that classical LP lower bounds, such as the KMW bound, hold verbatim in quantum-LOCAL. Additionally, the authors construct an LCL where quantum-LOCAL is strictly weaker than deterministic SLOCAL, and show a separation in the opposite direction between SLOCAL and the non-signaling model, establishing incomparability in the LCL landscape. The results hinge on a lift-based construction and the maximal matching problem to transfer fractional-LP limits to LCL separations, with implications for understanding model hierarchies and the true reach of quantum advantage in distributed computation.

Abstract

In this work, we give two results that put new limits on distributed quantum advantage in the context of the LOCAL model of distributed computing. First, we show that there is no distributed quantum advantage for any linear program. Put otherwise, if there is a quantum-LOCAL algorithm $\mathcal{A}$ that finds an $α$-approximation of some linear optimization problem $Π$ in $T$ communication rounds, we can construct a classical, deterministic LOCAL algorithm $\mathcal{A}'$ that finds an $α$-approximation of $Π$ in $T$ rounds. As a corollary, all classical lower bounds for linear programs, including the KMW bound, hold verbatim in quantum-LOCAL. Second, using the above result, we show that there exists a locally checkable labeling problem (LCL) for which quantum-LOCAL is strictly weaker than the classical deterministic SLOCAL model. Our results extend from quantum-LOCAL also to finitely dependent and non-signaling distributions, and one of the corollaries of our work is that the non-signaling model and the SLOCAL model are incomparable in the context of LCL problems: By prior work, there exists an LCL problem for which SLOCAL is strictly weaker than the non-signaling model, and our work provides a separation in the opposite direction.

Figures (3)

• Figure 1: Overview of the results and relevant models of computing.
• Figure 2: (a) A tree-like gadget at the top, and an octopus gadget at the bottom. (b) A proper instance at the bottom and, at the top, the graph obtained by contracting each octopus gadget into a single node.
• Figure 3: On the left, we show a solution to the maximal matching problem, where black nodes represent hyperedges of rank 2 and blue edges are in the matching. On the right, the same solution is encoded as a solution to $\Pi^{\mathsf {linearizable}}$.

Theorems & Definitions (34)

• Definition 2.1: Labeled graph balliu2025quantum-lcl
• Definition 2.2: Set of constraints
• Definition 2.3: Satisfying a set of constraints
• Definition 2.4: Locally Checkable Labeling (LCL) problems
• Definition 2.5: Outcome
• Definition 2.6: Non-signaling outcome
• Theorem 3.1
• Lemma 3.2
• Lemma 3.3
• proof : Proof of \ref{['thm:ns-lp']}