On the Limits of Distributed Quantum Computing

TL;DR

The paper surveys the quantum-LOCAL model, a quantum extension of LOCAL where information propagates in rounds, and contrasts it with classical LOCAL across several frameworks: non-signaling, bounded-dependence, and online-LOCAL. It highlights known quantum advantages—most notably through the GHZ game and Iterated GHZ constructions that achieve $O(1)$ rounds in quantum-LOCAL while classical LOCAL requires non-constant rounds—yet also delineates robust lower-bound techniques and their limits in bounding quantum gains. The work also clarifies how online-LOCAL, SLOCAL, and rooted-tree results relate to non-signaling models, showing strong connections that allow lower bounds in online-LOCAL to inform quantum-LOCAL limits, and it identifies major open questions about which LCLs can be separated by quantum enhancements in the LOCAL setting. Overall, the article maps current techniques, presents concrete quantum advantages, and outlines open problems guiding future exploration of distributed quantum computation in distance-constrained networks, with implications for both theory and potential implementations.

Abstract

Quantum advantage is well-established in centralized computing, where quantum algorithms can solve certain problems exponentially faster than classical ones. In the distributed setting, significant progress has been made in bandwidth-limited networks, where quantum distributed networks have shown computational advantages over classical counterparts. However, the potential of quantum computing in networks that are constrained only by large distances is not yet understood. We focus on the LOCAL model of computation (Linial, FOCS 1987), a distributed computational model where computational power and communication bandwidth are unconstrained, and its quantum generalization. In this brief survey, we summarize recent progress on the quantum-LOCAL model outlining its limitations with respect to its classical counterpart: we discuss emerging techniques, and highlight open research questions that could guide future efforts in the field.

Figures (6)

• Figure 1: Landscape of computational models. An arrow between model $X$ and $Y$, that is $X \to Y$ means that model $Y$ is stronger than model $X$, unless otherwise specified. Black arrows are trivial implications (by construction), blue arrows are known results, and red arrows are recent results.
• Figure 2: The no-signaling principle illustrated in the problem of 2-coloring cycles.
• Figure 3: Visual representation of the failure probability boost: We take two odd cycles $C^{(1)}_m$ and $C^{(2)}_m$ and subdivide the nodes in two slightly overlapping regions (the colored dashed regions). For $T = \left \lceil \frac{m-2}{4} \right \rceil-2$, a $T$-round algorithm $\mathcal{A}$ must fail both in $C^{(1)}_m$ and $C^{(2)}_m$ (it does not catch that we are in odd cycles). For each cycle, in at least one region $\mathcal{A}$ must fail with probability at least 1/2. Wlog, we assume that this happens in the red and the blue regions. Now we create a new cycle $C_n$ with $n = 2m$ nodes where we copy the radius$T$ neighborhoods of the red and the blue region, and we add the remaining nodes. The failing probability of $\mathcal{A}$ over $C_n$ is, by independence, at least $3/4$, and we get a lower bound on the locality of magnitude $T = \left \lceil \frac{n-4}{8} \right \rceil - 2$.
• Figure 4: No-signaling property. Alice's lab contains the red nodes, Bob's lab contains the blue nodes. By running any $2$-rounds synchronous distributed algorithm, the red nodes cannot distinguish between $G$ and $H$: Bob has freedom to change the topology outside the red region without being detected by Alice.
• Figure 5: Bounded-dependence property. When running any $2$-round synchronous distributed algorithm (which does not rely on shared resources), the output labeling distribution of the red nodes is independent of that of the blue nodes.

Theorems & Definitions (20)

• Definition 2.1: Labeling problem
• Definition 2.2: Locally Checkable Labeling problem
• Definition 3.1: Outcome
• Definition 3.2: Non-signaling outcome
• Theorem 3.3: gavoille2009
• Theorem 3.4: coiteuxroy2023
• Definition 4.1: $T$-dependent distribution
• Definition 4.2: Bounded-dependent outcome
• Theorem 4.3: akbari2024
• Theorem 5.1: akbari2024