Distributed Quantum Advantage in Locally Checkable Labeling Problems

TL;DR

The paper addresses whether quantum resources can speed up solving locally checkable labeling (LCL) problems in distributed graph settings. It constructs an LCL that achieves a genuine asymptotic quantum advantage, solvable in $O(\log n)$ rounds with quantum-LOCAL while any classical randomized-LOCAL algorithm requires at least $\Omega(\log n \cdot \log^{0.99} \log n)$ rounds, by linearizing a GHZ-based problem and using tree-like and octopus gadgets with padding. Beyond the separation, it establishes a universal limit: if an LCL is solvable in $T(n)$ quantum rounds, then it is solvable in $\tilde{O}(\sqrt{nT(n)})$ rounds classically, implying that strictly global LCLs remain almost-global even under quantum-LOCAL. A corollary shows that finitely dependent distributions cannot exist for global LCLs, tightening the link between distributional sampling and distributed computation. Overall, the work resolves a major open question by presenting the first LCL with super-constant quantum speedup and clarifying the attainable scale of quantum advantage in distributed LCLs.

Abstract

In this paper, we present the first known example of a locally checkable labeling problem (LCL) that admits asymptotic distributed quantum advantage in the LOCAL model of distributed computing: our problem can be solved in $O(\log n)$ communication rounds in the quantum-LOCAL model, but it requires $Ω(\log n \cdot \log^{0.99} \log n)$ communication rounds in the classical randomized-LOCAL model. We also show that distributed quantum advantage cannot be arbitrarily large: if an LCL problem can be solved in $T(n)$ rounds in the quantum-LOCAL model, it can also be solved in $\tilde O(\sqrt{n T(n)})$ rounds in the classical randomized-LOCAL model. In particular, a problem that is strictly global classically is also almost-global in quantum-LOCAL. Our second result also holds for $T(n)$-dependent probability distributions. As a corollary, if there exists a finitely dependent distribution over valid labelings of some LCL problem $Π$, then the same problem $Π$ can also be solved in $\tilde O(\sqrt{n})$ rounds in the classical randomized-LOCAL and deterministic-LOCAL models. That is, finitely dependent distributions cannot exist for global LCL problems.

Figures (6)

• Figure 1: On the left, a tree-like gadget. On the right, a tree-like gadget labeled with labels from $\Sigma^{\mathsf {tree}}$ such that the constraints $\mathcal{C}^{\mathsf {tree}}$ are satisfied.
• Figure 2: An example of solution of $\Pi^{\mathsf {badTree}}$ in which, since one node is marked (i.e., it has input $1$), all nodes output something different from $\bot$ (i.e., they all output pointers). Informally, the problem $\Pi^{\mathsf {badTree}}$ allows outputting pointers, and in order to not allow cheating (i.e., by creating local cycles with pointers), some specific sequences of pointers are forbidden in the definition of $\Pi^{\mathsf {badTree}}$. For example, a pointer going up or down is not allowed to appear after a pointer going left or right, and pointing to the left child is always forbidden.
• Figure 3: An example of octopus gadget. The connected component induced by yellow nodes and their incident orange edges is the head gadget. Each connected component induced by gray nodes and their incident orange edges is a port gadget. Blue edges properly connect these tree-like gadgets to create a proper octopus gadget. Nodes and edges will be labeled such that this structure is locally checkable. For example, blue edges will have special labels to denote that they do not belong to the tree-like gadgets.
• Figure 4: This graph contains a yellow node that has no children (i.e., no incident edges labeled $\mathsf {Ch_L}$ and $\mathsf {Ch_R}$) and is missing a blue edge, i.e., this graph is not a valid octopus gadget. On the left it is shown the output of the nodes in the first instance of $\Pi^{\mathsf {badTree}}$, where port gadgets do not see any error and output $\bot$, but the nodes that are part of the broken head gadget prove that there is some error by using pointers. On the right it is shown the output of the nodes in the next instance of $\Pi^{\mathsf {badTree}}$, where now the roots of the port gadgets are marked, and hence all the nodes in the port gadgets can now prove that there is an error.
• Figure 5: A graph $G$ (above) and a possible proper instance $G'$ (below) that can be constructed starting from $G$. Black nodes in $G'$ correspond to inter-octopus nodes, and black edges correspond to inter-octopus edges.

Theorems & Definitions (69)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1: Labeled graph
• Definition 2.2: Centered graph
• Definition 2.3: Set of constraints
• Definition 2.4: Labeled graph satysfying a set of constraints
• Definition 2.5: Locally checkable labeling (LCL) problems
• Definition 3.1: Linearizable problem
• Definition 4.1: Outcome
• Definition 4.2: Non-signaling outcome