Distributed Quantum Advantage for Local Problems (2411.03240v1)

Abstract: We present the first local problem that shows a super-constant separation between the classical randomized LOCAL model of distributed computing and its quantum counterpart. By prior work, such a separation was known only for an artificial graph problem with an inherently global definition [Le Gall et al. 2019]. We present a problem that we call iterated GHZ, which is defined using only local constraints. Formally, it is a family of locally checkable labeling problems [Naor and Stockmeyer 1995]; in particular, solutions can be verified with a constant-round distributed algorithm. We show that in graphs of maximum degree $\Delta$, any classical (deterministic or randomized) LOCAL model algorithm will require $\Omega(\Delta)$ rounds to solve the iterated GHZ problem, while the problem can be solved in $1$ round in quantum-LOCAL. We use the round elimination technique to prove that the iterated GHZ problem requires $\Omega(\Delta)$ rounds for classical algorithms. This is the first work that shows that round elimination is indeed able to separate the two models, and this also demonstrates that round elimination cannot be used to prove lower bounds for quantum-LOCAL. To apply round elimination, we introduce a new technique that allows us to discover appropriate problem relaxations in a mechanical way; it turns out that this new technique extends beyond the scope of the iterated GHZ problem and can be used to e.g. reproduce prior results on maximal matchings [FOCS 2019, PODC 2020] in a systematic manner.

• The paper’s main contribution demonstrates that the iterated GHZ problem exhibits a super-constant quantum advantage by enabling a one-round solution in the quantum-LOCAL model versus Ω(∆) rounds classically.
• It leverages innovative round elimination techniques to establish that classical randomized algorithms cannot replicate the efficiency of quantum approaches for this locally checkable labeling problem.
• The findings have practical implications for optimizing quantum network designs and set new theoretical benchmarks for distributed computing limits.

Distributed Quantum Advantage in Local Problems: An Examination

The paper "Distributed Quantum Advantage for Local Problems" introduces a compelling exploration into the domain of distributed quantum computing, specifically targeting an elusive question: which local problems admit a quantum advantage in distributed settings? This paper is anchored on the delineation between the classical randomized LOCAL model and its quantum counterpart, where computation proceeds in synchronous rounds and node outputs must satisfy the stipulated local constraints. The quintessential contribution of this work is the identification and analysis of the iterated GHZ problem, which is demonstrated to exhibit a super-constant separation between classical and quantum-LOCAL models—an unprecedented result for locally checkable labeling (LCL) problems.

Key Contributions

1. Iterated GHZ Problem: Defined as an LCL problem on (∆,3)-regular graphs, the iterated GHZ problem proves to be solveable in one round under the quantum-LOCAL model, as opposed to requiring Ω(∆) rounds in classical settings. This decisively showcases a distributed quantum advantage within the LOCAL model for LCL problems, answering longstanding open questions.
2. Groundbreaking Use of Round Elimination: By employing innovative techniques in round elimination—a method traditionally utilized in classical distributed computing—the paper establishes that classical algorithms cannot attain solutions in less than Ω(∆) rounds for the iterated GHZ problem. This separates quantum capabilities from classical ones in this context, as round elimination fails to derive a similar lower bound for quantum algorithms.
3. Notion of Non-Signaling and Its Implications: Networks of quantum games, conceptualized through non-signaling games such as the CHSH and GHZ games, reinforce the paper’s premise. These constructions illustrate that classical complexity for such game networks may scale with degree but remains constant with respect to the number of nodes.

Theoretical and Practical Implications

This research importantly broadens the understanding of distributed computing, demonstrating a case where quantum computation significantly reduces the time complexity of particular problems intrinsically bound by classical limitations. The findings challenge existing paradigms and suggest further inquiry into quantum-LOCAL models for other LCL problems.

Practically, these results can inform the future design and optimization of quantum networks, steering how distributed tasks might leverage quantum communication channels for efficiency gains. Meanwhile, theoretical implications include setting a new precedent for studying quantum advantages in classical computational frameworks, fostering novel algorithmic strategies that could circumvent the rigid bounds previously thought impassable.

Future Directions

The discovery of distributed quantum advantage for local problems begets numerous avenues for subsequent research. Extending these methodologies to global problems and other network topologies could unveil larger spectra of quantum advantages. Another prospective path lies in delineating finer distinctions among different classes of graph problems, potentially discovering more stark separations that can redefine computational limits.

In conclusion, this paper represents a significant leap in distributed quantum computing research, laying foundational principles for appreciating and harnessing quantum advantages in distributed systems. The iterated GHZ problem not only sets a benchmark for future work but also catalyzes interdisciplinary engagements that explore the amalgamation of classical bounds with quantum prospects. As quantum technologies mature, these insights will doubtlessly inform and enhance computational methodologies across diverse sectors.