Exact Quantum Algorithms for the Leader Election Problem

Many other algorithms such as Grover’s search [26,14,35] and quantum walk [18,4], and protocols [17,37,16,9] have been proposed to give separations in terms of the amount of computational resources (e.g., computational steps, communicated bits or work space) needed to compute some functions.

From the view point of computability, there are many results on languages recognizable by quantum automata [6,2,47,7,46]; they showed that there are some languages that quantum automata can recognize but their classical counterparts cannot. This gives the separation of quantum and classical models in terms of computability, instead of in terms of the amount of required computational resources, when placing a sort of restriction on computational ability (i.e., the number of internal states) of the models.

In the cryptographic field, the most remarkable quantum result would be the quantum key distribution protocols [13,12] that have been proved unconditionally secure [32,40,41,42,43]. In contrast, no unconditionally secure key distribution protocol is possible in classical settings. Many other studies demonstrate the superiority of quantum computation and communication for cryptography [23,3,20,19,10,5,11] This paper gives the first separation of quantum and classical abilities for a pure (i.e., non-cryptographic) computational task with no restriction on the amount of available computing resources; its key advance is to consider the exact solvability of a celebrated unsolvable problem in classical distributed computing, the “leader election problem” in anonymous networks.

The leader election problem is a core problem in traditional distributed computing in the sense that, once it is solved, it becomes possible to efficiently solve many substantial problems in distributed computing such as finding the maximum value and constructing a spanning tree (see, e.g., [31]). The goal of the leader election problem is to elect a unique leader from among distributed parties. When each party has a unique identifier, the problem can be deterministically solved by selecting the party that has the largest identifier as the leader; many classical deterministic algorithms in this setting have been proposed [22,36,25,24,45]. As the number of parties grows, however, it becomes difficult to preserve the uniqueness of the identifiers. Thus, other studies have examined the cases wherein each party is anonymous, i.e., each party has the same identifier [8,28,48,49], as an extreme case. In this setting, every party has to be in a common initial state and run a common algorithm; if there are two parties who are in different initial states or who run different algorithms, they can be distinguished by regarding their initial states or algorithms as their identifiers. A simple algorithm meets this condition: initially, all parties are eligible to be the unique leader, and repeats common subroutines that drop eligible parties until only one party is eligible. In the subroutines, (1) every eligible party independently generates a random bit, (2) all parties then collaborate to check if all eligible parties have the same bit, and (3) if not, the eligible parties having bit “0” are made ineligible (otherwise nothing changes). Thus, the problems can be solved probabilistically. Obviously, there is a positive probability that all parties get identical values from independent random number generators. In fact, the problem cannot be solved exactly (i.e., in bounded time and with zero error) on networks having symmetric structures such as rings, even if every party can have unbounded computational power or perform analogue computation with infinite precision. The situation is unchanged if every party is allowed to share infinitely many random strings. Strictly speaking, no classical exact algorithm (i.e., an algorithm that runs in bounded time and solves the problem with zero error) exists for a broad class of network topologies including regular graphs, even if the network topology (and thus the number of parties) is known to each party prior to algorithm invocation [48]. Moreover, no classical zero-error algorithm exists in such cases for any topology that has a cycle as its subgraph [28], if

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