Quantum of solitude

Ishkhanyan, Hayk, Mekhitarian, Vahram

M. Grant Norton

Seiichiro Tani

Solitude verification is arguably one of the simplest fundamental problems in distributed computing, where the goal is to verify that there is a unique contender in a network. This paper devises a quantum algorithm that exactly solves the problem on an anonymous network, which is known as a network model with minimal assumptions [Angluin, STOC’80]. The algorithm runs in O(N) rounds if every party initially has the common knowledge of an upper bound N on the number of parties. This implies that all solvable problems can be solved in O(N) rounds on average without error (i.e., with zero-sided error) on the network. As a generalization, a quantum algorithm that works in O(N log_2 (max{k, 2})) rounds is obtained for the problem of exactly computing any symmetric Boolean function, over n distributed input bits, which is constant over all the n bits whose sum is larger than k for k belongs to {0, 1, . . . , N −1}. All these algorithms work with the bit complexities bounded by a polynomial in N.

Konstantin Fedotov

We reformulate the Lonely Runner Conjecture, an open problem in Diophantine approximation, as a problem of Ψ-field coherence in an ensemble of autonomous agents. We introduce the Ψ-Hamiltonian of the runner ensemble, define a metric of Ψ-solitude via phase decoherence, and apply the Spiked Voting protocol (K=7 of M=10) to detect lonely states with robustness to noise. We further propose a GRAPE-optimization scheme for maximizing solitude probability and outline a numerical algorithm for arbitrary N. This preprint bridges SHOA’s Nonlinear Ψ-Theology with classical number theory and network scheduling.10.5281/zenodo.21810234

Ian S. Osborne, Robert Coontz