Computing on Anonymous Quantum Network

This paper considers distributed computing on an anonymous quantum network, a network in which no party has a unique identifier and quantum communication and computation are available. It is proved that the leader election problem can exactly (i.e., without error in bounded time) be solved with at most the same complexity up to a constant factor as that of exactly computing symmetric functions (without intermediate measurements for a distributed and superposed input), if the number of parties is given to every party. A corollary of this result is a more efficient quantum leader election algorithm than existing ones: the new quantum algorithm runs in O(n) rounds with bit complexity O(mn^2), on an anonymous quantum network with n parties and m communication links. Another corollary is the first quantum algorithm that exactly computes any computable Boolean function with round complexity O(n) and with smaller bit complexity than that of existing classical algorithms in the worst case over all (computable) Boolean functions and network topologies. More generally, any n-qubit state can be shared with that complexity on an anonymous quantum network with n parties.Comment: 25 page

Distributed Computing on Anonymous Hypercube Networks

We consider the bit-complexity (i.e.a, total number of bits transmitted) of computing boolean functions on an anonymous canonically labeled n-dimensional hypercube network and give a characterization of the boolean functions computable on such a network as exactly those boolean functions which are invariant under all bit-complement automorphisms of the hyercube. We provide an efficient algorithm for computing all such functions with bit complexity O(N · log4 N). For the case of symmetric boolean functions we give an algorithm with bit complexity O(N · log2 N)