Sublinear-Time Quantum Computation of the Diameter in CONGEST Networks
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The computation of the diameter is one of the most central problems in distributed computation. In the standard CONGEST model, in which two adjacent nodes can exchange O( n) bits per round (here n denotes the number of nodes of the network), it is known that exact computation of the diameter requires Ω̃(n) rounds, even in networks with constant diameter. In this paper we investigate quantum distributed algorithms for this problem in the quantum CONGEST model, where two adjacent nodes can exchange O( n) quantum bits per round. Our main result is a Õ(√(nD))-round quantum distributed algorithm for exact diameter computation, where D denotes the diameter. This shows a separation (up to quadratic for networks with small diameter) between the computational power of quantum and classical algorithms in the CONGEST model. We also show an unconditional lower bound Ω̃(√(n)) on the round complexity of any quantum algorithm computing the diameter, and furthermore show a tight lower bound Ω̃(√(nD)) for any distributed quantum algorithm in which each node can use only O( n) quantum bits of memory.
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