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We present a quantum Viterbi algorithm (QVA) with better than classicalperformance under certain conditions. In this paper the proposed algorithm isapplied to decoding classical convolutional codes, for instance; largeconstraint length $Q$ and short decode frames $N$. Other applications of theclassical Viterbi algorithm where $Q$ is large (e.g. speech processing) couldexperience significant speedup with the QVA. The QVA exploits the fact that thedecoding trellis is similar to the butterfly diagram of the fast Fouriertransform, with its corresponding fast quantum algorithm. The tensor-productstructure of the butterfly diagram corresponds to a quantum superposition thatwe show can be efficiently prepared. The quantum speedup is possible becausethe performance of the QVA depends on the fanout (number of possibletransitions from any given state in the hidden Markov model) which is ingeneral much less than $Q$. The QVA constructs a superposition of states whichcorrespond to all legal paths through the decoding lattice, with phase afunction of the probability of the path being taken given received data. Aspecialized amplitude amplification procedure is applied one or more times torecover a superposition where the most probable path has a high probability ofbeing measured.

A quantum algorithm for Viterbi decoding of classical convolutional codes