Efficient tests for equivalence of hidden Markov processes and quantum random walks

While two hidden Markov process (HMP) resp. quantum random walk (QRW) parametrizations can differ from one another, the stochastic processes arising from them can be equivalent. Here a polynomial-time algorithm is presented which can determine equivalence of two HMP parametrizations ${\text{cM}}_1,{\text{cM}}_2$ resp. two QRW parametrizations ${\text{cQ}}_1,{\text{cQ}}_2$ in time $O(\mathrm{\mid }\mathrm{\S }\mathrm{\mid }\max (N_1,N_2{)}^4)$, where $N_1,N_2$ are the number of hidden states in ${\text{cM}}_1,{\text{cM}}_2$ resp. the dimension of the state spaces associated with ${\text{cQ}}_1,{\text{cQ}}_2$, and $\mathrm{\S }$ is the set of output symbols. Previously available algorithms for testing equivalence of HMPs were exponential in the number of hidden states. In case of QRWs, algorithms for testing equivalence had not yet been presented. The core subroutines of this algorithm can also be used to efficiently test hidden Markov processes and quantum random walks for ergodicity.