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 M₁, M₂ resp. two QRW parametrizations Q₁, Q₂ in time O(|Σ| max(N₁, N₂)⁴), where N₁,N₂ are the number of hidden states in M₁, M₂ resp. the dimension of the state spaces associated with Q₁, Q₂, and Σ is the set of output symbols. Previously avail able 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 subrou tines of this algorithm can also be used to efficiently test hidden Markov processes and quantum random walks for ergodicity.