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

TL;DR

This paper introduces a polynomial-time algorithm to determine the equivalence of hidden Markov process and quantum random walk parametrizations, improving efficiency over previous methods and enabling ergodicity testing.

Contribution

It presents the first polynomial-time algorithm for testing equivalence of QRW parametrizations and improves HMP equivalence testing from exponential to polynomial time.

Findings

Polynomial-time algorithm for HMP equivalence testing

First efficient algorithm for QRW equivalence testing

Algorithm can also test ergodicity of processes

Abstract

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 $\cM_1,\cM_2$ resp. two QRW parametrizations $\cQ_1,\cQ_2$ in time $O(|\S|\max(N_1,N_2)^{4})$, where $N_1,N_2$ are the number of hidden states in $\cM_1,\cM_2$ resp. the dimension of the state spaces associated with $\cQ_1,\cQ_2$, and $\S$ 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.