Quantum Markov chains: Description of hybrid systems, decidability of equivalence, and model checking linear-time properties

Abstract

In this paper, we study a model of quantum Markov chains that is a quantum analogue of Markov chains and is obtained by replacing probabilities in transition matrices with quantum operations. We show that this model is very suited to describe hybrid systems that consist of a quantum component and a classical one, although it has the same expressive power as another quantum Markov model proposed in the literature. Indeed, hybrid systems are often encountered in quantum information processing; for example, both quantum programs and quantum protocols can be regarded as hybrid systems. Thus, we further propose a model called hybrid quantum automata (HQA) that can be used to describe these hybrid systems that receive inputs (actions) from the outer world. We show the language equivalence problem of HQA is decidable in polynomial time. Furthermore, we apply this result to the trace equivalence problem of quantum Markov chains, and thus it is also decidable in polynomial time. Finally, we discuss model checking linear-time properties of quantum Markov chains, and show the quantitative analysis of regular safety properties can be addressed successfully.

An Introduction to Quantum Model Checking

“…The only difference between classical Markov chains and quantum Markov chains is the use of super-operators instead of probability values to decorate the transitions; this difference, however, has a considerable effect on the model checking algorithm, as pointed out in, e.g., [49,72]. Consider the two Markov chains, enriched with a parity acceptance condition, shown in Figure 2.…”

“…Let M = (S, Q, pri) be a PQMC on H; for each s, t ∈ S, let E s,t = { E s,t j | j ∈ J s,t } be the set of Kraus operators with J s,t as set of indexes for the super-operator E s,t = Q(s, t). Following [72], let E M = { |t s| ⊗ E s,t j | s, t ∈ S, j ∈ J s,t } be the super-operator acting on the extended Hilbert space H c ⊗ H, where H c is the |S|-dimensional Hilbert space with orthonormal basis { |s | s ∈ S }.…”

Quantum hidden Markov models based on transition operation matrices

“…(20) Thus, our model can be expressed in the language proposed by Monras et al However, formalism proposed in this paper has three notable advantages. First, it presents a hybrid quantum-classical model similar to the one presented in [14] therefore has similar field of applications. Our model intuitively generalizes both classical and quantum models.…”

Checking Continuous Stochastic Logic against Quantum Continuous-Time Markov Chains

“…Checking on quantum DTMC. A quantum discrete-time Markov chain (quantum DTMC) is a composite model on classical state space (a finite set) and quantum state space (a continuum), on which the evolution is discrete-time given by quantum operations [LF15]. Gay et al [GNP08] restricted the quantum operations to Clifford group gates (i.e., Hadamard, CNOT and the phase gates) and the whole state space as a finite set of describable states, named stabilizer states, that are closed under those Clifford group gates.…”