Simulation of quantum walks and fast mixing with classical processes

Abstract

We compare discrete-time quantum walks on graphs to their natural classical equivalents, which we argue are lifted Markov chains, that is, classical Markov chains with added memory. We show that these can simulate quantum walks, allowing us to answer an open question on how the graph topology ultimately bounds their mixing performance, and that of any stochastic local evolution. The results highlight that speedups in mixing and transport phenomena are not necessarily diagnostic of quantum effects, although superdiffusive spreading is more prominent with quantum walks.

Characterizing limits and opportunities in speeding up Markov chain mixing

“…The invariance requirement (i) may appear like a natural "stabilizing" requirement in an algorithmic setting, but it differs from the typical LMC setting in [15], and if added to the (s) assumption of [15] it would allow no speedup at all. Properly identifying such properties gains particular importance when lifted Markov chains are to be compared to other acceleration strategies, like the discretetime quantum walks [35] which we are addressing elsewhere [8]. On the positive side, our results clarify, with explicit lift constructions, that some properties come essentially for free.…”

“…The results are also of independent interest for the LMC literature itself, as they clarify how the mixing time bound depends precisely on the assumptions of the setting, and how some traditional assumptions relate to existing algorithmic approaches. Finally, one of the motivations for the present work is to pave the way towards a quantitative and fair comparison between lifted chains and quantum walks: although very different from an internal dynamics perspective, both these models rely on non-Markovian effects and share many similarities that can be made precise within the proposed framework [8].…”

“…It arguably imposes to "avoid unnecessary work"; it can also play an essential role towards interfacing the Markov chain with other algorithmic elements, in particular implementing the key task of amplification, i.e., boosting the success probability of a randomized algorithm (in our case the closeness to the stationary distribution) by rerunning the algorithm on its own output, see e.g. [48]; and in a similar spirit it allows to ensure that the LMC stabilizes the system towards π, even in presence of perturbations (see [8]). We thus identify two possible scenarios: (i) We impose C x(t) = π for all t > 0 whenever C x(0) = π , for all x(0) ∈ S. (I) We allow C x(t) ̸ = π even when C x(0) = π .…”

“…P(1) p, where all the P(t) satisfy the locality constraints imposed by G, see e.g. [8,16,23,52]. We call this sequence {P(t)} D G t=1 a stochastic bridge from p to p ′ .…”

“…We call this sequence {P(t)} D G t=1 a stochastic bridge from p to p ′ . The existence of such a bridge, as well as its construction, can be derived using a max-flow min-cut argument [8]. A concrete example of such construction can be found in Example 1.1 in relation with the gather-and-distribute technique: during the time steps T /2 to T , we use a bridge that maps e 0 onto the target uniform distribution.…”