Quantum Fast-Forwarding Markov Chains

Simon Apers, Alain Sarlette

We introduce a new tool for quantum algorithms called quantum fast-forwarding (QFF). The tool uses quantum walks as a means to quadratically fast-forward a reversible Markov chain. More specifically, with $P$ the Markov chain transition matrix and $D = \sqrt{P\circ P^T}$ its discriminant matrix ($D=P$ if $P$ is symmetric), we construct a quantum walk algorithm that for any quantum state $|v\rangle$ and integer $t$ returns a quantum state $ε$-close to the state $D^t|v\rangle/\|D^t|v\rangle\|$. The algorithm uses $O\Big(\|D^t|v\rangle\|^{-1}\sqrt{t\log(ε\|D^t|v\rangle\|)^{-1}}\Big)$ expected quantum walk steps and $O(\|D^t|v\rangle\|^{-1})$ expected reflections around $|v\rangle$. This shows that quantum walks can accelerate the transient dynamics of Markov chains, complementing the line of results that proves the acceleration of their limit behavior. We show that this tool leads to speedups on random walk algorithms in a very natural way. Specifically we consider random walk algorithms for testing the graph expansion and clusterability, and show that we can quadratically improve the dependency of the classical property testers on the random walk runtime. Moreover, our quantum algorithm exponentially improves the space complexity of the classical tester to logarithmic. As a subroutine of independent interest, we use QFF for determining whether a given pair of nodes lies in the same cluster or in separate clusters. This solves a robust version of $s$-$t$ connectivity, relevant in a learning context for classifying objects among a set of examples. The different algorithms crucially rely on the quantum speedup of the transient behavior of random walks.

Luigi Accardi

David Sutter

Avah Banerjee, Asim Sharma, Sooraj Soman

Quantum fast-forwarding (QFF) is usually formulated for reversible Markov chains, where the projected quantum walk evolution is exactly governed by Chebyshev polynomials of a Hermitian discriminant matrix. We study whether this framework can be extended to nonreversible dynamics for an $α$-perturbed $n$-cycle Markov chain, which preserves circulant structure while introducing controlled irreversibility. We show that the nonreversible case has a fundamental obstruction: for $α\neq 0$, the eigenvalues of $P_α$ leave the interval $[-1,1]$, so $T_m(P_α)$ is not uniformly bounded and cannot arise as an exact unitary compression for all times. Thus, exact Chebyshev-based QFF does not extend directly beyond reversibility. Nevertheless, we obtain a finite-time approximation result using truncated Chebyshev and LCU techniques. The evolution $P_α^t$ can be approximated with degree $τ=O\left(|α|t+\sqrt{t\log(t/η)}\right),$ which recovers the reversible $O(\sqrt t)$ behavior only in the perturbative regime $|α|=O(t^{-1/2})$. This identifies a nearly reversible regime where QFF survives perturbatively and quantifies how irreversibility degrades the speedup.

Simon Apers, Alain Scarlet

We introduce a new tool for quantum algorithms called quantum fast-forwarding (QFF). The tool uses quantum walks as a means to quadratically fast-forward a reversible Markov chain. More specifically, with P the Markov chain transition matrix and D = \sqrt{P\circ P^T} its discriminant matrix (D=P if P is symmetric), we construct a quantum walk algorithm that for any quantum state |v> and integer t returns a quantum state \epsilon-close to the state D^t|v>/\|D^t|v>. The algorithm uses O(|D^t|v>|^{-1}\sqrt{t\log(\epsilon\|D^t|v>})^{-1}}) expected quantum walk steps and O(\|D^t|v>|^{-1}) expected reflections around |v>. This shows that quantum walks can accelerate the transient dynamics of Markov chains, complementing the line of results that proves the acceleration of their limit behavior. We show that this tool leads to speedups on random walk algorithms in a very natural way. Specifically we consider random walk algorithms for testing the graph expansion and clusterability, and show that we can quadratically improve the dependency of the classical property testers on the random walk runtime. Moreover, our quantum algorithm exponentially improves the space complexity of the classical tester to logarithmic. As a subroutine of independent interest, we use QFF for determining whether a given pair of nodes lies in the same cluster or in separate clusters. This solves a robust version of s-t connectivity, relevant in a learning context for classifying objects among a set of examples. The different algorithms crucially rely on the quantum speedup of the transient behavior of random walks.

David Sutter