Quantum routing with fast reversals

TL;DR

This paper introduces quantum routing protocols that outperform swap-based methods by utilizing rapid reversal techniques, achieving faster permutation times on various architectures and for different permutation types.

Contribution

It presents the first quantum advantage over swap-based routing and provides algorithms with improved routing times for realistic quantum architectures.

Findings

Quantum routing time is at most (1-ε)n, with ε≈0.034, outperforming the n-1 lower bound for swap-based protocols.

Expected routing time approaches 2n/3 for random permutations, faster than the n time for swap-based methods.

Algorithms for sparse permutations achieve routing times of n/3 + O(k^2) on paths and 2r/3 + O(k^2) on general graphs.

Abstract

We present methods for implementing arbitrary permutations of qubits under interaction constraints. Our protocols make use of previous methods for rapidly reversing the order of qubits along a path. Given nearest-neighbor interactions on a path of length $n$, we show that there exists a constant $\epsilon \approx 0.034$ such that the quantum routing time is at most $(1-\epsilon)n$, whereas any swap-based protocol needs at least time $n-1$. This represents the first known quantum advantage over swap-based routing methods and also gives improved quantum routing times for realistic architectures such as grids. Furthermore, we show that our algorithm approaches a quantum routing time of $2n/3$ in expectation for uniformly random permutations, whereas swap-based protocols require time $n$ asymptotically. Additionally, we consider sparse permutations that route $k \le n$ qubits and give algorithms with quantum routing time at most $n/3 + O(k^2)$ on paths and at most $2r/3 + O(k^2)$ on general graphs with radius $r$.