Quantum error correction via multi-particle discrete-time quantum walk

Abstract

We propose a scheme of quantum error correction that employs a multi-particle quantum walk defined on nested squares, each hosting a single particle. In this model, each particle moves within its own distinct square through iterations of three discrete-time operations: (i) $\mathcal{C}$: each particle updates its two-level internal {\it coin} state, (ii) $\mathcal{S}$: it either shifts to an adjacent vertex or stays put, depending on the coin state, (iii) $\mathcal{N}$: it interacts with another particle if these particles arrive at the nearest-neighbor vertices of the two adjacent squares, acquiring a phase factor of $-1$. Because a single particle represents a three-qubit state through its position and coin state, Shor's nine-qubit code is implemented using only three particles, with two additional particles for syndrome measurement. Notably, our proposal would lead to ultrafast and resource-efficient quantum error correction by taking the continuous limit of the discrete-time iterations of $\mathcal{C}\rightarrow\mathcal{S}\rightarrow\mathcal{N}\rightarrow\mathcal{C}\rightarrow\cdots$. Note that the scheme is also resilient against a unified correctable noise model presented in the companion paper~(arXiv:2604.25747).