Permutation-invariant codes: a numerical study and qudit constructions
This work addresses the problem of designing efficient quantum error-correcting codes for fault-tolerant quantum computing, with incremental contributions to understanding scaling bounds and qudit constructions.
The paper investigates permutation-invariant quantum error-correcting codes for qubits and qudits, extending conditions for deletion errors and analyzing block length scaling. Key results include a conjectured lower bound on block length for qubit codes, numerical evidence that increasing physical qudit dimension reduces block length toward the quantum Singleton bound, and a semi-analytic method for constructing qudit codes.
We investigate Permutation-Invariant (PI) quantum error-correcting codes encoding a logical qudit of dimension $\mathrm{d}_\mathrm{L}$ in PI states using physical qudits of dimension $\mathrm{d}_\mathrm{P}$. We extend the Knill--Laflamme (KL) conditions for $d-1$ deletion errors from qubits to qudits and investigate numerically both qubit ($\mathrm{d}_\mathrm{L} = \mathrm{d}_\mathrm{P} = 2$) and qudit ($\mathrm{d}_\mathrm{L} > 2$ or $\mathrm{d}_\mathrm{P} > 2$) PI codes. We analyze the scaling of the block length $n$ in terms of the code distance $d$, and compare to existing families of PI codes due to Ouyang, Aydin--Alekseyev--Barg (AAB) and Pollatsek--Ruskai (PR). Our three main findings are: (i) We conjecture that qubit PI codes correcting up to $d-1$ deletion errors have block length $n(d) \geq (3d^2 + 1) / 4$, which implies an upper bound $d \leq \sqrt{12n-3}/3$ on their code distance, and that PR codes can saturate this bound. (ii) For qudit PI codes encoding a single qudit we numerically observe that increasing $\mathrm{d}_\mathrm{P}$ results in $n$ monotonically decreasing and approaching the quantum Singleton bound $n(d) \geq 2d-1$. (iii) We propose a semi-analytic extension of the qubit AAB construction to qudits that finds explicit solutions by solving a linear program. Our results therefore provide key insights into lower bounds on the block length scaling of both qubit and qudit PI codes, and demonstrate the benefit of increased physical local dimension in the context of PI codes.