AI summaryⓘ
The authors study special quantum error-correcting codes called permutation-invariant (PI) codes, which work with quantum bits (qubits) or higher-dimensional quantum units (qudits). They extend known conditions from qubits to qudits to correct certain errors where parts of the data are lost. Their main findings include a proposed formula limiting how short these codes can be and showing that increasing the dimension of the physical units helps shorten the code length toward an important theoretical limit. They also introduce a new method to create these codes for qudits using a mathematical approach called linear programming. This work helps understand how code length scales with error protection for both qubits and qudits.
Permutation-invariant codesQuantum error correctionQuditsKnill-Laflamme conditionsDeletion errorsCode distanceQuantum Singleton boundLinear programmingBlock length
Authors
Liam J. Bond, Jiří Minář, Māris Ozols, Arghavan Safavi-Naini, Vladyslav Visnevskyi
Abstract
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.