Permutation-invariant codes: a numerical study and qudit constructions

2026-03-11

AI Summary

This work investigates the relationship between block length and code distance for permutation-invariant (PI) quantum error-correcting codes against deletion errors, extending the analysis from qubits to higher-dimensional qudit systems. By generalizing the Knill–Laflamme conditions and employing numerical optimization and linear programming, the study systematically evaluates the performance of single-logical-qudit encodings and compares prominent PI code families, including Ouyang, AAB, and PR codes. Key contributions include the observation that, in the qudit setting, block length decreases monotonically with increasing physical dimension and asymptotically approaches the quantum Singleton bound; the formulation of a conjectured lower bound on block length for qubit PI codes, which is shown to be achievable by PR codes; and a semi-analytic generalization of the AAB construction to qudits, yielding explicit qudit PI code solutions.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

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.

Problem

• Permutation-Invariant codes
• quantum error correction
• qudit
• code distance
• block length

Innovation

• Permutation-Invariant codes
• quantum error correction
• qudit
• Knill-Laflamme conditions
• Singleton bound