Lower Bounds on Pauli Manipulation Detection Codes
Keiya Ichikawa, Kenji Yasunaga
TL;DR
This work addresses the absence of lower bounds for Pauli Manipulation Detection (PMD) codes and derives the first quantitative trade-off between error tolerance and redundancy for these codes. By exploiting that the Pauli group forms a unitary 1-design, the authors bound the average Pauli-induced leakage to the code space and obtain the central result $\varepsilon \ge \sqrt{(q^{2n-\lambda}-1)/(q^{2n}-1)}$, which implies $\lambda \ge 2 \log_q(1/\varepsilon) - \log_q 2$. This establishes a fundamental limit on PMD code performance and clarifies the redundancy required to detect nontrivial Pauli errors, with comparisons to existing constructions showing a gap in certain regimes. The findings advance PMD code theory and enable construction of quantum erasure and tamper-detection schemes with provable reliability by linking error detection strength to redundancy.
Abstract
We present a lower bound for Pauli Manipulation Detection (PMD) codes, which enables the detection of every Pauli error with high probability and can be used to construct quantum erasure and tamper-detection codes. Our lower bound reveals the first trade-off between the error and the redundancy parameters in PMD codes.