Phantom Codes: Hardness, Rate Optimal qLDPC Constructions, and Distance Limits
Abstract
An $[[n,k,d]]$ stabilizer code is phantom if every in-block logical CNOT gate can be realized by a permutation of its physical qubits. This eliminates the large and complicated physical overhead normally required for logical entangling gates. Yet this symmetry is highly restrictive: phantom codes are rare, the number of logical qubits is limited to k=O(log n), and no phantom qLDPC family with growing logical dimension was previously known. We make three contributions. 1. We prove that recognizing phantomness of a given stabilizer code is at least as hard as Graph Isomorphism, even for CSS codes encoding only k=2 logical qubits. 2. We propose the first rate optimal phantom qLDPC families: for every fixed D, our CSS families achieve the maximal logical scaling k=Θ(log n) and distance d≥ D. 3. We prove a distance no-go theorem: every phantom code family with k=ω(√(log n)) and check weight w=O(1) satisfies d≤ w for sufficiently large n. Thus our fixed-distance qLDPC families are distance-scaling optimal at the maximal logical scaling k=Θ(log n).
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BibTeX
@misc{mao2026phantom,
title = {{Phantom Codes: Hardness, Rate Optimal qLDPC Constructions, and Distance Limits}},
author = {Rui Mao and Weixiao Sun and Shengyu Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16542},
eprint = {2609.16542},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16542},
}
RIS
TY - UNPB
TI - Phantom Codes: Hardness, Rate Optimal qLDPC Constructions, and Distance Limits
AU - Rui Mao
AU - Weixiao Sun
AU - Shengyu Zhang
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16542
UR - https://arxiv.org/abs/2609.16542
AB - An $[[n,k,d]]$ stabilizer code is phantom if every in-block logical CNOT gate can be realized by a permutation of its physical qubits. This eliminates the large and complicated physical overhead normally required for logical entangling gates. Yet this symmetry is highly restrictive: phantom codes are rare, the number of logical qubits is limited to k=O(log n), and no phantom qLDPC family with growing logical dimension was previously known. We make three contributions. 1. We prove that recognizing phantomness of a given stabilizer code is at least as hard as Graph Isomorphism, even for CSS codes encoding only k=2 logical qubits. 2. We propose the first rate optimal phantom qLDPC families: for every fixed D, our CSS families achieve the maximal logical scaling k=Θ(log n) and distance d≥ D. 3. We prove a distance no-go theorem: every phantom code family with k=ω(√(log n)) and check weight w=O(1) satisfies d≤ w for sufficiently large n. Thus our fixed-distance qLDPC families are distance-scaling optimal at the maximal logical scaling k=Θ(log n).
ER -