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 $d\geq D$. 3. We prove a distance no-go theorem: every phantom code family with $k=ω(\sqrt{\log n})$ and check weight $w=O(1)$ satisfies $d\leq w$ for sufficiently large $n$. Thus our fixed-distance qLDPC families are distance-scaling optimal at the maximal logical scaling $k=Θ(\log n)$.