Native Non-Clifford Gates in Quantum LDPC Codes: Conditions, Synthesis, and Scaling Limits

Abstract

Native constant-depth non-Clifford gates on quantum low-density parity-check (qLDPC) codes can substantially reduce the space-time overhead of magic-state distillation. This paper investigates the underlying parity and structural conditions governing parallel non-Clifford gate implementations. By analyzing their invariance, algebraic forms, and circuit synthesis, we establish fundamental limits on code scaling: strict saturated implementations cannot achieve constant-depth realization as code distance grows due to tight distance-depth bounds. We evaluate alternative scaling routes, showing that scalable designs must either relax strict subspace requirements or manage gate congestion. Through analytical bounds and code searches, we demonstrate that reconciling parallel non-Clifford operations with linear distance requires navigating these fundamental structural trade-offs.

11 pages, 2 figures, 3 tables, to be presented and appear in the proceedings of IEEE QCE'26 (IEEE Quantum Week 2026)