2024/10/10 by Alexander Barg, Nolan J. Coble, Barg, Alexander +5 · 1 citation
Computer Science · Engineering · #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #Information Theory (cs.IT) #Quantum Physics (quant-ph) #Quantum-Dot Cellular Automata #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2410.07595
openalex publication_date 2024/10/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Designing efficient and noise-tolerant quantum computation protocols generally begins with an understanding of quantum error-correcting codes and their native logical operations. The simplest class of native operations are transversal gates, which are naturally fault-tolerant. In this paper, we aim to characterize the transversal gates of quantum Reed-Muller (RM) codes by exploiting the well-studied properties of their classical counterparts. We start our work by establishing a new geometric characterization of quantum RM codes via the Boolean hypercube and its associated subcube complex. More specifically, a set of stabilizer generators for a quantum RM code can be described via transversal X and Z operators acting on subcubes of particular dimensions. This characterization leads us to define subcube operators composed of single-qubit π/2k Z-rotations that act on subcubes of given dimensions. We first characterize the action of subcube operators on the code space: depending on the dimension of the subcube, these operators either (1) act as a logical identity on the code space, (2) implement non-trivial logic, or (3) rotate a state away from the code space. Second, and more remarkably, we uncover that the logic implemented by these operators corresponds to circuits of multi-controlled-Z gates that have an explicit and simple combinatorial description. Overall, this suite of results yields a comprehensive understanding of a class of natural transversal operators for quantum RM codes.