Geometric structure and transversal logic of quantum Reed-Muller codes
<|reference_start|>Geometric structure and transversal logic of quantum Reed-Muller codes: 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 $\pi/2^k$ $Z$-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.<|reference_end|>
@article{barg2024geometric,
    title={Geometric structure and transversal logic of quantum Reed-Muller codes},
    author={Alexander Barg, Nolan J. Coble, Dominik Hangleiter, Christopher Kang},
    journal={arXiv preprint arXiv:2410.07595},
    year={2024},
    archivePrefix={arXiv},
    eprint={2410.07595},
    primaryClass={quant-ph cs.IT math.CO math.IT}
}