Construction of the full logical Clifford group for high-rate quantum Reed-Muller codes using only transversal and fold-transversal gates

Abstract

To build large-scale quantum computers while minimizing resource requirements, one may want to use high-rate quantum error-correcting codes that can efficiently encode information. However, realizing an addressable gate$\unicode{x2014}$a logical gate on a subset of logical qubits within a high-rate code$\unicode{x2014}$in a fault-tolerant manner can be challenging and may require ancilla qubits. Transversal and fold-transversal gates could provide a means to fault-tolerantly implement logical gates using a constant-depth circuit without ancilla qubits, but available gates of these types could be limited depending on the code and might not be addressable. In this work, we study a family of $[[n=2^m,k=\left(\genfrac{}{}{0ex}{}{m}{m/2}\right)\approx n/\sqrt{\pi {\log }_2(n)/2},d=2^{m/2}=\sqrt{n}]]$ self-dual quantum Reed$\unicode{x2013}$Muller codes, where $m$ is a positive even number. For any code in this family, we construct a generating set of the full logical Clifford group comprising only transversal and fold-transversal gates, thus enabling the implementation of any addressable Clifford gate. To our knowledge, this is the first known construction of the full logical Clifford group using only transversal and fold-transversal gates without requiring ancilla qubits for a family of codes in which $k$ grows near-linearly in $n$ up to a $1/\sqrt{\log n}$ factor.

38 pages, 3 figures. V2: added discussions on relations between the studied quantum Reed-Muller codes and phantom codes. V3: minor revision