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 $[![n=2^m,k={m \choose m/2}\approx n/\sqrt{π\log_2(n)/2},d=2^{m/2}=\sqrt{n}]!]$ 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 for a family of codes in which $k$ grows near-linearly in $n$ up to a $1/\sqrt{\log n}$ factor that uses only transversal and fold-transversal gates without requiring ancilla qubits.