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

TL;DR

This work addresses fault-tolerant quantum computation with high-rate quantum error-correcting codes by focusing on self-dual quantum Reed–Muller codes $\mathrm{QRM}(m)$ with parameters $[\![n=2^m,k={m \choose m/2},d=2^{m/2}]\!]$ for even $m$. The authors construct fold-transversal gates from automorphisms of classical Reed–Muller codes and, together with a transversal $\mathsf{H}^{\otimes n}$, realize a suite of addressable Clifford gates—$\overline{\mathsf{S}}$, $\overline{\mathsf{H}}$, $\overline{\mathsf{SW}}$, and $\overline{\mathsf{C_{00}Z}}$—sufficient to generate the full logical Clifford group $\overline{C}_k$ on the code. They prove a fundamental depth limitation: for any code family where $\overline{C}_k$ is achievable via transversal and fold-transversal gates, there exists a Clifford gate whose circuit depth is at least $\Omega\left(\dfrac{n}{(\log n)^2}\right)$, implying constant-depth implementations are impossible for their high-rate Reed–Muller codes. The paper also provides an open-source Python package for constructing and verifying addressable gates and discusses the implications for fault-tolerant operation, architecture designs, and potential extensions to non-Clifford gates via magic states or code switching. Overall, the results demonstrate that one can fault-tolerantly implement any addressable Clifford circuit on this high-rate code family using only transversal and fold-transversal gates, without ancilla blocks, with concrete depth-characteristics and practical tooling for verification.

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.

Figures (3)

• Figure 1: Circuit diagrams for the Hadamard gate $\mathsf{H}$, the phase gate $\mathsf{S}$, the swap gate $\mathsf{SW}$, the controlled-NOT gate $\mathsf{CX}$, and two types of controlled-$\mathsf{Z}$ gates $\mathsf{C_{00}Z}$ and $\mathsf{C_{11}Z}$.
• Figure 2: Examples of swap-type fold-transversal gates for the quantum Reed--Muller code $\mathrm{QRM}(4)$.
• Figure 3: Examples of phase-type fold-transversal gates for the quantum Reed--Muller code $\mathrm{QRM}(4)$.

Theorems & Definitions (54)

• Theorem 9: The full logical Clifford group of quantum Reed--Muller codes, informal
• Theorem 10: The circuit depth of an arbitrary logical Clifford gate, informal
• Definition 1
• Definition 2
• Example 1
• Proposition 1
• proof
• Definition 3
• Example 2
• Proposition 2