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...

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Bibliographic Details
Published in:arXiv.org 2024-10
Main Authors: Barg, Alexander, Coble, Nolan J, Hangleiter, Dominik, Kang, Christopher
Format: Article
Language:English
Subjects:
Codes
Combinatorial analysis
Error correcting codes
Error correction
Fault tolerance
Gates
Hypercubes
Logic
Operators
Quantum computing
Qubits (quantum computing)
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Summary: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.
ISSN:2331-8422