Geometric structure and transversal logic of quantum Reed-Muller codes
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908799273533440 |
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| author | Barg, Alexander Coble, Nolan J. Hangleiter, Dominik Kang, Christopher |
| author_facet | Barg, Alexander Coble, Nolan J. Hangleiter, Dominik Kang, Christopher |
| contents | 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 $π/2^k$ $Z$-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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_07595 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Geometric structure and transversal logic of quantum Reed-Muller codes Barg, Alexander Coble, Nolan J. Hangleiter, Dominik Kang, Christopher Quantum Physics Information Theory Combinatorics 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 $π/2^k$ $Z$-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. |
| title | Geometric structure and transversal logic of quantum Reed-Muller codes |
| topic | Quantum Physics Information Theory Combinatorics |
| url | https://arxiv.org/abs/2410.07595 |