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Paper 1

On Optimality of CSS Codes for Transversal $T$

Narayanan Rengaswamy, Robert Calderbank, Michael Newman, Henry D. Pfister

Year

2019

Journal

arXiv preprint

DOI

arXiv:1910.09333

arXiv

1910.09333

In order to perform universal fault-tolerant quantum computation, one needs to implement a logical non-Clifford gate. Consequently, it is important to understand codes that implement such gates transversally. In this paper, we adopt an algebraic approach to characterize all stabilizer codes for which transversal $T$ and $T^{-1}$ gates preserve the codespace. Our Heisenberg perspective reduces this to a finite geometry problem that translates to the design of certain classical codes. We prove three corollaries: (a) For any non-degenerate $[[ n,k,d ]]$ stabilizer code supporting a physical transversal $T$, there exists an $[[ n,k,d ]]$ CSS code with the same property; (b) Triorthogonal codes are the most general CSS codes that realize logical transversal $T$ via physical transversal $T$; (c) Triorthogonality is necessary for physical transversal $T$ on a CSS code to realize the logical identity. The main tool we use is a recent efficient characterization of certain diagonal gates in the Clifford hierarchy (arXiv:1902.04022). We refer to these gates as Quadratic Form Diagonal (QFD) gates. Our framework generalizes all existing code constructions that realize logical gates via transversal $T$. We provide several examples and briefly discuss connections to decreasing monomial codes, pin codes, generalized triorthogonality and quasitransversality. We partially extend these results towards characterizing all stabilizer codes that support transversal $π/2^{\ell}$ $Z$-rotations. In particular, using Ax's theorem on residue weights of polynomials, we provide an alternate characterization of logical gates induced by transversal $π/2^{\ell}$ $Z$-rotations on a family of quantum Reed-Muller codes. We also briefly discuss a general approach to analyze QFD gates that might lead to a characterization of all stabilizer codes that support any given physical transversal $1$- or $2$-local diagonal gate.

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Paper 2

ADaPT: Adaptive-window Decoding for Practical fault-Tolerance

Tina Oberoi, Joshua Viszlai, Frederic T. Chong

Year

2026

Journal

arXiv preprint

DOI

arXiv:2605.01149

arXiv

2605.01149

Window decoding, first proposed to reduce decoding complexity for real-time decoding, is an essential component to realize scalable, universal-fault tolerant computation. Prior work has focused on improving throughput through parallelization and reducing reaction time via speculation on window boundaries. However, these methods use a fixed window size d, paying a fixed decoding time overhead for each window. In practice, we find this overhead of a fixed window size unnecessary in many cases due to the sparsity of average-case errors in QEC. Leveraging this insight, in this paper we propose an adaptive window decoding technique based on decoder confidence. This technique reduces the overhead in decoding time thus reducing reaction time without compromising on logical error rates. We benchmark adaptive window decoding across different codes and hardware inspired noise models. Our results show that this adaptive technique reaches the target error rate while maintaining a low decoding time overhead across different codes, and under different noise models.

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