T-count optimization and Reed-Muller codes

In this paper, we study the close relationship between Reed-Muller codes and single-qubit phase gates from the perspective of $T$-count optimization. We prove that minimizing the number of $T$ gates in an $n$-qubit quantum circuit over CNOT and $T$, together with the Clifford group powers of $T$, corresponds to finding a minimum distance decoding of a length $2^n-1$ binary vector in the order $n-4$ punctured Reed-Muller code. Moreover, we show that the problems are polynomially equivalent in the length of the code. As a consequence, we derive an algorithm for the optimization of $T$-count in quantum circuits based on Reed-Muller decoders, along with a new upper bound of $O(n^2)$ on the number of $T$ gates required to implement an $n$-qubit unitary over CNOT and $T$ gates. We further generalize this result to show that minimizing small angle rotations corresponds to decoding lower order binary Reed-Muller codes. In particular, we show that minimizing the number of $R_Z(2\pi/d)$ gates for any integer $d$ is equivalent to minimum distance decoding in $\mathcal{RM}(n - k - 1, n)^*$, where $k$ is the highest power of $2$ dividing $d$.