Xinyu Tan

We study the error correcting properties of Haar random codes, in which a $K$-dimensional code space $\bC \subseteq \C^N$ is chosen at random from the Haar distribution. Our main result is that Haar random codes can approximately correct errors up to the quantum Hamming bound, meaning that a set of $m$ Pauli errors can be approximately corrected so long as $mK \ll N$. This is the strongest bound known for any family of quantum error correcting codes (QECs), and continues a line of work showing that approximate QECs can significantly outperform exact QECs [LNCY97,CGS05,BGG24]. Our proof relies on a recent matrix concentration result of Bandeira, Boedihardjo, and van Handel [BBV23].