Fermi Ma, Xinyu Tan, John Wright (Oct 09 2025).
Abstract: We study the error correcting properties of Haar random codes, in which a
K-dimensional code space
C⊆CN 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≪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.