How Error-Correcting Codes Raise Quantum Gate Complexity

Søren Fuglede Jørgensen of Kvantify has described a matrix with a CNOT-complexity exceeding that of cyclic permutations on symbols, a benchmark previously held by the most difficult known family of reversible operators. The research demonstrates a new family of matrices, constructed from parity-check matrices of error-correcting codes, with a CNOT-complexity that asymptotically surpasses the cyclic permutations’ complexity. This advance stems from extending lower bounds from linear operators that are not necessarily reversible to the reversible setting, enabling new approaches to analyzing CNOT-complexity. The work establishes a new lower bound, potentially impacting…