Lower bounds for the CNOT-complexity of linear reversible operators

Abstract

The CNOT-complexity of an invertible matrix over $\mathbb{F}_2$ is the minimum number of CNOT gates needed to synthesize the corresponding linear reversible operator. While the maximum CNOT-complexity over all $n \times n$ matrices is known to be $Θ(n^2 / \log n)$, no explicit family of matrices requiring a superlinear number of CNOT gates is known, and until now the hardest explicitly known family has been the cyclic permutations, with CNOT-complexity $3(n-1)$. We show that lower bounds for the additive complexity of not-necessarily-reversible linear operators can be lifted to the reversible setting with only a small loss. As an application, we use this to describe an explicit family of matrices, constructed from parity-check matrices of error-correcting codes, with CNOT-complexity at least $4n - o(n)$, asymptotically surpassing the cyclic permutations. Moreover, this construction yields an explicit matrix $A \in \mathrm{GL}_{n}(\mathbb{F}_2)$, $n = 17167$, whose CNOT-complexity exceeds that of the cyclic permutation on $n$ symbols.