Brown-Fawzi Clifford-circuit code[1]

Description

An \([[n,k]]\) stabilizer code whose encoder is a random Clifford circuit of depth of order \(O(\log^3 n)\).

An \(n\)-qubit quantum encoding circuit with \(O(n^2 \log n)\)) random two-qubit Clifford gates applied to pairs of randomly chosen qubits yields a code with distance \(d\) with probability \(1 - \Omega(1/n^8)\), granted that \begin{equation} \frac{k}{n} < 1 - \frac{d}{n}\log_2 3 - h\left(\frac{d}{n}\right)~, \tag*{(1)}\end{equation} where \(h\) is the entropy function. Noting that two gates acting on disjoint qubits could be executed simultaneously, the depth of a circuit with such a size is typically of order \(O(\log^3 n)\).'