Subsystem QPC[1]

Subsystem version of the QPC which has the same parameters as the subspace version, but requires significantly fewer stabilizer measurements, resulting in a much simpler error recovery routine.

This \([[n_1n_2,k_1k_2,\min(d_1,d_2)]]\) code can be defined [2] via the CSS construction applied to two binary linear codes, \(C_X\) and \(C_Z\), satisfying \(C_X^{\perp}\subset C_Z\). These codes are in turn constructed from two more binary linear codes, \(C_1 = [n_1, k_1, d_1]\) and \(C_2 = [n_2, k_2, d_2]\), with parity-check matrices \(H_{1,2}\) and generator matrices \(G_{1,2}\), respectively. The parity-check matrices of \(C_X\) and \(C_Z\) are then \begin{align} \begin{split} H_X &= H_1 \otimes I_{n_2}\\ H_Z &= G_1 \otimes H_2~. \end{split} \tag*{(1)}\end{align}

Based on the above construction, the Hilbert space on \(n_1n_2\) qubits can be decomposed into a multiple direct sums of multiple tensor products of Hilbert spaces of lower dimensions, as outlined in [3].