Optimal quantum subsystem codes in two dimensions

Given any two classical codes with parameters $[n_1,k,d_1]$ and $[n_2,k,d_2]$, we show how to construct a quantum subsystem code in two dimensions with parameters $[[N,K,D]]$ satisfying $N\le 2n_1{n}_2, K=k$, and $D=min(d_1,d_2)$. These quantum codes are in the class of generalized Bacon-Shor codes introduced by Bravyi [Phys. Rev. A 83, 012320 (2011)]. We note that constructions of good classical codes can be used to construct quantum codes that saturate Bravyi's bound $KD=O\left(N\right)$ on the code parameters of two-dimensional subsystem codes. One of these good constructions uses classical expander codes. This construction has the additional advantage of a linear time quantum decoder based on the classical Sipser-Spielman flip decoder. Finally, while the subsystem codes we create do not have asymptotic thresholds, we show how they can be gauge fixed to certain hypergraph product codes that do.

Figure 1

A $[[6,2,2]]$ Bravyi-Bacon-Shor code corresponding to $A=\left(\begin{array}{ccc}1& 1& 0\\ 1& 0& 1\\ 0& 1& 1\end{array}\right)$ [9]. In panel (a), we encircle the supports of $X$-type (red, square, solid) and $Z$-type (blue, rounded, dashed) gauge operators. In panel (b), we show the supports of $X$- and $Z$-type logical operators for the two encoded qubits. We do not show them, but there are just two stabilizers, $X^{\otimes 6}$ and $Z^{\otimes 6}$.

Figure 2

Generating sets of (a) $X$-type and (b) $Z$-type gauge operators for the augmented version of the code from Fig. 1. Type-0 qubits are shown as large, filled circles, while type-1 and 2 qubits are small and unfilled. The generating sets consist entirely of two-qubit (dark) and single-qubit (light) operators.

Figure 3

The two lattices of qubits that make up a hypergraph product code.

Figure 4

The region of possible [9, 25] two-dimensional subsystem $[[N,K,D]]$ code families. We can construct families at all these points by appealing to constructions of good classical codes and using Theorem t6.

Figure 5

Gauge fixing an aBBS code to HGP codes takes place on four lattices of qubits. Each code involved is supported on two lattices: $\text{aBBS}\left(A\right)$ is supported on $L_1$ and $L_2, \text{HGP}(H_R,H_2)$ on $L_1$ and $l_1$, and $\text{HGP}(H_1,H_R)$ on $L_2$ and $l_2$.

Figure 6

The surface code (a) with boundary and (b) on the torus drawn on the $L$ (filled qubits) and $l$ (unfilled qubits) lattices of the hypergraph product. Some example $X$- (red, solid) and $Z$-type (blue, dashed) stabilizers are shown. These example stabilizers correspond to select rows of the matrices $S_X$ and $S_Z$ of the appropriate hypergraph products.