Error and loss tolerances of surface codes with general lattice structures

We propose a family of surface codes with general lattice structures, where the error tolerances against bit and phase errors can be controlled asymmetrically by changing the underlying lattice geometries. The surface codes on various lattices are found to be efficient in the sense that their threshold values universally approach the quantum Gilbert-Varshamov bound. We find that the error tolerance of the surface codes depends on the connectivity of the underlying lattices; the error chains on a lattice of lower connectivity are easier to correct. On the other hand, the loss tolerance of the surface codes exhibits an opposite behavior; the logical information on a lattice of higher connectivity has more robustness against qubit loss. As a result, we come upon a fundamental trade-off between error and loss tolerances in the family of surface codes with different lattice geometries. We also provide the physical aspects of the present results from the viewpoint of statistical physics, which leads to an equality that captures well both the error and loss tolerances of these surface codes.

Figure 1

(a) A surface code with a general lattice structure. (b) The surface code on the hexagonal lattice and the logical operators. (c) Incorrect error syndromes against $Z$ (top) and $X$ (bottom) errors, which are associated with the chains on the primal and dual lattices respectively. Here, the $Z$ (top) and $X$ (bottom) errors are located at the same qubits, depicted by the solid circles. (d) The remaining edges (qubits) on the primal (top) and dual (bottom) lattices after qubit loss. The logical $Z$ and $X$ operators are associated with the noncontractible loop on the primal and dual lattices, respectively.

Figure 2

The threshold values $(p_X^{\mathrm{th}},p_Z^{\mathrm{th}})$ for the square, kagome, hexagonal, and tri-hexa lattices are plotted (red/gray dots) with the quantum GVB (green/light gray line). The blue/dark gray dots are the threshold values for the surface codes defined on the duals of the kagome, hexagonal, and tri-hexa lattices (from left to right).

Figure 3

(a) The threshold curves $(p^{\mathrm{loss}},p^{\mathrm{th}})$ for logical $X$ (solid line) and $Z$ (dotted line) information [17]. Top to bottom: Tri-hexa ($X$), hexagonal ($X$), kagome ($X$), square ($X$ and $Z$), kagome ($Z$), hexagonal ($Z$), and tri-hexa ($Z$). (b) A recursive way to construct a fractal-like lattice ${\mathcal{L}}_n$ of higher connectivity. Its dual lattice ${\overline{\mathcal{L}}}_n$ has lower connectivity.