Abstract
The surface code is one of the most successful approaches to topological quantum error-correction. It boasts the smallest known syndrome extraction circuits and correspondingly largest thresholds. Defect-based logical encodings of a new variety called twists have made it possible to implement the full Clifford group without state distillation. Here we investigate a patch-based encoding involving a modified twist. In our modified formulation, the resulting codes, called triangle codes for the shape of their planar layout, have only weight-four checks and relatively simple syndrome extraction circuits that maintain a high, near surface-code-level threshold. They also use 25% fewer physical qubits per logical qubit than the surface code. Moreover, benefiting from the twist, we can implement all Clifford gates by lattice surgery without the need for state distillation. By a surgical transformation to the surface code, we also develop a scheme of doing all Clifford gates on surface code patches in an atypical planar layout, though with less qubit efficiency than the triangle code. Finally, we remark that logical qubits encoded in triangle codes are naturally amenable to logical tomography, and the smallest triangle code can demonstrate high-pseudothreshold fault-tolerance to depolarizing noise using just 13 physical qubits.
Highlights
The surface code [1,2,3] is a dominating proposal for nearest-neighbor quantum error-correction in a plane
We show how triangle codes are related by lattice surgery to the traditional surface code, and use this to deduce a surgical method for implementing S on patches of surface code in a nonstandard planar layout
The signature element of a triangle code is a central twist defect, and so to begin, we show in Fig. 1 how the triangle codes arise from the dislocation codes [24, 25] by lattice surgery
Summary
This is a difficult claim to justify completely in any rigorous sense. There are a plethora of other topological coding strategies offering advantages and disadvantages. Computing with surface code “patches”, where each logical qubit is localized to a square grid of physical qubits, can be done with lattice surgery [16, 23] and comparatively few qubits Still, both these strategies use state distillation for the Clifford phase gate, S = diag(1, i). Four (or three) twists in a surface code lattice with uniform boundary can encode one qubit [25] This twist encoding appears to have advantages over the hole encoding, as the full Clifford group can be faulttolerantly implemented by local operations and without state distillation for the phase gate S [25].
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