Lesson plan
Surface code
The surface code is a topological quantum error correcting code, and an example of a stabilizer code, defined on a two-dimensional spin lattice. The first type of surface code introduced by Alexei Kitaev in 1997 was the toric code, which gets its name from its periodic boundary conditions, giving it the shape of a torus. These conditions give the model translational invariance, which is useful for analytic study. The toric code is the simplest and most well studied of the quantum double models. It is also the simplest example of topological order—Z 2 topological order (first studied in the context of Z 2 spin liquid in 1991). The toric code can also be considered to be a Z 2 lattice gauge theory in a particular limit. However, on many quantum computation platforms, experimental realization of a surface code is much easier if the code can be embedded on a 2D plane. This motivated the design of another type of surface code with open boundary conditions, the planar code. As of 2025, Google Quantum AI has implemented a distance-7 planar code on their newest generation of superconducting quantum processors, Willow processor, demonstrating a below-threshold physical error rate.
Article summary
The surface code is a topological quantum error correcting code, and an example of a stabilizer code, defined on a two-dimensional spin lattice. The first type of surface code introduced by Alexei Kitaev in 1997 was the toric code, which gets its name from its periodic boundary conditions, giving it the shape of a torus. These conditions give the model translational invariance, which is useful for analytic study. The toric code is the simplest and most well studied of the quantum double models. It is also the simplest example of topological order—Z 2 topological order (first studied in the context of Z 2 spin liquid in 1991). The toric code can also be considered to be a Z 2 lattice gauge theory in a particular limit. However, on many quantum computation platforms, experimental realization of a surface code is much easier if the code can be embedded on a 2D plane. This motivated the design of another type of surface code with open boundary conditions, the planar code. As of 2025, Google Quantum AI has implemented a distance-7 planar code on their newest generation of superconducting quantum processors, Willow processor, demonstrating a below-threshold physical error rate.
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Discussion questions
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