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I understand how the Pauli X and Z logicals on the planar surface code work - they are stringlike operators between either the rough boundaries (in case of the Z logical) or between smooth boundaries (for X logical). I choose these boundaries in this order because I want the Z logical to commute with the X-type stabilizers and vice-versa.

My question is - how can one similarly identify the Pauli logicals for hyperbolic surface codes? What do these Pauli operators visually look like, given a tesselation of the hyperbolic plane? Since hyperbolic codes have constant rate, there must be multiple pairs.

clearski
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To begin, note that the hyperbolic surface codes have to be defined on periodic hyperbolic surfaces, so you should think of them analogously to toric codes, and not planar codes with rough/smooth boundaries.

If you look at section IIID of https://arxiv.org/abs/1506.04029, you'll notice something unusual about the hyperbolic codes they construct: for a given tiling, if you try to make a larger hyperbolic surface, the only way to make this tiling consistent with periodic boundary conditions and the hyperbolic metric is to increase the genus of the surface.

What this means is that for, e.g., the {5,4} tiling, if you increase the number of tiles you increase the number of non-contractable loops. The logical operators are still an independent set of non-contractable loops of X and Z operators, just like for the euclidean toric code. This is why the rate is constant--the number of non-contractable loops is proportional to the number of tiles.

Jahan Claes
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