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The surface code is a leading candidate quantum error correcting code, owingto its high threshold, and compatibility with existing experimentalarchitectures. Bravyi et al. (2006) showed that encoding a state in the surfacecode using local unitary operations requires time at least linear in thelattice size $L$, however the most efficient known method for encoding anunknown state, introduced by Dennis et al. (2002), has $O(L^2)$ timecomplexity. Here, we present an optimal local unitary encoding circuit for theplanar surface code that uses exactly $2L$ time steps to encode an unknownstate in a distance $L$ planar code. We further show how an $O(L)$ complexitylocal unitary encoder for the toric code can be found by enforcing locality inthe $O(\log L)$-depth non-local renormalisation encoder. We relate thesetechniques by providing an $O(L)$ local unitary circuit to convert between atoric code and a planar code, and also provide optimal encoders for therectangular, rotated and 3D surface codes. Furthermore, we show how ourencoding circuit for the planar code can be used to prepare fermionic states inthe compact mapping, a recently introduced fermion to qubit mapping that has astabiliser structure similar to that of the surface code and is particularlyefficient for simulating the Fermi-Hubbard model.

Optimal local unitary encoding circuits for the surface code