Optimal local unitary encoding circuits for the surface code

Oscar Higgott1, Matthew Wilson1,2, James Hefford1,2, James Dborin1,3, Farhan Hanif1, Simon Burton1, and Dan E. Browne1

1Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom
2Department of Computer Science, University of Oxford, Oxford OX1 3QD, United Kingdom
3London Centre for Nanotechnology, University College London, Gordon St., London WC1H 0AH, United Kingdom

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The surface code is a leading candidate quantum error correcting code, owing to its high threshold, and compatibility with existing experimental architectures. Bravyi {et al.} [7] showed that encoding a state in the surface code using local unitary operations requires time at least linear in the lattice size $L$, however the most efficient known method for encoding an unknown state, introduced by Dennis {et al.} [18], has $O(L^2)$ time complexity. Here, we present an optimal local unitary encoding circuit for the planar surface code that uses exactly $2L$ time steps to encode an unknown state in a distance $L$ planar code. We further show how an $O(L)$ complexity local unitary encoder for the toric code can be found by enforcing locality in the $O(\log L)$-depth non-local renormalisation encoder. We relate these techniques by providing an $O(L)$ local unitary circuit to convert between a toric code and a planar code, and also provide optimal encoders for the rectangular, rotated and 3D surface codes. Furthermore, we show how our encoding circuit for the planar code can be used to prepare fermionic states in the compact mapping, a recently introduced fermion to qubit mapping that has a stabiliser structure similar to that of the surface code and is particularly efficient for simulating the Fermi-Hubbard model.

Kitaev’s seminal toric code and surface code have become cornerstones of both quantum computing theory and theoretical condensed matter physics. However, the most efficient known methods for preparing an unknown state in these codes using local unitary operations are far from optimal: Bravyi et al. [PRL 97, 050401 (2006)] showed that any method must take time at least linear in the lattice size L, whereas the fastest known method by Dennis et al. [Journal of Mathematical Physics 43, 4452 (2002)] takes time quadratic in L.

In our work, we construct unitary encoding circuits for the toric and surface code that are asymptotically optimal, with a number of time steps scaling linearly in L.
Our circuits encode rotated, planar and rectangular surface codes, as well as the 3D surface code. We further show how our encoding circuits can be used to encode states in the compact mapping, a fermion-to-qubit mapping that is particularly efficient for simulating fermionic lattice models.

Since encoding circuits do not require stabiliser measurements, our methods do not require ancilla qubits, and can be implemented on devices that do not yet support mid-circuit measurements. Our work therefore reduces the resources required to experimentally demonstrate topological quantum order, and has applications in the simulation of fermionic systems.

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