Abstract
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. (2006) 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. (2002), has O(L2) 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(logL)-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.
Highlights
One of the most promising error correcting codes for achieving fault-tolerant quantum computing is the surface code, owing to its high threshold and low weight check operators that are local in two dimensions [18, 29]
We present local unitary encoding circuits for both the planar and toric code that take time linear in the lattice size to encode an unknown state, achieving the Ω(L) lower bound given by Bravyi et al [7]
We have presented local unitary circuits for encoding an unknown state in the surface code that take time linear in the lattice size L
Summary
One of the most promising error correcting codes for achieving fault-tolerant quantum computing is the surface code, owing to its high threshold and low weight check operators that are local in two dimensions [18, 29]. We present local unitary encoding circuits for both the planar and toric code that take time linear in the lattice size to encode an unknown state, achieving the Ω(L) lower bound given by Bravyi et al [7]. On many Noisy IntermediateScale Quantum (NISQ) [41] devices, which are often restricted to local unitary operations, our techniques provide an optimal method for experimentally realising topological quantum order Another advantage of using a unitary encoding circuit is that it does not require the use of ancillas to measure stabilisers, providing a more qubit efficient method of preparing topologically ordered states (2× fewer qubits are required to prepare a surface code state of a given lattice size). We show how our unitary encoding circuits for the planar code can be used to construct O(L) depth circuits to encode a Slater determinant state in the compact mapping [19], which can be used for the simulation of fermionic systems on quantum computers
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