Optimal local unitary encoding circuits for the surface code

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(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.

Quantum circuits for toric code and X-cube fracton model

“…To simulate a toric code with length L, using local unitary gates requires at least linear size O(L) depth circuits [2], and constant depth is achievable if measurement operations are allowed [3]. A recent work provided a systematic method to simulate an unknown toric code in 3L + 2 depth [7]. In comparison, we can simulate a known toric code state like |00 in 2L + 2 depth and an unknown toric code Φ in 2L + log 2 (L) + 2 depth.…”

Implementation of Fault-Tolerant Encoding Circuit Based on Stabilizer Implementation and “Flag” Bits in Steane Code

“…In 2019, Christian Kraglund Andersen et al initialized the cardinal states of the encoded logical qubit with an average logical fidelity of 96.1%, demonstrating the practicability of implementing quantum error correction in surface codes [ 18 ]. In 2020, Oscar Higgott et al proposed a linear design for local surface code encoding [ 19 ], Fan Jihao et al proposed asymmetric quantum tandem and tensor product codes [ 20 ]. Rui Chao et al presented surface code error-correction schemes using only Pauli measurements on single qubits and pairs of nearest-neighbor qubits.…”