Fault-tolerant quantum error correction (QEC) is crucial for unlocking the true power of quantum computers. QEC codes use multiple physical qubits to encode a logical qubit, which is protected against errors at the physical qubit level. Here we use a trapped ion system to experimentally prepare m-qubit GHZ states and sample the measurement results to construct m×m logical states of the [[m2,1,m]] Shor code, up to m=7. The synthetic logical fidelity shows how deeper encoding can compensate for additional gate errors in state preparation for larger logical states. However, the optimal code size depends on the physical error rate and we find that m=5 has the best performance in our system. We further realize the direct logical encoding of the [[9,1,3]] Shor code on nine qubits in a thirteen-ion chain for comparison, with 98.8(1)% and 98.5(1)% fidelity for state ∣±⟩L, respectively.
@article{arxiv.2104.01205,
title = {Demonstration of Shor encoding on a trapped-ion quantum computer},
author = {Nhung H. Nguyen and Muyuan Li and Alaina M. Green and Cinthia Huerta Alderete and Yingyue Zhu and Daiwei Zhu and Kenneth R. Brown and Norbert M. Linke},
journal= {arXiv preprint arXiv:2104.01205},
year = {2021}
}