Demonstration of Shor Encoding on a Trapped-Ion Quantum Computer

Fault-tolerant quantum error correction (QEC) is crucial for unlocking the true power of computers. QEC codes use multiple physical qubits to encode a logical qubit, which protected against errors at qubit level. Here we trapped ion system experimentally prepare $m$-qubit GHZ states and sample measurement results construct $m\times m$ $[[m^2,1,m]]$ Shor code, up $m=7$. The synthetic fidelity shows how deeper encoding can compensate additional gate in state preparation larger states. However, optimal code size depends on rate find that $m=5$ has best performance our system. We further realize direct $[[9,1,3]]$ nine thirteen-ion chain comparison, with $98.8(1)\%$ $98.5(1)\%$ $\left\vert\pm\right\rangle_L$, respectively.