Demonstration of Shor encoding on a trapped-ion quantum computer

TL;DR

This paper demonstrates the experimental implementation of Shor quantum error correction codes on a trapped-ion system, showing how logical fidelity varies with code size and physical error rates, and achieving high-fidelity logical states.

Contribution

First experimental realization of the $[[m^2,1,m]]$ Shor code on a trapped-ion quantum computer, including direct encoding of the $[[9,1,3]]$ code with high fidelity.

Findings

Optimal code size depends on physical error rate, with $m=5$ performing best in this system.

Achieved over 98.5% fidelity for logical states using the $[[9,1,3]]$ Shor code.

Demonstrated scalable encoding of logical qubits with up to 7 physical qubits.

Abstract

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 $m\times m$ logical states of the $[[m^2,1,m]]$ 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 $\left\vert\pm\right\rangle_L$, respectively.