Demonstration of Shor Encoding on a Trapped-Ion Quantum Computer

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 Greenberger-Horne-Zeilinger 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 13-ion chain for comparison, with $98.8(1)\mathrm{\%}$ and $98.5(1)\mathrm{\%}$ fidelity for state ${|\pm ⟩}_L$, respectively.

Figure 1

Circuits for fault-tolerant preparation of logical states (a) ${|+⟩}_L$ and (b) ${|-⟩}_L$ of the $[[m^2,1,m]]$ Shor code for $m=3$. The circuit separates into $m$ groups, each preparing a $|{\mathrm{GHZ}}_m^{\pm }⟩$ state.

Figure 2

The circuit to prepare a $|{\mathrm{GHZ}}_m^{\pm }⟩$ on a trapped-ion quantum computer, with $\varphi =0$ for $|{\mathrm{GHZ}}_m^{+}⟩$ and $\varphi =\pi$ for $|{\mathrm{GHZ}}_m^{-}⟩$.

Figure 3

The measurement of $|{\mathrm{GHZ}}_3^{\pm }⟩$ in the (a) $Z$ basis and (b) $X$ basis. In the $Z$ basis, $|{\mathrm{GHZ}}_3^{\pm }⟩$ have the same measurement outcomes. The dashed line gives the ideal target population of $0.25$.

Figure 4

(a) The $|{\mathrm{GHZ}}_m^{\pm }⟩$ fidelity measured on $m$ trapped-ion qubits. (b) The up-sampled logical state fidelity of $[[m^2,1,m]]$ Shor codes after majority voting. For even $m$, ties are assigned randomly. The dashed yellow (blue) line is the SPAM fidelity for state $|+⟩$ ($|-⟩$) of the physical qubit. Note that the vertical ranges for (a) and (b) are different. The increase in logical fidelity from $m=3$ to $m=5$ shows how deeper encoding can offer increased protection against physical errors.

Figure 5

The scaling of the $[[m^2,1,m]]$ Shor code given by a simple depolarizing error model, given in Eq. (5).

Figure 6

A full $[[9,1,3]]$ Shor-code logical-state measurement with nine trapped-ion qubits (left). We also show the average fidelity of the $|{\mathrm{GHZ}}_3^{\pm }⟩$ states. For comparison, we again show the up-sampled results with three qubits from Fig. 4 (right). The dashed yellow (blue) line is the SPAM fidelity for state $|+⟩$ ($|-⟩$) of the physical qubit.