Demonstration of logical qubits and repeated error correction with better-than-physical error rates

TL;DR

The paper demonstrates fault-tolerant quantum computation on a commercial trapped-ion QCCD (H2) by encoding logical qubits with the Steane [[7,1,3]] and Carbon [[12,2,4]] codes and performing both single-shot and repeated error correction. Logical Bell-state preparation achieves error rates orders of magnitude below physical levels, showcasing 9.8× to 800× improvements depending on pre-/post-selection. It further shows repeated fault-tolerant error correction with up to three rounds where the logical error rate per round remains well below the corresponding physical baselines, signaling a practical path toward scalable fault-tolerant quantum computation. Collectively, these results illustrate a transition from NISQ-like behavior to reliable quantum processing on现-stage hardware, informing hardware-software co-design for future quantum architectures.

Abstract

The promise of quantum computers hinges on the ability to scale to large system sizes, e.g., to run quantum computations consisting of more than 100 million operations fault-tolerantly. This in turn requires suppressing errors to levels inversely proportional to the size of the computation. As a step towards this ambitious goal, we present experiments on a trapped-ion QCCD processor where, through the use of fault-tolerant encoding and error correction, we are able to suppress logical error rates to levels below the physical error rates. In particular, we entangled logical qubits encoded in the [[7,1,3]] code with error rates 9.8 times to 500 times lower than at the physical level, and entangled logical qubits encoded in a [[12,2,4]] code based on Knill's C4/C6 scheme with error rates 4.7 times to 800 times lower than at the physical level, depending on the judicious use of post-selection. Moreover, we demonstrate repeated error correction with the [[12,2,4]] code, with logical error rates below physical circuit baselines corresponding to repeated CNOTs, and show evidence that the error rate per error correction cycle, which consists of over 100 physical CNOTs, approaches the error rate of two physical CNOTs. These results signify a transition from noisy intermediate scale quantum computing to reliable quantum computing, and demonstrate advanced capabilities toward large-scale fault-tolerant quantum computing.

Figures (12)

• Figure 1: High-level depiction of the logical program of the Bell resource-state preparation using the Steane code. The blue dashed box indicates the pre-selected portion of the circuit where both the verification of $\ket{0}$ and trivial measurement results of syndrome extraction rounds (the boxes labeled $Syn$) are used to verify the creation of a resource state. The experiments are analyzed using error correction (with pre-selection) or error detection (with pre- and post-selection) independently. Post-selection accepts experiments where the syndrome inferred from the logical measurements (red dashed boxes) is trivial.
• Figure 2: Results for the Bell state preparation in the $[[7,1,3]]$ Steane code comparing physical level, error correction, and error detection experiments. There is a statistically significant separation between the physical and encoded results given the errors bars, which indicate $95\%$ confidence intervals.
• Figure 3: Effective logical circuit for Bell state preparation using the $[[12,2,4]]$ Carbon code. The top and bottom pair of lines correspond to separate blocks. The pre-selected portion of the circuit (blue dashed box) include verification measurements at the physical level which are used for pre-selection (not shown). When post-selection is performed, it is based only on the syndrome information in the transversal measurements (red dashed boxes), and a separate decision is made for each code block. The logical observables $A$ and $B$ can be either $X$ or $Z$ independently, but for the experiments discussed here we focus on the scenario where $A=B$.
• Figure 4: Comparison between physical and logical error rates of the $[[12,2,4]]$ Carbon code for the Bell state preparation circuit in \ref{['fig:carbon-bell-circuit']}. Precise numerical values can be found in \ref{['tab:bell-carbon-results']}. The difference in the error rates of the physical versus logical levels is statistically significant, as illustrated by the separation between the $95\%$ confidence intervals.
• Figure 5: Syndrome information can be obtained by performing teleportation at the logical level, as described by Knill Knill2003. Taking the realization of the original teleportation circuit for two logical qubits encoded in a Carbon block, which required 3 encoded blocks (left), it is possible to rearrange commuting circuit components to arrive at a circuit that uses a sequence of two 1-bit teleportations Zhou2000 to extract syndrome information requiring only 2 encoded blocks at any given time (right) (see \ref{['app:carbon']}).