Exponential suppression of bit or phase flip errors with repetitive error correction (2102.06132v1)

Abstract: Realizing the potential of quantum computing will require achieving sufficiently low logical error rates. Many applications call for error rates in the $10^{-15}$ regime, but state-of-the-art quantum platforms typically have physical error rates near $10^{-3}$. Quantum error correction (QEC) promises to bridge this divide by distributing quantum logical information across many physical qubits so that errors can be detected and corrected. Logical errors are then exponentially suppressed as the number of physical qubits grows, provided that the physical error rates are below a certain threshold. QEC also requires that the errors are local and that performance is maintained over many rounds of error correction, two major outstanding experimental challenges. Here, we implement 1D repetition codes embedded in a 2D grid of superconducting qubits which demonstrate exponential suppression of bit or phase-flip errors, reducing logical error per round by more than $100\times$ when increasing the number of qubits from 5 to 21. Crucially, this error suppression is stable over 50 rounds of error correction. We also introduce a method for analyzing error correlations with high precision, and characterize the locality of errors in a device performing QEC for the first time. Finally, we perform error detection using a small 2D surface code logical qubit on the same device, and show that the results from both 1D and 2D codes agree with numerical simulations using a simple depolarizing error model. These findings demonstrate that superconducting qubits are on a viable path towards fault tolerant quantum computing.

• The paper demonstrates exponential reduction of logical errors by employing 1D repetition codes, achieving over a 100-fold error rate drop with increased qubits.
• It introduces high-precision error correlation analysis to quantify localized error mechanisms such as crosstalk and leakage over multiple correction rounds.
• The study validates both 1D repetition and 2D surface codes through experiments and simulations, underscoring a clear path toward fault-tolerant quantum computing.

Exponential Suppression of Bit or Phase Flip Errors with Repetitive Error Correction

Quantum error correction (QEC) is fundamental to realizing practical quantum computing, bridging the gap between high physical error rates and the low logical error rates required by many quantum algorithms. Current quantum platforms exhibit physical error rates near $10^{-3}$, yet many applications necessitate error rates in the $10^{-15}$ range. This paper investigates approaches to QEC, particularly through repetition codes, to exponentially suppress logical errors as the number of qubits increases, provided errors remain localized and correction performance is stable over many error correction rounds.

In this paper, superconducting qubits utilizing 1D repetition codes were implemented within a 2D grid to suppress bit or phase-flip errors exponentially. The research highlights that by increasing the qubit count from 5 to 21, the logical error rate can be reduced by over 100-fold. This error suppression is steady over 50 corrective rounds. For detailed analysis, a method for high-precision error correlation evaluation is introduced alongside a characterization of error locality. Furthermore, QEC was executed using a small, 2D surface code logical qubit, confirming that both 1D and 2D code results align with numerical simulations based on a straightforward depolarizing error model.

The findings outlined here demonstrate the viability of superconducting qubits for achieving fault-tolerant quantum computing. The paper meticulously explores two stabilizer codes: a 1D repetition code and a 2D surface code. Within the repetition code, qubits are arranged in a chain while alternately concerning measure and data qubits. Each measure qubit tests the parity of adjacent data qubits in the same error basis, protecting the logical qubit from either bit-flip ($X$ errors) or phase-flip ($Z$ errors), but not both. These codes serve as probes for exponential error suppression relative to the number of qubits; the setup includes a small ($d=2$) surface code for observing 2D code performance against an existing error model.

Utilizing the Sycamore processor with improved qubit design and configurations allows substantial error mitigation and analysis. Key advancements in gate calibration, the use of a reset protocol, and strategic use of controlled-Z gates underscore the experimental setup's robustness. Simulations were integral to understanding and projecting QEC performance, employing a depolarizing noise model aligned with component errors derived from experimental measures. The meticulously documented error budgets were pivotal in comparing projected versus measured $1/\Lambda$ values, highlighting discrepancies possibly due to factors like crosstalk and leakage, not present in simulations.

The extension of the work into understanding correlations between detection events further fortifies its contribution to QEC. By calculating and analyzing two-point detection event correlations, the authors identify and quantify both expected and unexpected correlations. These include crosstalk errors and state leakages, providing insights into possible avenues for future research and error mitigation.

The implications of this research are multifaceted: in the short term, imperative developments are necessary to reduce CZ gate errors and improve data qubit resilience during crucial operational cycles. While achieving current surface code thresholds is an immediate milestone, practical quantum computing efficiency will demand substantial enhancements in these areas, addressing novel error mechanisms such as high-energy particle-induced errors.

In conclusion, this research presents a clear path forward for enhancing quantum error correction through repetition codes and surface codes within superconducting qubit architectures. The work's comprehensive approach to error analysis and the vigorous investigation into component contributions to overall logical error rate fortify the path toward scalable, fault-tolerant quantum computing, presenting a significant step towards the eventual realization of large-scale practical quantum computations.