Quantum error correction below the surface code threshold (2408.13687v1)

Abstract: Quantum error correction provides a path to reach practical quantum computing by combining multiple physical qubits into a logical qubit, where the logical error rate is suppressed exponentially as more qubits are added. However, this exponential suppression only occurs if the physical error rate is below a critical threshold. In this work, we present two surface code memories operating below this threshold: a distance-7 code and a distance-5 code integrated with a real-time decoder. The logical error rate of our larger quantum memory is suppressed by a factor of $\Lambda$ = 2.14 $\pm$ 0.02 when increasing the code distance by two, culminating in a 101-qubit distance-7 code with 0.143% $\pm$ 0.003% error per cycle of error correction. This logical memory is also beyond break-even, exceeding its best physical qubit's lifetime by a factor of 2.4 $\pm$ 0.3. We maintain below-threshold performance when decoding in real time, achieving an average decoder latency of 63 $\mu$s at distance-5 up to a million cycles, with a cycle time of 1.1 $\mu$s. To probe the limits of our error-correction performance, we run repetition codes up to distance-29 and find that logical performance is limited by rare correlated error events occurring approximately once every hour, or 3 $\times$ 10$^9$ cycles. Our results present device performance that, if scaled, could realize the operational requirements of large scale fault-tolerant quantum algorithms.

• The paper demonstrates exponential error suppression by employing a distance-7 surface code with a Λ of 2.14 ± 0.02.
• The researchers utilized real-time decoding with a 63 µs latency to extend the logical qubit lifetime by a factor of 2.4 compared to the best physical qubit.
• The experimental setup on 72- to 105-qubit processors confirms the viability of fault-tolerant quantum computing under practical conditions.

Quantum Error Correction Below the Surface Code Threshold: An Examination

Introduction

Quantum error correction (QEC) aims to stabilize quantum information and facilitate practical quantum computations. The presented paper discusses the experimental realization of quantum error correction performance using surface codes that operate below the threshold error rate, thereby achieving exponential error suppression. By utilizing surface code memories with real-time decoding mechanisms, this work marks a significant advancement in maintaining quantum coherence and fault tolerance.

Theoretical Rationale

Surface codes, formed by combining many physical qubits into logical qubits, provide a structured method for managing qubit errors. Theoretical predictions postulate that as long as the physical error rate remains below a critical threshold, logical errors can be exponentially suppressed by increasing code distance. The suppression factor is fundamentally tied to the code distance, $d$, and the ratio of the physical error rate, $p$, to the threshold error rate, $p_\text{thr}$, as outlined in the paper's Equation (1): $$\varepsilon_d \propto \left(\frac{p}{p_\text{thr}}\right)^{(d + 1) / 2}$$.

Experimental Results

This paper utilizes superconducting transmon qubits to implement surface codes on two processors: one with 72 qubits and another with 105 qubits. The researchers deployed a distance-7 surface code comprising 101 qubits and achieved a logical error rate significantly below the threshold with an error suppression factor, $\Lambda = 2.14 \pm 0.02$. Notably, the lifetime of the distance-7 logical qubit exceeded the best physical qubit's lifetime by a factor of 2.4, which is a noteworthy achievement in quantum error correction beyond break-even.

The implemented system also maintained performance consistently under real-time decoding conditions with an average latency of 63 µs, demonstrating its applicability in real-world quantum algorithm execution.

Implications

By demonstrating below-threshold operation and error suppression at scale, this research paves the path for large-scale fault-tolerant quantum computations. Achieving logical qubit performance beyond break-even is a crucial milestone towards deployment of quantum processors for complex calculations in fields such as cryptography and quantum simulations.

The significant error suppression observed, coupled with the ability to decode in real time, fulfills various operational requirements necessary for viable fault-tolerant quantum computations. Moreover, by running successful long-cycle experiments, the researchers have shown that maintaining device stability over extended periods is achievable, addressing one of the significant challenges in scaling up quantum processors.

Logical Error Sensitivity and Challenges

The paper also scrutinizes the sensitivity of the logical error to external noise, codec variations, and unexpected error bursts. The analysis showed that addressing leakage, a substantial source of logical error introduced through higher energy quantum states, is pivotal to operating surface codes effectively. Notably, error correlation and unexpected bursts present challenges that must be resolved to achieve even lower error rates.

Repetition code experiments further illustrated the potential error floors at extremely low error rates, highlighting the need for further investigation into correlated error events that pose roadblocks in implementing error-tolerant operations at the requisite scales.

Real-Time Decoding and Future Prospects

The paper also focused on the implementation of real-time decoding to meet rapid processing requirements. The ability to sustain such decoding efficacy underpins the scalability of quantum processors and is critical for enabling real-time feedback in performing non-Clifford operations.

Considering future developments, advancements in error correction protocols and decoding methods promise to augment the efficiency and performance of quantum systems. Overcoming the existing correlated error barriers and improving error throughputs would significantly push the boundaries of practical quantum computation.

Conclusion

The work presented in this paper significantly advances the foundational capability of quantum error correction operating below the surface code threshold. While certain challenges remain, stringent system coherence, real-time processing, and strategic error management demonstrate that scalable, fault-tolerant quantum computing might soon be an achievable reality. The promise of quantum computation to solve complex problems hinges on paths charted by research investigating the threshold behaviors and stability of logical qubits as demonstrated in this paper.