Constant overhead quantum fault-tolerance with quantum expander codes (1808.03821v1)

Abstract: We prove that quantum expander codes can be combined with quantum fault-tolerance techniques to achieve constant overhead: the ratio between the total number of physical qubits required for a quantum computation with faulty hardware and the number of logical qubits involved in the ideal computation is asymptotically constant, and can even be taken arbitrarily close to 1 in the limit of small physical error rate. This improves on the polylogarithmic overhead promised by the standard threshold theorem. To achieve this, we exploit a framework introduced by Gottesman together with a family of constant rate quantum codes, quantum expander codes. Our main technical contribution is to analyze an efficient decoding algorithm for these codes and prove that it remains robust in the presence of noisy syndrome measurements, a property which is crucial for fault-tolerant circuits. We also establish two additional features of the decoding algorithm that make it attractive for quantum computation: it can be parallelized to run in logarithmic depth, and is single-shot, meaning that it only requires a single round of noisy syndrome measurement.

• The paper introduces a fault-tolerance scheme that achieves constant overhead by integrating quantum expander codes with Gottesman’s framework.
• It presents an efficient, single-shot, and parallelizable decoding algorithm robust against noisy syndrome measurements.
• The approach significantly reduces the space overhead in quantum computing, paving the way for scalable, fault-tolerant architectures.

Constant Overhead Quantum Fault-Tolerance with Quantum Expander Codes

The paper presented by Omar Fawzi, Antoine Grospellier, and Anthony Leverrier addresses a significant advancement in the domain of quantum error correction by demonstrating how quantum expander codes, combined with quantum fault-tolerance techniques, can achieve constant overhead for quantum computing. This work builds upon Gottesman's framework to show that the total number of physical qubits required for a computation with faulty hardware can maintain an asymptotically constant relation to the logical qubits involved in the ideal computation. The authors leverage the properties of quantum expander codes, which belong to the class of low-density parity-check (LDPC) codes, to construct fault-tolerant circuits that reduce the overhead previously scaled polylogarithmically as prescribed by the threshold theorem.

Technical Contributions

The primary contribution of this work is the analysis of an efficient decoding algorithm for quantum expander codes which is robust in the presence of noisy syndrome measurements. The decoding algorithm is designed to correct stochastic errors and is parallelizable to logarithmic depth, thus offering significant practical advantages for quantum computation. It is noteworthy that the decoding process is single-shot, requiring only a single round of noisy syndrome measurement.

The authors establish strong theoretical results by examining specific properties of quantum expander codes derived from expander graphs. These properties include constant rate, robustness against stochastic noise, and a minimum distance scaling with the square root of the block length. The research capitalizes on Gottesman's insight that constant rate quantum codes can facilitate fault-tolerant computation with an overhead arbitrarily close to one, provided the physical error rate is sufficiently low.

Implications and Future Directions

The theoretical implications of this research are profound, as it addresses one of the significant limitations of conventional quantum error correction mechanisms — the space overhead scaling factor. By achieving a constant overhead, quantum computations can be expected to become more viable in practice, especially as quantum technologies advance and larger systems are developed. However, practical implementation remains constrained by challenges such as constructing bipartite expander graphs with sufficient vertex expansion and addressing the small threshold values produced by this scheme.

A notable practical application of this work would be in the design of more efficient quantum circuits for large-scale quantum computers. The research opens avenues for exploring quantum error correction codes that offer scalable solutions without sacrificing the fault-tolerant integrity of the computations. Additionally, the robust decoding algorithm could be of interest for further investigations into reducing computational errors during quantum processes.

Future research may involve optimizing the threshold value through simulations or statistical methods to better understand its behavior across different computational contexts. Additionally, refining the construction of expander graphs and exploring alternative code families other than surface codes or hypergraph product codes may enhance the applicability of quantum expander codes.

Conclusion

This paper contributes significantly to the field of quantum error correction by providing a detailed framework for constant overhead quantum fault-tolerance using quantum expander codes. By advancing the decoding strategies and leveraging the properties of LDPC codes, the authors present a pathway for more efficient quantum computing architectures. While certain limitations require addressing, the paper lays a foundation for further practical and theoretical advancements in fault-tolerant quantum computation.