Efficient Single-shot Decoding of Quantum Tanner Codes under Adversarial Noise

Core Concepts

Quantum Tanner codes facilitate single-shot quantum error correction under adversarial noise, where a single round of constant-weight parity check measurements suffices to reliably correct errors even in the presence of measurement errors.

Abstract

The content discusses the problem of decoding quantum Tanner codes, a family of quantum low-density parity-check (QLDPC) codes with good parameters, in the presence of measurement errors. Key highlights: Quantum Tanner codes admit computationally efficient decoding algorithms, such as the sequential and parallel mismatch decomposition decoders. The authors show that these decoders are single-shot, meaning they can reliably correct errors using a single round of noisy syndrome measurements. Specifically, the sequential decoder is shown to be (α=0, β)-single-shot, while the parallel decoder is (α, β)-single-shot for any α>0, with α decreasing exponentially with the number of parallel decoding iterations. The authors further analyze the performance of these single-shot decoders under multiple rounds of errors and decoding, showing that the residual error can be kept bounded under mild assumptions on the error and syndrome noise weights. The single-shot decoding property and the constant-time overhead of the parallel decoder make quantum Tanner codes attractive for fault-tolerant quantum computing protocols.

Stats

The content does not provide specific numerical data or metrics. It focuses on theoretical analysis and properties of the decoding algorithms.

Quotes

"Quantum Tanner codes also facilitate single-shot quantum error correction (QEC) of adversarial noise, where one measurement round (consisting of constant-weight parity checks) suffices to perform reliable QEC even in the presence of measurement errors." "There exists a constant β such that the sequential decoder (Algorithm 1) is (α = 0, β)-single-shot." "There exists a constant β such that for all α > 0, the O(log(1/α))-iteration parallel decoder (Algorithm 3) is (α, β)-single-shot."

Key Insights Distilled From

Single-shot decoding of good quantum LDPC codes

by Shouzhen Gu,... at arxiv.org 04-15-2024

https://arxiv.org/pdf/2306.12470.pdf

Deeper Inquiries

How do the performance guarantees of the single-shot decoders for quantum Tanner codes compare to other approaches for handling measurement errors, such as repeated syndrome measurements or fault-tolerant syndrome preparation

The performance guarantees of the single-shot decoders for quantum Tanner codes offer a significant advantage over other approaches for handling measurement errors, such as repeated syndrome measurements or fault-tolerant syndrome preparation. Efficiency: Single-shot decoding requires only one round of measurements, making it more efficient than repeated measurements, which can incur significant time overhead. This efficiency is crucial for practical implementations where minimizing the time required for error correction is essential. Robustness: Despite the presence of measurement errors, single-shot decoders for quantum Tanner codes can reliably correct errors with a bounded residual error weight. This robustness is crucial for ensuring the accuracy of error correction in the presence of noisy measurements. Simplicity: Single-shot decoding simplifies the error correction process by reducing the number of measurements needed to determine the error syndrome. This simplicity can lead to easier implementation and reduced complexity in fault-tolerant quantum computing protocols. Optimality: The guarantees provided by single-shot decoders for quantum Tanner codes ensure that errors can be corrected effectively even in the presence of measurement errors. This optimality is essential for maintaining the integrity of quantum information during error correction processes. Overall, the performance guarantees of single-shot decoders for quantum Tanner codes outperform other approaches by providing efficient, robust, simple, and optimal error correction capabilities in the presence of measurement errors.

What are the practical implications of the constant-time overhead of the parallel decoder for quantum Tanner codes in the context of fault-tolerant quantum computing protocols

The constant-time overhead of the parallel decoder for quantum Tanner codes has significant practical implications in the context of fault-tolerant quantum computing protocols. Scalability: The constant-time overhead of the parallel decoder ensures that the error correction process can scale efficiently with the size of the quantum system. This scalability is crucial for implementing fault-tolerant protocols on larger quantum computers. Real-time Error Correction: The constant-time overhead allows for real-time error correction, enabling quantum systems to maintain stability and accuracy during computation. This real-time capability is essential for ensuring the reliability of quantum computations. Resource Efficiency: The constant-time overhead minimizes the resources required for error correction, making the quantum computing system more resource-efficient. This efficiency is important for optimizing the use of qubits and minimizing computational costs. Fault Tolerance: The constant-time overhead contributes to the fault tolerance of the quantum system by providing rapid and effective error correction mechanisms. This fault tolerance is essential for ensuring the reliability and robustness of quantum computations. In conclusion, the constant-time overhead of the parallel decoder for quantum Tanner codes enhances the scalability, real-time error correction, resource efficiency, and fault tolerance of fault-tolerant quantum computing protocols.

Are there any other families of QLDPC codes that admit single-shot decoders with similar properties to the quantum Tanner codes discussed in this work

While quantum Tanner codes exhibit impressive properties in terms of single-shot decoding and error correction, there are other families of QLDPC codes that also admit single-shot decoders with similar properties. Some examples include: Quantum Expander Codes: These codes exhibit high expansion in the associated factor graphs, allowing for efficient error correction with minimal measurements. Quantum expander codes can achieve reliable error correction in a single-shot scenario, similar to quantum Tanner codes. 3D Subsystem Toric Code: This code incorporates geometrically local redundancies among constant-weight parity checks, enabling effective error correction with a single round of measurements. The 3D subsystem toric code shares similarities with quantum Tanner codes in terms of single-shot decoding capabilities. Gauge Color Code: Another family of QLDPC codes that offers robust error correction properties with single-shot decoding capabilities. The gauge color code leverages geometrically local redundancies to ensure efficient error correction in the presence of measurement errors. Overall, these families of QLDPC codes share similarities with quantum Tanner codes in terms of their ability to facilitate single-shot decoding and robust error correction in the presence of measurement errors. Each of these code families offers unique advantages and may be suitable for different applications in quantum computing protocols.