Decoding quantum color codes with MaxSAT (2303.14237v3)

Abstract: In classical computing, error-correcting codes are well established and are ubiquitous both in theory and practical applications. For quantum computing, error-correction is essential as well, but harder to realize, coming along with substantial resource overheads and being concomitant with needs for substantial classical computing. Quantum error-correcting codes play a central role on the avenue towards fault-tolerant quantum computation beyond presumed near-term applications. Among those, color codes constitute a particularly important class of quantum codes that have gained interest in recent years due to favourable properties over other codes. As in classical computing, decoding is the problem of inferring an operation to restore an uncorrupted state from a corrupted one and is central in the development of fault-tolerant quantum devices. In this work, we show how the decoding problem for color codes can be reduced to a slight variation of the well-known LightsOut puzzle. We propose a novel decoder for quantum color codes using a formulation as a MaxSAT problem based on this analogy. Furthermore, we optimize the MaxSAT construction and show numerically that the decoding performance of the proposed decoder achieves state-of-the-art decoding performance on color codes. The implementation of the decoder as well as tools to automatically conduct numerical experiments are publicly available as part of the Munich Quantum Toolkit (MQT) on GitHub.

• The paper introduces a MaxSAT-based decoding method that reformulates quantum color code error correction as a combinatorial optimization problem.
• It leverages Microsoft's Z3 solver to manage complex parity constraints, achieving noise thresholds close to 10% for fault-tolerant quantum computing.
• The work paves the way for integrating classical optimization techniques into quantum error correction, suggesting scalable approaches for future quantum systems.

Decoding Quantum Color Codes with MaxSAT: A Comprehensive Overview

Quantum error correction (QEC) is fundamental for advancing quantum computing toward fault-tolerant processing. A significant portion of research in QEC is dedicated to developing efficient decoders that can manage the complexity of quantum error correction codes (QECCs). The paper under discussion presents a novel approach to decoding quantum color codes utilizing a MaxSAT (Maximum Satisfiability Problem) formulation. This work is particularly concerned with improving the decoding efficiency and performance of color codes, a class of QECCs recognized for their advantageous properties over other quantum codes.

Overview of Quantum Color Codes and MaxSAT

Color codes, classified under CSS (Calderbank-Shor-Steane) codes, are defined on a two-dimensional lattice satisfying specific combinatorial properties, such as being three-colorable. Due to their rich structure, color codes offer efficient error correction with potentially lower resource demands compared to surface codes. The decoding process involves identifying and correcting quantum errors deduced from the violation of stabilizer conditions, known through a set of parity checks. Decoding these codes efficiently pushes the boundaries of feasible fault-tolerant quantum computing.

The authors propose a novel MaxSAT-based decoder for quantum color codes, leveraging the analogy between decoding and a combinatorial puzzle known as the puzzle. MaxSAT is a well-studied optimization problem where the goal is to satisfy as many clauses of a given boolean formula as possible. In this context, the decoding problem is transformed into identifying a minimal set of errors that satisfy the constraints, hence minimizing the qubits involved in errors.

Numerical Performance and Methodology

The paper provides comprehensive numerical evaluations demonstrating that the MaxSAT decoder achieves state-of-the-art decoding performance. For the noise model considered, the decoder almost reaches the threshold values, which gauge the quantum code's robustness against noise. With thresholds approximating 10% under specific conditions, this decoder closely aligns with optimal solutions.

The authors meticulously address the challenge of efficiently formulating multi-variable parity constraints inherent in the MaxSAT encoding. This formulation not only optimizes performance but ensures re-usability, vital for practical implementations where repeated decoding is required.

The implementation, built on Microsoft's Z3 solver for MaxSAT, is shown to be both versatile and effective. Compared with other decoders such as the tensor network decoder, the proposed MaxSAT decoder demonstrates noteworthy runtime efficiency, especially at lower error rates crucial for fault-tolerant quantum computing operations.

Implications and Future Directions

The significance of this work lies in its potential applications and implications. The proposed methodology opens up pathways for leveraging sophisticated classical constraint optimization techniques in quantum computing contexts. It exemplifies how advanced classical algorithms can be repurposed to meet the demands of quantum applications, thus fostering a symbiotic relationship between classical and quantum computing research domains.

Reflecting on future directions, this MaxSAT decoder's general framework suggests pathways for adaptations to more complex noise models, including circuit-level noise where physical errors are not limited to simple bit-flip scenarios. Furthermore, exploring hybrid approaches combining MaxSAT with other decoders, such as union-find or belief-propagation, may offer enhanced performance and broader applicability in fault-tolerant quantum systems.

Optimization strategies for MaxSAT solvers could further enhance performance and make them more suited to different types of quantum coding challenges. As robust and scalable quantum computing technologies evolve, methods such as those presented in this paper will undoubtedly be integral in overcoming the current limitations hindering the realization of practical quantum computing systems.

In conclusion, this work underscores the evolving landscape of quantum error correction by introducing a versatile and efficient decoding approach, motivating further explorations into integrating classical optimization techniques within the quantum field. The open-source nature of the implementation ensures wide accessibility, fostering collaborative advancements in quantum decoding research. The proposed MaxSAT approach represents a step forward in the continuing effort to achieve fault-tolerant quantum computing.