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, d e c o d i n g 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.
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, d e c o d i n g 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.
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.
Quantum Error Correction (QEC) is an essential field of research towards the realization of large-scale quantum computers. On the theoretical side, a lot of effort is put into designing error-correcting codes that protect quantum data from errors, which inevitably happen due to the noisy nature of quantum hardware and quantum bits (qubits). Protecting data with an error-correcting code necessitates means to recover the original data, given a potentially corrupted data set---a task referred to as decoding. It is vital that decoding algorithms can recover error-free states in an efficient manner. While theoretical properties of certain QEC methods have been extensively studied, good techniques to analyze their performance in practically more relevant settings is still a widely unexplored area. In this work, we propose a set of software tools that facilitate numerical experiments with so-called Quantum Low-Density Parity-Check codes (QLDPC codes)---a broad class of codes, some of which have recently been shown to be asymptotically good. Based on that, we provide an implementation of a general decoder for QLDPC codes. On top of that, we propose a highly efficient heuristic decoder that eliminates the runtime bottlenecks of the general QLDPC decoder while still maintaining comparable decoding performance. These tools eventually make it possible to confirm theoretical results around QLDPC codes in a more practical setting and showcase the value of software tools (in addition to theoretical considerations) for investigating codes for practical applications. The resulting tool, which is publicly available at https://github.com/cda-tum/qecc as part of the Munich Quantum Toolkit (MQT), is meant to provide a playground for the search for "practically good" quantum codes.
The constituent parts of a quantum computer are inherently vulnerable to errors. To this end, we have developed quantum error-correcting codes to protect quantum information from noise. However, discovering codes that are capable of a universal set of computational operations with the minimal cost in quantum resources remains an important and ongoing challenge. One proposal of significant recent interest is the gauge color code. Notably, this code may offer a reduced resource cost over other well-studied fault-tolerant architectures by using a new method, known as gauge fixing, for performing the non-Clifford operations that are essential for universal quantum computation. Here we examine the gauge color code when it is subject to noise. Specifically, we make use of single-shot error correction to develop a simple decoding algorithm for the gauge color code, and we numerically analyse its performance. Remarkably, we find threshold error rates comparable to those of other leading proposals. Our results thus provide the first steps of a comparative study between the gauge color code and other promising computational architectures.
Practical large-scale quantum computation requires both efficient error correction and robust implementation of logical operations. Three-dimensional (3D) color codes are a promising candidate for fault-tolerant quantum computation due to their transversal non-Clifford gates, but efficient decoding remains challenging. In this work, we extend previous decoders for two-dimensional color codes [1], which are based on the restriction of the decoding problem to a subset of the qubit lattice, to three dimensions. Including boundaries of 3D color codes, we demonstrate that the 3D restriction decoder achieves optimal scaling of the logical error rate and a threshold value of 1.55(6)% for code-capacity bit- and phase-flip noise, which is almost a factor of two higher than previously reported for this family of codes [2, 3]. We furthermore present qCodePlot3D, a Python package for visualizing 2D and 3D color codes, error configurations, and decoding paths, which supports the development and analysis of such decoders. These advancements contribute to making 3D color codes a more practical option for exploring fault-tolerant quantum computation.
One of the most significant obstacles for implementing quantum computing is noise, corrupting quantum information stored in qubits. Quantum error-correcting codes can be utilized to overcome this by distributing quantum information into multiple qubits. The color code is one of the representative error-correcting codes, which allows relatively straightforward implementations of logical gates. In this article, I describe how to perform fault-tolerant quantum computing using the color code, including correcting errors via decoding and implementations of logical gates.
Quantum computation promises to revolutionize information processing by exploiting superposition and entanglement, enabling applications such as quantum simulation, combinatorial optimization, and secure quantum communication. Beyond individual processors, the quantum internet aims to connect quantum nodes through entanglement distribution, enabling distributed computation. However, quantum states are fragile: environmental interactions, imperfect gate operations, and measurement errors can quickly degrade encoded information. Reliable quantum computation therefore requires carefully designed quantum error-correcting codes and fault-tolerant protocols. Unlike classical redundancy schemes, quantum error correction must preserve delicate phase relationships and relies on indirect error detection through syndrome extraction with ancillary qubits, enabling error identification without collapsing encoded states. Topological codes, in particular, emerged early as the preferred choice for many architectures due to their planar structure, local interactions, and natural support for fault-tolerant logical operations. Color codes, for instance, allow transversal implementation of all Clifford gates, whereas non-Clifford operations require magic-state injection. More recent constructions, including quantum low-density parity-check codes, offer higher encoding rates and reduced resource overhead, but require long-range qubit connectivity and more complex protocols. Despite experimental progress, several challenges remain. Many platforms exhibit biased noise, motivating codes that exploit channel asymmetry. Real-time decoding remains a critical bottleneck, as fault-tolerant computation requires low-latency processing of rapidly generated syndrome data. Fault-tolerant logical state initialization also remains resource-intensive and scales poorly with code distance. Addressing these challenges, this thesis develops a unified framework spanning theory and experiment. It provides precise performance analysis of quantum error-correcting codes in the low physical error-rate regime, introduces codes tailored to biased noise, develops fast low-complexity decoding algorithms for real-time operation, and presents modular and efficient protocols for fault-tolerant state preparation of CSS codes. The latter is demonstrated experimentally through the preparation of a logical zero state of the Golay code on the Quantinuum H2 trapped-ion quantum computer.