We investigate a class of quantum error-correcting codes to correct errors on both qubits and higher-state quantum systems represented as qudits. These codes arise from an original graph-theoretic representation of sets of quantum errors. In this framework, we represent the algebraic conditions for error correction in terms of edge avoidance between graphs providing a visual representation of the interplay between errors and error-correcting codes. Most importantly, this framework supports the development of quantum codes that correct against a predetermined set of errors, in contrast to current methods. A heuristic algorithm is presented, providing steps to develop codes that correct against an arbitrary noisy channel. We benchmark the correction capability of reflexive stabilizer codes for the case of single-qubit errors by comparison to existing stabilizer codes that are widely used. In addition, we present two instances of optimal encodings: an optimal encoding for fully correlated noise, which achieves a higher encoding rate than previously known, and a minimal encoding for single-qudit errors on a four-state system..
We investigate a class of quantum error-correcting codes to correct errors on both qubits and higher-state quantum systems represented as qudits. These codes arise from an original graph-theoretic representation of sets of quantum errors. In this framework, we represent the algebraic conditions for error correction in terms of edge avoidance between graphs providing a visual representation of the interplay between errors and error-correcting codes. Most importantly, this framework supports the development of quantum codes that correct against a predetermined set of errors, in contrast to current methods. A heuristic algorithm is presented, providing steps to develop codes that correct against an arbitrary noisy channel. We benchmark the correction capability of reflexive stabilizer codes for the case of single-qubit errors by comparison to existing stabilizer codes that are widely used. In addition, we present two instances of optimal encodings: an optimal encoding for fully correlated noise, which achieves a higher encoding rate than previously known, and a minimal encoding for single-qudit errors on a four-state system..
Long distance communication of digital data, whether through a physical medium or a broadcast signal, is often subjected to noise. To deliver data reliably through noisy communication channels, one must use codes that can detect and correct the particular noise of the channel. For transmission of classical data, error correcting schemes can be as simple as the sending of replicates. For quantum data, and in tandem the development of machines that can process quantum data, quantum error correcting codes must be developed. In addition to a larger set of possible errors, quantum error correcting schemes must contend with other peculiarities of quantum mechanics, such as the no-cloning theorem which can prevent the sending of replicate messages. Stabilizer codes are one family of quantum error correcting codes which can protect and correct errors expressed in terms of the Pauli group, exploiting its group structure and utilizing classical codes and the corresponding duals. We develop and examine a family of quantum stabilizer codes which arise from reflexive stabilizers. Moreover, we provide a mapping from our reflexive stabilizer codes to the well-known CSS codes developed by Calderbank, Shor, and Steane. For the case of a 4-state system we show that these codes can obtain the minimal embedding for code which can correct any flip or phase error. We also provide heuristic algorithms for creating reflexive stabilizer codes starting from the noise of a quantum channel. Furthermore, we show that the problem can be posed in terms of finding maximal Cayley subgraphs with restrictions imposed by the set of potential errors.
Abstract We propose a systematic procedure for the construction of graphs associated with binary quantum stabilizer codes. The procedure is characterized by means of the following three step process. First, the stabilizer code is realized as a codeword-stabilized (CWS) quantum code. Second, the canonical form of the CWS code is determined and third, the input vertices are attached to the graphs. In order to verify the effectiveness of the procedure, we implement the Gottesman stabilizer code characterized by multi-qubit encoding operators for the resource-efficient error correction of arbitrary single-qubit errors. Finally, the error-correcting capabilities of the Gottesman eight-qubit quantum stabilizer code is verified in graph-theoretic terms as originally advocated by Schlingemann and Werner.
This paper investigates the application of topological optimization techniques to quantum error correction codes. The core claim is that optimizing the topological structure of quantum error correction codes can significantly reduce error correction resource consumption and enhance the resilience of quantum computations. This is achieved through the utilization of topological optimization algorithms to identify optimal sub-graph structures within the topological network of a quantum error correction code, minimizing resource usage while maintaining sufficient error correction capability. We present a novel approach to quantum error correction design, moving beyond traditional code selection and offering a framework for dynamic code adaptation. The theoretical framework incorporates concepts from graph theory, linear algebra, and quantum information theory. The primary objective is to demonstrate a method for systematically reducing the overhead associated with quantum error correction, a critical step towards building fault-tolerant quantum computers. Our approach focuses on minimizing the number of physical qubits required to encode a logical qubit, a key metric for evaluating the efficiency of any quantum error correction scheme. The results indicate a substantial potential for optimization, suggesting a path toward more practical and scalable quantum computation.
Quantum Error-Correcting Codes (QECCs) play a crucial role in enhancing the robustness of quantum computing and communication systems against errors. Within the realm of QECCs, stabilizer codes, and specifically graph codes, stand out for their distinct attributes and promising utility in quantum technologies. This study underscores the significance of devising expansive QECCs and adopts the ZX-calculus a graphical language adept at quantum computational reasoning-to depict the encoders of graph codes effectively. Through the integration of ZX-calculus with established encoder frameworks, we present a nuanced approach that leverages this graphical representation to facilitate the construction of large-scale QECCs. Our methodology is rigorously applied to examine the intricacies of concatenated graph codes and the development of holographic codes, thus demonstrating the practicality of our graphical approach in addressing complex quantum error correction challenges. This research contributes to the theoretical understanding of quantum error correction and offers practical tools for its application, providing objective advancements in the field of quantum computing.
Noise is a major obstacle in the development of practical schemes for quantum computation and communication. Similar to the case of classical communication, this noise can be protected against by employing a code, which provides a means for encoding quantum states prior to transmission and allows for errors to be inferred, and hopefully corrected, by a decoder at the receiver. Unfortunately, designing good codes and decoders is typically a difficult problem. This thesis focuses on developing low-complexity heuristic approaches to three such problems: the design of modified belief propagation decoders for quantum low-density parity-check codes, the design of stabilizer codes for asymmetric channels, and the design of codeword stabilized codes. Quantum low-density parity-check codes are stabilizer codes with low-weight generators. Such codes permit low-complexity decoding via the use of belief propagation, which is an iterative message passing algorithm that takes place on a factor graph defined by the code. However, the performance of such a decoder is limited both by code structure and the degenerate nature of quantum errors. To overcome these limitations, at least in part, a number of modifications to belief propagation are developed. Central among these is the augmented decoder, which in the case of a decoding error, iteratively reattempts decoding using modified factor graph. This heuristic modification simply involves the duplication of a randomly selected subset of the graph’s check nodes, which are in one-to-one correspondence with the code’s stabilizer generators. Across a range of codes, it is shown that the decoders developed perform as well as or better than other modified decoders presented in literature. For a number of channels of physical interest, phase-flip errors occur far more frequently than bit-flip errors. When transmitting across these so-called asymmetric channels, the decoding error rate can be minimized by tailoring the code used to the channel. However, assessing the performance of codes on a given channel is made difficult by the #P-completeness of optimal decoding. To address this complexity, it is shown that the decoding error rate can be accurately approximated using only a small fraction of the possible errors caused by the channel. This approximation is then used to identify a number of cyclic stabilizer codes that perform well on two different asymmetric channels. To further build on this, a heuristic is demonstrated for assessing code performance based on the decoding error rate of an associated classical code. The complexity of calculating this classical error rate is relatively low, and it is shown that it can be used as the basis for a hill-climbing search algorithm. Such searches have yielded a large number of highly performant codes satisfying various structure constraints. The family of codeword stabilized codes encompasses both the stabilizer codes as well as many of the best known nonadditive codes. Constructing a standard form codeword stabilized code is a matter of selecting a simple undirected graph and a binary classical code. This makes designing optimal codes difficult as the number of possible graphs grows exponentially with code length, and the clique search required to construct the classical code is NP-hard. To address the exponential growth of the search space, a heuristic is developed for assessing graphs. This heuristic is then employed by a genetic algorithm that also makes use of a novel crossover operation based on spectral bisection, which is show to be superior to more standard crossover operations. With a graph selected, it is demonstrated that the complexity of the clique search required to construct the associated classical code can be mitigated through the use of a heuristic clique finding algorithm. A number of best known codes are presented that have been found using this approach.
We propose a systematic scheme for the construction of graphs associated with binary stabilizer codes. The scheme is characterized by three main steps: first, the stabilizer code is realized as a codeword-stabilized (CWS) quantum code; second, the canonical form of the CWS code is uncovered; third, the input vertices are attached to the graphs. To check the effectiveness of the scheme, we discuss several graphical constructions of various useful stabilizer codes characterized by single and multi-qubit encoding operators. In particular, the error-correcting capabilities of such quantum codes are verified in graph-theoretic terms as originally advocated by Schlingemann and Werner. Finally, possible generalizations of our scheme for the graphical construction of both (stabilizer and nonadditive) nonbinary and continuous-variable quantum codes are briefly addressed.