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量子 Reed-Müller 碼

鄭閔鴻

Published Jan 1, 2011• 0 citations
Publisher

Abstract

In this thesis, we start from the theory of classical error-correcting codes. We also introduce classical Reed-Muller codes along with its properties. We discuss the encoding and decoding methods for classical Reed-Muller codes using the majority rule. Then we turn to the topic of quantum error-correcting codes.We talk about stabilizer codes and discuss how classical Reed-Muller codes can be transformed into quantum Reed-Muller codes using Steane's method. Finally, we give a construction of a quantum code with parameter [[8,3,3]], and discuss its encoding and decoding process.

DOI
10.6844/ncku.2011.00411

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In this thesis, we start from the theory of classical error-correcting codes. We also introduce classical Reed-Muller codes along with its properties. We discuss the encoding and decoding methods for classical Reed-Muller codes using the majority rule. Then we turn to the topic of quantum error-correcting codes.We talk about stabilizer codes and discuss how classical Reed-Muller codes can be transformed into quantum Reed-Muller codes using Steane's method. Finally, we give a construction of a quantum code with parameter [[8,3,3]], and discuss its encoding and decoding process.
In this thesis some families of linear error correcting codes are presented. The reader will find a general description of binary codes and more specific details about linear codes such as Hamming, repetition codes, Reed-Muller codes, etc. To fully immerse ourselves in the methods of coding and decoding, we will introduce examples in order to contribute to the understanding of the theories. In these times of much communication through computer technology, our daily lives involve a substantial amount of data transmission. It is essential that these data are transmitted without errors through the communication channels. Therefore, the scientific field of error-correcting codes holds a significant importance in many aspects of todays society. The main goal of this thesis is to study linear block codes which belong to the class of binary codes. In this case we will attribute a more prominent role to first order Reed-Muller codes.
Good quantum error-correcting codes that fulfill practical considerations, such as simple encoding circuits and efficient decoders, are essential for functional quantum information processing systems. Quantum polar codes satisfy some of these requirements but lack certain critical features, thereby hindering their widespread use. Existing constructions either require entanglement assistance to produce valid quantum codes, suffer from poor finite-size performance, or fail to tailor polar codes to the underlying channel properties. Meanwhile, quantum Reed-Muller (RM) codes demonstrate strong performance, though no known efficient decoding algorithm exists for them. In this work, we propose strategies to interpolate between quantum polar codes and quantum RM codes, thus addressing the challenges of designing valid quantum polar codes without entanglement assistance and improving finite-size code performance.
Reed-Muller (RM) codes are classical codes that have enjoyed unabated interest since their introduction in 1954 due to their simple recursive structure and low-complexity decoding. The main goal of the thesis is to develop new coding techniques that can improve the performance of RM codes. In this work, we wish to keep the recursive low-complexity structure of RM codes while extending the sets of possible parameters and decoding algorithms. We address this task using two different methods. First, we consider new code constructions that extend code parameters using puncturing. Second, we design new decoding algorithms that repeatedly use the gradient properties of the classic recursive structure to improve code performance of RM codes. In the first part of the thesis, we propose a new class of codes that enjoy the recursive structure of RM codes, but reduce the exponential length of RM codes to a polynomial in m. These codes are obtained by puncturing classical RM codes, i.e. by restricting binary RM codes to small subsets of their positions. We study new recursive properties of these codes and analyze their parameters. In the second part of the thesis we develop new decoding algorithms for binary RM codes of second order that achieve near optimal performance for short and moderate code lengths. RM codes enjoy some simple decoding procedures; however, these algorithms fall short of optimal performance on moderate block lengths. We propose a suboptimal algorithm that reduces the complexity of maximum-likelihood decoding from super-polynomial order O(n(m+1)/2) to polynomial order O(mn2). At the same time, our numerical simulations show that the new algorithm performs very closely to its maximum-likelihood counterpart for blocks of up to several thousand bits.
Reed-Muller (RM) codes form a classic family studied for its interesting algebraic and combinatorial properties as well as from the perspective of information transmission. They achieve Shannon capacity of the basic binary channel models such as channels with independent erasures or flip errors. They have numerous applications to the theory of computational complexity and have well-understood local testability. They also give rise to a family of quantum codes admitting transversal logical operators in increasing levels of the Clifford hierarchy, a property of codes considered necessary for future fault-tolerant quantum computations. In this dissertation, we show that, despite decades of research surrounding RM codes, new directions sprouting from their simple definition continue to be possible. We will begin by considering an alternate description of RM codes in terms of faces of the Boolean hypercube, and will see that by replacing the hypercube with a more general algebraic object we obtain a large class of previously undiscovered classical error-correcting codes sharing many structural properties with RM codes. These classical codes directly lead to an even larger class of new quantum error-correcting codes with explicitly determined parameters; we will demonstrate that these quantum codes admit many natural transversal logical gates. We finish by specializing to the class of quantum Reed-Muller codes and provide a complete characterization of the logic implemented by these transversal operators, yielding a broad generalization of prior work on the subject.