Andrew Steane
A set of quantum error correcting codes based on classical Reed-Muller codes is described. The codes have parameters [[n,k,d]] = [[2^r, 2^r - C(r,t) - 2 sum_{i=0}^{t-1} C(r,i), 2^t + 2^{t-1} ]].
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This paper presents a set of quantum Reed-Muller codes which are typically 100 times more effective than existing quantum Reed-Muller codes.
In this paper we show that there is a direct correspondence between quantum Boolean operations and certain forms of classical (non-quantum) logic known as Reed-Muller expansions. This allows us to readily convert Boolean circuits into their quantum equivalents. A direct result of this is that the problem of synthesis and optimization of quantum Boolean logic can be tackled within the field of Reed-Muller logic.
We construct nonbinary quantum codes from classical generalized Reed-Muller codes and derive the conditions under which these quantum codes can be punctured. We provide a partial answer to a question raised by Grassl, Beth and Roetteler on the existence of q-ary quantum MDS codes of length n with q\le n\le q^2-1.
Local distillation from Reed Muller codes unfolding
本文通过揭示展开所基于的代数结构，推广了Ruiz等人提出的Reed-Muller蒸馏厂展开方法，展示了距离4的Reed-Muller蒸馏厂的2D本地布局及距离4和7的3D本地布局，实现了高保真度的CCZ态和T态制备。
Steane's 7-qubit quantum error-correcting code admits a set of fault-tolerant gates that generate the Clifford group, which in itself is not universal for quantum computation. The 15-qubit Reed-Muller code also does not admit a universal fault-tolerant gate set but possesses fault-tolerant T and control-control-Z gates. Combined with the Clifford group, either of these two gates generate a universal set. Here, we combine these two features by demonstrating how to fault-tolerantly convert between these two codes, providing a new method to realize universal fault-tolerant quantum computation. On…
In this paper, we study the close relationship between Reed-Muller codes and single-qubit phase gates from the perspective of $T$-count optimization. We prove that minimizing the number of $T$ gates in an $n$-qubit quantum circuit over CNOT and $T$, together with the Clifford group powers of $T$, corresponds to finding a minimum distance decoding of a length $2^n-1$ binary vector in the order $n-4$ punctured Reed-Muller code. Moreover, we show that the problems are polynomially equivalent in the length of the code. As a consequence, we derive an algorithm for the optimization of $T$-count in q…
Efficient recursive encoders for quantum Reed-Muller codes towards Fault tolerance
本文为允许横向门的量子Reed-Muller码设计高效递归编码器，实现对数深度和更少门数，并证明CNOT门数与纠缠熵相等，展示最优性及纠缠提取能力。
We propose families of protocols for magic state distillation -- important components of fault tolerance schemes --- for systems of odd prime dimension. Our protocols utilize quantum Reed-Muller codes with transversal non-Clifford gates. We find that, in higher dimensions, small and effective codes can be used that have no direct analogue in qubit (two-dimensional) systems. We present several concrete protocols, including schemes for three-dimensional (qutrit) and five-dimensional (ququint) systems. The five-dimensional protocol is, by many measures, the best magic state distillation scheme ye…
We design forward and backward fault-tolerant conversion circuits, which convert between the Steane code and the 15-qubit Reed-Muller quantum code so as to provide a universal transversal gate set. In our method, only 7 out of total 14 code stabilizers need to be measured, and we further enhance the circuit by simplifying some stabilizers; thus, we need only to measure eight weight-4 stabilizers for one round of forward conversion and seven weight-4 stabilizers for one round of backward conversion. For conversion, we treat random single-qubit errors and their influence on syndromes of gauge op…
Concatenation of two quantum error correcting codes with complementary sets of transversal gates can provide a means towards universal fault-tolerant computation. We first show that it is generally preferable to choose the inner code with the higher pseudo-threshold in order to achieve lower logical failure rates. We then explore the threshold properties of a wide range of concatenation schemes. Notably, we demonstrate that the concatenation of complementary sets of Reed-Muller codes can increase the code capacity threshold under depolarizing noise when compared to extensions of previously pro…
Convolutional stabilizer codes promise to make quantum communication more reliable with attractive online encoding and decoding algorithms. This paper introduces a new approach to convolutional stabilizer codes based on direct limit constructions. Two families of quantum convolutional codes are derived from generalized Reed-Solomon codes and from Reed- Muller codes. A Singleton bound for pure convolutional stabilizer codes is given.
Entanglement-assisted Quantum Reed-Muller Tensor Product Codes
本文研究经典Reed-Muller码的纠缠辅助量子编码构造，证明CSS构造导致零速率，而张量积码构造可实现正编码速率和正催化速率，从而建立该码族的编码超可加性。
Previous work has provided methods for decomposing unitary matrices to series of quantum multiplexers, but the multiplexers created in this way are highly non-minimal. This paper presents a new approach for optimizing quantum multiplexers with arbitrary single-qubit quantum target functions. For quantum multiplexers, we define standard forms and two types of new forms: fixed polarity quantum forms (FPQF) and Kronecker quantum forms (KQF), which are analogous to Minterm Sum of Products forms, Fixed Polarity Reed-Muller (FPRM) forms, and Kronecker Reed-Muller (KRM) forms, respectively, for class…
Interpolation of Quantum Polar Codes and Quantum Reed-Muller Codes
本文提出在量子极化码与量子Reed-Muller码之间进行插值的策略，以解决无需纠缠辅助构造有效量子极化码并提升其有限长度性能的问题。
Coxeter codes: Extending the Reed-Muller family
本文提出了一类基于Coxeter群的二元线性码，扩展了Reed-Muller码家族，并展示了其在量子编码中的应用。
Reed-Muller Codes for Quantum Pauli and Multiple Access Channels
本文通过统一框架将 RM 码分析扩展到 Q-MAC 与量子 Pauli 信道，推导可达速率区域并建立二者联系，证明量子 RM 码在多类 Pauli 信道参数下的速率最优普适性。
Reed-Muller Codes on CQ Channels via a New Correlation Bound for Quantum Observables
研究二元输入对称经典-量子（BSCQ）信道上Reed-Muller码与Holevo容量，通过对最小均方误差（MSE）可观测量的正交分解推导码元外在MMSE递归式，使速率低于Holevo容量固定值的码序列中规定比特集可顺序解码且错误概率趋于零。
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