Maxim Raginsky
Inspired by Kitaev's argument that physical error correction is possible in a system of interacting anyons, we demonstrate that such "self-correction" is fairly common in spin systems with classical Hamiltonians that admit the Peierls argument and where errors are modelled by quantum perturbations.
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In this paper we study the geometrical structures of multi-qubit states based on symplectic toric manifolds. After a short review of symplectic toric manifolds, we discuss the space of a single quantum state in terms of these manifolds. We also investigate entangled multipartite states based on moment map and Delzant's construction of toric manifolds and algebraic toric varieties.
Quantum error correction requires decoders that are both accurate and efficient. To this end, union-find decoding has emerged as a promising candidate for error correction on the surface code. In this work, we benchmark a weighted variant of the union-find decoder on the toric code under circuit-level depolarizing noise. This variant preserves the almost-linear time complexity of the original while significantly increasing the performance in the fault-tolerance setting. In this noise model, weighting the union-find decoder increases the threshold from 0.38% to 0.62%, compared to an increase fr…
We analyze the properties of a 2D topological code derived by concatenating the [[4, 2, 2]] code with the toric/surface code, or alternatively by removing check operators from the 2D square-octagon or 4.8.8 color code. We show that the resulting code has a circuit-based noise threshold of $\sim 0.41\%$ (compared to $\sim 0.6\%$ for the toric code in a similar scenario), which is higher than any known 2D color code. We believe that the construction may be of interest for hardware in which one wants to use both long-range two-qubit gates as well as short-range gates between small clusters of qub…
We analyze the four dimensional toric code in a hyperbolic space and show that it has a classical error correction procedure which runs in almost linear time and can be parallelized to almost constant time, giving an example of a quantum LDPC code with linear rate and efficient error correction.
Measurement-induced phase transitions in the toric code
本文利用 parton 构造将 toric code 上的随机 Pauli 测量映射为经典环模型和混合电路，揭示未测量边界的测量诱导纠缠相变，并提出可由线性观测量诊断的混合态有序相。
ParaToric 1.0: Continuous-time quantum Monte Carlo for the toric code in a parallel field
ParaToric 1.0 提供了在平行磁场中模拟拓扑码的连续时间量子蒙特卡洛算法，支持多种晶格结构和边界条件，适用于生成其他方法的训练数据。
We have studied the dynamics of a generalized toric code based on qudits at finite temperature by finding the master equation coupling the code's degrees of freedom to a thermal bath. As a consequence, we find that for qutrits new types of anyons and thermal processes appear that are forbidden for qubits. These include creation, annihilation and diffusion throughout the system code. It is possible to solve the master equation in a short-time regime and find expressions for the decay rates as a function of the dimension $d$ of the qudits. Although we provide an explicit proof that the system re…
Universal representation of the long-range entanglement in the family of Toric Code states
该研究针对拓扑量子态长程纠缠的普适表示需求，以Kitaev's Ladders为模块，通过非局域解纠缠子将不同几何Toric Code态转化为其张量积，揭示了长程纠缠的普适非局域模式，为拓扑序描述提供了新方法。
We propose a diagnostic for finite temperature topological order using `topological entanglement negativity', the long-range component of a mixed-state entanglement measure. As a demonstration, we study the toric code model in $d$ spatial dimension for $d$=2,3,4, and find that when topological order survives thermal fluctuations, it possesses a non-zero topological entanglement negativity, whose value is equal to the topological entanglement entropy at zero temperature. Furthermore, we show that the Gibbs state of 2D and 3D toric code at any non-zero temperature, and that of 4D toric code abov…
In order to build a large scale quantum computer, one must be able to correct errors extremely fast. We design a fast decoding algorithm for topological codes to correct for Pauli errors and erasure and combination of both errors and erasure. Our algorithm has a worst case complexity of $O(n α(n))$, where $n$ is the number of physical qubits and $α$ is the inverse of Ackermann's function, which is very slowly growing. For all practical purposes, $α(n) \leq 3$. We prove that our algorithm performs optimally for errors of weight up to $(d-1)/2$ and for loss of up to $d-1$ qubits, where $d$ is th…
We provide a basic introduction to discrete-variable, rotor, and continuous-variable quantum phase spaces, explaining how the latter two can be understood as limiting cases of the first. We extend the limit-taking procedures used to travel between phase spaces to a general class of Hamiltonians (including many local stabilizer codes) and provide six examples: the Harper equation, the Baxter parafermionic spin chain, the Rabi model, the Kitaev toric code, the Haah cubic code (which we generalize to qudits), and the Kitaev honeycomb model. We obtain continuous-variable generalizations of all mod…
Abelian multi-cycle codes for single-shot error correction
本文提出了一种量子低密度奇偶校验码，局部等价于高维量子超图积码，通过高冗余低权重稳定器生成器提高容错解码精度，并具有单次错误校正特性。
Floquet implementation of a 3d fermionic toric code with full logical code space
本文提出一种3D Floquet码，实现费米拓扑码并保持全部逻辑量子比特，通过设计特定晶格几何结构避免逻辑信息崩溃，同时引入监控Kitaev模型家族。
Progressive-Proximity Bit-Flipping for Decoding Surface Codes
针对拓扑量子码现有译码器难以兼顾低复杂度、低延迟和低功耗的问题，提出基于邻近向量的渐进邻近比特翻转译码器，可高效校正相邻量子比特简并错误，在二进制对称信道下实现了7.5%（二维环面码）和7%（旋转平面码）的译码阈值。
For large d, we study quantum channels on C^d obtained by selecting randomly N independent Kraus operators according to a probability measure mu on the unitary group U(d). When mu is the Haar measure, we show that for N>d/epsilon^2$, such a channel is epsilon-randomizing with high probability, which means that it maps every state within distance epsilon/d (in operator norm) of the maximally mixed state. This slightly improves on a result by Hayden, Leung, Shor and Winter by optimizing their discretization argument. Moreover, for general mu, we obtain a epsilon-randomizing channel provided N > …
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