Sergey Bravyi, Robert Raussendorf
We study measurement-based quantum computation (MQC) using as quantum resource the planar code state on a two-dimensional square lattice (planar analogue of the toric code). It is shown that MQC with the planar code state can be efficiently simulated on a classical computer if at each step of MQC the sets of measured and unmeasured qubits correspond to connected subsets of the lattice.
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Three dimensional (3D) toric codes are a class of stabilizer codes with local checks and come under the umbrella of topological codes. While decoding algorithms have been proposed for the 3D toric code on a cubic lattice, there have been very few studies on the decoding of 3D toric codes over arbitrary lattices. Color codes in 3D can be mapped to toric codes. However, the resulting toric codes are not defined on cubic lattice. They are arbitrary lattices with triangular faces. Decoding toric codes over an arbitrary lattice will help in studying the performance of color codes. Furthermore, gaug…
Toric codes and color codes are two important classes of topological codes. Kubica, Yoshida, and Pastawski showed that any $D$-dimensional color code can be mapped to a finite number of toric codes in $D$-dimensions. In this paper we propose an alternate map of 3D color codes to 3D toric codes with a view to decoding 3D color codes. Our approach builds on Delfosse's result for 2D color codes and exploits the topological properties of these codes. Our result reduces the decoding of 3D color codes to that of 3D toric codes. Bit flip errors are decoded by projecting on one set of 3D toric codes w…
We introduce an efficient decoder of the color code in $d\geq 2$ dimensions, the Restriction Decoder, which uses any $d$-dimensional toric code decoder combined with a local lifting procedure to find a recovery operation. We prove that the Restriction Decoder successfully corrects errors in the color code if and only if the corresponding toric code decoding succeeds. We also numerically estimate the Restriction Decoder threshold for the color code in two and three dimensions against the bit-flip and phase-flip noise with perfect syndrome extraction. We report that the 2D color code threshold $…
Fault-Tolerant Constant-Depth Clifford Gates on Toric Codes
针对2D环面码，提出并模拟了结合折叠横向门、Dehn扭转和单次逻辑Pauli测量的容错常深度逻辑门组，其能生成完整Clifford群并具备容错性与常深度特性。
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…
A decade ago Kitaev's toric code model established the new paradigm of topological quantum computation. Due to remarkable theoretical and experimental progress, the quantum simulation of such complex many-body systems is now within the realms of possibility. Here we consider the question, to which extent the ground states of small toric code systems differ from LU-equivalent graph states. We argue that simplistic (though experimentally attractive) setups obliterate the differences between the toric code and equivalent graph states; hence we search for the smallest setups on the square- and tri…
Efficient encoding of the 2D toric code logical state using local Clifford gates
提出一种仅用局域 $CX$ 和 $H$ 门、深度 $2L+1$ 的电路高效编码二维环面码逻辑态的算法。
The toric code is a simple and exactly solvable example of topological order realising Abelian anyons. However, it was shown to support non-local lattice defects, namely twists, which exhibit non-Abelian anyonic behaviour [1]. Motivated by this result, we investigated the potential of having non-Abelian statistics from puncture defects on the toric code. We demonstrate that an encoding with mixed-boundary punctures reproduces Ising fusion, and a logical Pauli-$X$ upon their braiding. Our construction paves the way for local lattice defects to exhibit non-Abelian properties that can be employed…
The surface code scheme for quantum computation features a 2d array of nearest-neighbor coupled qubits yet claims a threshold error rate approaching 1% (NJoP 9:199, 2007). This result was obtained for the toric code, from which the surface code is derived, and surpasses all other known codes restricted to 2d nearest-neighbor architectures by several orders of magnitude. We describe in detail an error correction procedure for the toric and surface codes, which is based on polynomial-time graph matching techniques and is efficiently implementable as the classical feed-forward processing step in …
Quantum computers are predicted to utilize quantum states to perform memory and to process tasks far faster than those of conventional classical computers. In this paper we show a new road towards building fault tolerance quantum computer by tuning quantum tunneling effect of the degenerate quantum states in topological order, instead of by braiding anyons. Using a designer Hamiltonian - the Wen-Plaquette model as an example, we study its quantum tunneling effect of the toric codes and show how to control the toric code to realize topological quantum computation (TQC). In particular, we give a…
We examine the performance of the single-mode GKP code and its concatenation with the toric code for a noise model of Gaussian shifts, or displacement errors. We show how one can optimize the tracking of errors in repeated noisy error correction for the GKP code. We do this by examining the maximum-likelihood problem for this setting and its mapping onto a 1D Euclidean path-integral modeling a particle in a random cosine potential. We demonstrate the efficiency of a minimum-energy decoding strategy as a proxy for the path integral evaluation. In the second part of this paper, we analyze and nu…
We consider the problem of preparing specific encoded resource states for the toric code by local, time-independent interactions with a memoryless environment. We propose a construction of such a dissipative encoder which converts product states to topologically ordered ones while preserving logical information. The corresponding Liouvillian is made up of four-local Lindblad operators. For a qubit lattice of size $L\times L$, we show that this process prepares encoded states in time $O(L)$, which is optimal. This scaling compares favorably with known local unitary encoders for the toric code w…
We apply deep reinforcement learning techniques to design high threshold decoders for the toric code under uncorrelated noise. By rewarding the agent only if the decoding procedure preserves the logical states of the toric code, and using deep convolutional networks for the training phase of the agent, we observe near-optimal performance for uncorrelated noise around the theoretically optimal threshold of 11%. We observe that, by and large, the agent implements a policy similar to that of minimum weight perfect matchings even though no bias towards any policy is given a priori.
Given their potential for fault-tolerant operations, topological quantum states are currently the focus of intense activity. Of particular interest are topological quantum error correction codes, such as the surface and planar stabilizer codes that are equivalent to the celebrated toric code. While every stabilizer state maps to a graph state under local Clifford operations, the graphs associated with topological stabilizer codes remain unknown. We show that the toric code graph is composed of only two kinds of subgraphs: star graphs (which encode Greenberger-Horne-Zeilinger states) and half g…
Decohered toric code under quantum damping noise and its mapping to a classical spin model
将广义振幅阻尼和压缩广义振幅阻尼通道下的退相干环面码映射到统计力学模型，并通过Pauli twirling转化为随机Pauli型错误，得到逻辑失败概率与温度和压缩的关系。
Quantum error correction beyond the toric code: dynamical systems meet encoding
本文在量子纠错背景下，受不可积经典台球不变积分曲面拓扑启发，构建了亏格大于一的表面码，显著提升了编码率与码距并具备抗噪免疫能力。
Analytically Exact Quantum Simulation of N-Body Interactions via Untunable Decentralized Hamiltonians for Implementing the Toric Code and Its Modifications
提出一种解析精确的量子模拟方法，通过迭代附加二体相互作用模拟 N 体相互作用，并应用于环面码及其变体的模拟，解决拓扑量子计算中的关键障碍。
The design of a four-dimensional toric code is explored with the goal of finding a lattice capable of implementing a logical $\mathsf{CCCZ}$ gate transversally. The established lattice is the octaplex tessellation, which is a regular tessellation of four-dimensional Euclidean space whose underlying 4-cell is the octaplex, or hyper-diamond. This differs from the conventional 4D toric code lattice, based on the hypercubic tessellation, which is symmetric with respect to logical $X$ and $Z$ and only allows for the implementation of a transversal Clifford gate. This work further develops the estab…
Non-equilibrium and equilibrium thermodynamic foundations of the 2D toric code within the SEAQT framework
本研究针对现有文献仅研究二维toric码平衡态的空白，在SEAQT框架下探究其向稳定平衡演化的动力学，描述了非平衡态下的多种信息关联，建立了非平衡动力学与信息度量的联系。
We suggest several techniques to improve the toric codes and the finite-rate generalized toric codes (quantum hypergraph-product codes) recently introduced by Tillich and Zémor. For the usual toric codes, we introduce the rotated lattices specified by two integer-valued periodicity vectors. These codes include the checkerboard codes, and the family of minimal single-qubit-encoding toric codes with block length $n=t^2+(t+1)^2$ and distance $d=2t+1$, $t=1,2,...$. We also suggest several related algebraic constructions which nearly quadruple the rate of the existing hypergraph-product codes.
Polynomial-Time Preparation of Low-Temperature Gibbs States for 2D Toric Code
提出一种多项式时间算法，通过结合局部Davies生成器的Lindblad动力学与全局跳跃算子，实现了任意温度下二维Toric Code哈密顿量Gibbs态的高效制备。
The topological color code and the toric code are two leading candidates for realizing fault-tolerant quantum computation. Here we show that the color code on a $d$-dimensional closed manifold is equivalent to multiple decoupled copies of the $d$-dimensional toric code up to local unitary transformations and adding or removing ancilla qubits. Our result not only generalizes the proven equivalence for $d=2$, but also provides an explicit recipe of how to decouple independent components of the color code, highlighting the importance of colorability in the construction of the code. Moreover, for …
We describe a computationally-efficient heuristic algorithm based on a renormalization-group procedure which aims at solving the problem of finding minimal surface given its boundary (curve) in any hypercubic lattice of dimension $D>2$. We use this algorithm to correct errors occurring in a four-dimensional variant of the toric code, having open as opposed to periodic boundaries. For a phenomenological error model which includes measurement errors we use a five-dimensional version of our algorithm, achieving a threshold of $4.35\pm0.1\%$. For this error model, this is the highest known thresho…
We demonstrate that multipartite entanglement, witnessed by the quantum Fisher information (QFI), can characterize topological quantum phase transitions in the spin-$\frac{1}{2}$ toric code model on a square lattice with external fields. We show that the QFI density of the ground state can be written in terms of the expectation values of gauge-invariant Wilson loops for different sizes of square regions and identify $\mathbb{Z}_2$ topological order by its scaling behavior. Furthermore, we use this multipartite entanglement witness to investigate thermalization and disorder-assisted stabilizati…
Single-shot error correction on toric codes with high-weight stabilizers
本文为环面码实现了基于高斯消元法的单次校验算符，在含噪测量模型下达到5.62%的阈值，优于传统多轮测量方案，但在基于门的错误模型中未发现单次阈值行为。
We explore the possibility of passive error correction in the toric code model. We first show that even coherent dynamics, stemming from spin interactions or the coupling to an external magnetic field, lead to logical errors. We then argue that Anderson localization of the defects, arising from unavoidable fluctuations of the coupling constants, provides a remedy. This protection is demonstrated using general analytical arguments that are complemented with numerical results which demonstrate that self-correcting memory can in principle be achieved in the limit of a nonzero density of identical…
A blindness property of the Min-Sum decoding for the toric code
研究toric码在最小和解码下的盲性限制，揭示局部信息传播的内在约束，并提出稳定子膨胀预处理方法以提升纠错性能。
We analyze the performance of decoders for the 2D and 4D toric code which are local by construction. The 2D decoder is a cellular automaton decoder formulated by Harrington which explicitly has a finite speed of communication and computation. For a model of independent $X$ and $Z$ errors and faulty syndrome measurements with identical probability we report a threshold of $0.133\%$ for this Harrington decoder. We implement a decoder for the 4D toric code which is based on a decoder by Hastings arXiv:1312.2546 . Incorporating a method for handling faulty syndromes we estimate a threshold of $1.5…
Surface codes are a promising method of quantum error correction and the basis of many proposed quantum computation implementations. However, their efficient decoding is still not fully explored. Recently, approaches based on machine learning techniques have been proposed by Torlai and Melko as well as Varsamopoulos et al. In these approaches, a so called high level decoder is used to post-correct an underlying decoder by correcting logical errors. A significant problem is that these methods require large amounts of training data even for relatively small code distances. The above-mentioned me…
Quantum error correction requires decoders that are both accurate and efficient. To this end, union-find decoding has emerged as a promising candidate for error correction on the surface code. In this work, we benchmark a weighted variant of the union-find decoder on the toric code under circuit-level depolarizing noise. This variant preserves the almost-linear time complexity of the original while significantly increasing the performance in the fault-tolerance setting. In this noise model, weighting the union-find decoder increases the threshold from 0.38% to 0.62%, compared to an increase fr…
Asymmetric quantum error-correcting codes are quantum codes defined over biased quantum channels: qubit-flip and phase-shift errors may have equal or different probabilities. The code construction is the Calderbank-Shor-Steane construction based on two linear codes. We present families of toric surfaces, toric codes and associated asymmetric quantum error-correcting codes.
Tailoring Bell inequalities to the qudit toric code and self testing
本文提出了一种针对$\mathbb{Z}_d$扭码的贝尔不等式构造方法，通过自测试验证三能级扭码子空间，为设备无关认证拓扑量子物质提供了新工具。
Fault-tolerant mixed boundary punctures on the toric code
本文证明环面码混合边界穿刺模型在反对称子空间中同样支持非阿贝尔统计，并验证其在两个子空间中均具有抵抗集体退相位和集体旋转噪声的容错能力，同时提出并验证了一种三体量子信息掩蔽方案。
The recent years have seen a growing interest in quantum codes in three dimensions (3D). One of the earliest proposed 3D quantum codes is the 3D toric code. It has been shown that 3D color codes can be mapped to 3D toric codes. The 3D toric code on cubic lattice is also a building block for the welded code which has highest energy barrier to date. Although well known, the performance of the 3D toric code has not been studied extensively. In this paper, we propose efficient decoding algorithms for the 3D toric code on a cubic lattice with and without boundaries and report their performance for …
In this paper we study the symmetry protected Majorana edge states for the Z2 topological order of the Wen-plaquette model and the toric-code model and calculate the dispersion of the Majorana edge states. For the system with translational symmetry, the Majorana edge states are gapless and have the nodal points at k=0 and k=pi. For the edge states of the toric-code model without translational symmetry, the edge modes become gapped.
We study Hamiltonians which have Kitaev's toric code as a ground state, and show how to construct a Hamiltonian which shares the ground space of the toric code, but which has gapless excitations with a continuous spectrum in the thermodynamic limit. Our construction is based on the framework of Projected Entangled Pair States (PEPS), and can be applied to a large class of two-dimensional systems to obtain gapless "uncle Hamiltonians".
Dynamic local single-shot checks for toric codes
本研究提出拓扑码的动态局部单次检查方法，通过约束检查权重减少测量轮次，提升解码性能，降低容错量子计算的时间开销。
In some quantum computing (QC) architectures, entanglement of an arbitrary number of qubits can be generated in a single operation. This property has many potential applications, and may specifically be useful for quantum error correction (QEC). Stabilizer measurements can then be implemented using a single multi-qubit gate instead of several two-qubit gates, thus reducing circuit depth. In this study, the toric code is used as a benchmark to compare the performance of two-qubit and five-qubit gates within parity-check circuits. We consider trapped ion qubits that are controlled via Raman tran…
Long linear codes constructed from toric varieties over finite fields, their multiplicative structure and decoding. The main theme is the inherent multiplicative structure on toric codes. The multiplicative structure allows for \emph{decoding}, resembling the decoding of Reed-Solomon codes and aligns with decoding by error correcting pairs. We have used the multiplicative structure on toric codes to construct linear secret sharing schemes with \emph{strong multiplication} via Massey's construction generalizing the Shamir Linear secret sharing shemes constructed from Reed-Solomon codes. We have…
To date, a great deal of attention has focused on characterizing the performance of quantum error correcting codes via their thresholds, the maximum correctable physical error rate for a given noise model and decoding strategy. Practical quantum computers will necessarily operate below these thresholds meaning that other performance indicators become important. In this work we consider the scaling of the logical error rate of the toric code and demonstrate how, in turn, this may be used to calculate a key performance indicator. We use a perfect matching decoding algorithm to find the scaling o…
We investigate the quantum phase diagram of the $K$-layer Ising toric code corresponding to $K$ layers of two-dimensional toric codes coupled by Ising interactions. While for small Ising interactions the system displays $\mathbb{Z}_2^K$ topological order originating from the toric codes in each layer, the system shows $\mathbb{Z}_2$ topological order in the high-Ising limit. The latter is demonstrated for general $K$ by deriving an effective low-energy model in $K^{\rm th}$-order degenerate perturbation theory, which is given as an effective anisotropic single-layer toric code in terms of coll…
PhD thesis investigating homological quantum codes derived from curved and higher dimensional geometries. In the first part we will consider closed surfaces with constant negative curvature. We show how such surfaces can be constructed and enumerate all quantum codes derived from them which have less than 10.000 physical qubits. For codes that are extremal in a certain sense we perform numerical simulations to determine the value of their threshold. Furthermore, we give evidence that these codes can be used for more overhead efficient storage as compared to the surface code by orders of magnit…
Qudit toric codes are a natural higher-dimensional generalization of the well-studied qubit toric code. However standard methods for error correction of the qubit toric code are not applicable to them. Novel decoders are needed. In this paper we introduce two renormalization group decoders for qudit codes and analyze their error correction thresholds and efficiency. The first decoder is a generalization of a "hard-decisions" decoder due to Bravyi and Haah [arXiv:1112.3252]. We modify this decoder to overcome a percolation effect which limits its threshold performance for high dimensions. The s…
Self-attention U-Net decoder for toric codes
本文提出自注意力 U-Net 量子解码器（SU-NetQD）用于环面码，在电路级噪声下优于 MWPM，获得 0.231 高阈值，结合低层与高层解码器及迁移学习实现可扩展高精度解码。
In this paper, we studied the doped Z2 topological spin liquid of the toric-code model. We found that the doped holes become supersymmetric particles. The ground state of the doped Z2 topological spin liquid becomes new matters of quantum states - supersymmetric Bose-Einstein condensation or supersymmetric superfluid. As a result, this system provides a unique example of manipulatable supersymmetric many-body system.
Self-testing the toric code against classical communication
该研究提出在量子网络中允许有限距离经典通信的放宽设置下，实现环面码码空间的噪声鲁棒自检验，并开发算法方法扩展至表面码和簇态。
Single-Shot Quantum Error Correction in Intertwined Toric Codes
构造三维交织Toric码(ITC)实现单次量子纠错，其物理起源明确、逻辑算符直观且相图简单，校验子与Kubica-Vasmer码相似并可用其解码方案，揭示了与Walker-Wang模型及测量基量子计算的联系。
We numerically study the effects of two forms of quenched disorder on the anyons of the toric code. Firstly, a new class of codes based on random lattices of stabilizer operators is presented, and shown to be superior to the standard square lattice toric code for certain forms of biased noise. It is further argued that these codes are close to optimal, in that they tightly reach the upper bound of error thresholds beyond which no correctable CSS codes can exist. Additionally, we study the classical motion of anyons in toric codes with randomly distributed onsite potentials. In the presence of …
In this work, we will show how the topological order of the Toric Code appears when the lattice on which it is defined discretizes a three-dimensional torus. In order to do this, we will present a pedagogical review of the traditional two-dimensional Toric Code, with an emphasis on how its quasiparticles are conceived and transported. With that, we want to make clear not only how all these same quasiparticle conception and transportation fit into this three-dimensional model, but to make it clear how topology controls the degeneracy of ground state in this new situation.
The toric code is known to be equivalent to free fermions. This paper presents explicit local unitary transformations that map the $\mathbb{Z}_2$ toric and surface code --- the open boundary equivalent of the toric code --- to fermions. Through this construction it is shown that the surface code can be mapped to a set of free fermion modes, while the toric code requires additional fermionic symmetry operators. Finally, it is demonstrated how the anyonic statistics of these codes are encoded in the fermionic representations.
We study the nonequilibrium dynamics of the extended toric code model (both ordered and disordered) to probe the existence of the dynamical quantum phase transitions (DQPTs). We show that in the case of the ordered toric code model, the zeros of Loschmidt overlap (generalized partition function) occur at critical times when DQPTs occur, which is confirmed by the nonanalyticities in the dynamical counter-part of the free-energy density. Moreover, we show that DQPTs occur for any non-zero field strength if the initial state is the excited state of the toric code model. In the disordered case, we…
We study quantum quenches in the two-dimensional Kitaev toric code model and compute exactly the time-dependent entanglement entropy of the non-equilibrium wave-function evolving from a paramagnetic initial state with the toric code Hamiltonian. We find that the area law survives at all times. Adding disorder to the toric code couplings makes the entanglement entropy per unit boundary length saturate to disorder-independent values at long times and in the thermodynamic limit. There are order-one corrections to the area law from the corners in the subsystem boundary but the topological entropy …
At zero temperature, a two-dimensional lattice of Majorana zero modes on mesoscopic superconducting islands exhibits a topologically-ordered toric code phase. Recently, a Landau field theory was used to describe the different phases of the aforementioned system and the phase-transitions separating them. While the field theory provides details on the properties of the system close to the phase-transitions, signatures of topological ordering in the different phases have not been computed. This is the primary goal of the current work. We describe a lattice gauge theory of the Majorana toric code …
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.
We present a generic framework for the emergence of the $\mathbb{Z}_2$ toric-code and the double-semion topological order in a wide class of hardcore Bose-Hubbard-type models governed by density-density interaction and in the strong-interaction regime. We fix fractional filling factor and study under which conditions the density-density interaction gives rise to topological degeneracy. We further specify which dynamics determines the toric-code and the double-semion topological order. Our results indicate that the specifics of the density-density interaction determine the long-range entangleme…
Characterization of $n$-Dimensional Toric and Burst-Error-Correcting Quantum Codes from Lattice Codes
本文推广量子交织方法，基于格码构造 $n\geq5$ 的新型环面量子码并导出突发错误纠正量子码，兼具良好码率与编码增益，且在 Golomb–Welch 猜想成立时具有唯一性。
In this paper we define homological stabilizer codes which encompass codes such as Kitaev's toric code and the topological color codes. These codes are defined solely by the graphs they reside on. This feature allows us to use properties of topological graph theory to determine the graphs which are suitable as homological stabilizer codes. We then show that all toric codes are equivalent to homological stabilizer codes on 4-valent graphs. We show that the topological color codes and toric codes correspond to two distinct classes of graphs. We define the notion of label set equivalencies and sh…
Quantum Ising model in a transverse field is of the simplest quantum many body systems used for studying universal properties of quantum phase transitions. Interestingly, it is well-known that such phase transitions can be mapped to topological phase transitions in Toric code models. Therefore, one can expect that well known properties of the transverse Ising model are used for characterizing topological phase transition in Toric code model. In this paper, we consider the magnetization of Ising model and show while it is a local order parameter, it is mapped to a non local order parameter in t…
Typical topological systems undergo a topological phase transition in the presence of a strong enough perturbation. In this paper, we propose an adjustable frustrated Toric code with a "topological line" at which no phase transition happens in the system and the topological order is robust against a non-linear perturbation of arbitrary strength. This important result is a consequence of the interplay between frustration and nonlinearity in our system, which also causes to the emergence of other interesting phenomena such as reentrant topological phases and survival of the topological order und…
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。
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