S. B. Bravyi, A. Yu. Kitaev
A new type of local-check additive quantum code is presented. Qubits are associated with edges of a 2-dimensional lattice whereas the stabilizer operators correspond to the faces and the vertices. The boundary of the lattice consists of alternating pieces with two different types of boundary conditions. Logical operators are described in terms of relative homology groups.
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The periodic boundary conditions changed the plane square-lattice Ising model to the torus-lattice system which restricts the spin-projection orientations. Only two of the three important spin-projection orientations, parallel to the x-axis or to the y-axis, are suited to the torus-lattice system. The infinitesimal difference of the free-energies of the systems between the two systems mentioned above makes their critical temperatures infinitely close to each other, but their topological fundamental groups are distinct.
We demonstrate how to use lattice surgery to enact a universal set of fault-tolerant quantum operations with color codes. Along the way, we also improve existing surface-code lattice-surgery methods. Lattice-surgery methods use fewer qubits and the same time or less than associated defect-braiding methods. Furthermore, per code distance, color-code lattice surgery uses approximately half the qubits and the same time or less than surface-code lattice surgery. Color-code lattice surgery can also implement the Hadamard and phase gates in a single transversal step---much faster than surface-code l…
Boundary-induced Phases in the Dissipative Dicke Lattice Model
本文研究少量格点耗散Dicke晶格模型，发现开放边界条件诱导出破缺平移对称性的超辐射相，而无限系统无此现象，凸显边界效应对稳态相的关键影响。
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…
We continue our analysis of the physics of quantum lattice gas automata (QLGA). Previous work has been restricted to periodic or infinite lattices; simulation of more realistic physical situations requires finite sizes and non-periodic boundary conditions. Furthermore, envisioning a QLGA as a nanoscale computer architecture motivates consideration of inhomogeneities in the `substrate'; this translates into inhomogeneities in the local evolution rules. Concentrating on the one particle sector of the model, we determine the various boundary conditions and rule inhomogeneities which are consisten…
We examine general Gottesman-Kitaev-Preskill (GKP) codes for continuous-variable quantum error correction, including concatenated GKP codes, through the lens of lattice theory, in order to better understand the structure of this class of stabilizer codes. We derive formal bounds on code parameters, show how different decoding strategies are precisely related, propose new ways to obtain GKP codes by means of glued lattices and the tensor product of lattices and point to natural resource savings that have remained hidden in recent approaches. We present general results that we illustrate through…
Fault-tolerant quantum computation relies on scaling up quantum error correcting codes in order to suppress the error rate on the encoded quantum states. Topological codes, such as the surface code or color codes are leading candidates for practical scalable quantum error correction and require efficient and scalable decoders. In this work, we propose and study the efficiency of a decoder for 2D topological color codes on the 4.8.8 lattice, by building on the work of [1] for color codes on hexagonal lattices. The decoder is based on a rescaling approach, in which syndrome information on a part…
Efficient and Expressive Boundary Conditions in Quantum Lattice Boltzmann Methods
针对量子格子玻尔兹曼方法中边界条件效率与表达力有限的问题，提出一种对整个边界施加单一连贯操作的新方法，在反弹和镜面反射边界条件下渐近和实际资源需求更少。
Boundary-Driven Exceptional Points in Photonic Waveguide Lattices
该研究通过侧耦合缺陷在无限长光子波导晶格中诱导边界驱动的异常点，揭示了非马尔可夫记忆核对共振轨迹和共扼条件的影响，为探索记忆增强的非Hermitian物理提供了实验平台。
We study the error threshold of topological color codes on Union Jack lattices that allow for the full implementation of the whole Clifford group of quantum gates. After mapping the error-correction process onto a statistical mechanical random 3-body Ising model on a Union Jack lattice, we compute its phase diagram in the temperature-disorder plane using Monte Carlo simulations. Surprisingly, topological color codes on Union Jack lattices have similar error stability than color codes on triangular lattices, as well as the Kitaev toric code. The enhanced computational capabilities of the topolo…
We have investigated how the dynamics of a quantum particle initially localized in the left boundary site under periodic driving can be manipulated via control of the right boundary site of a lattice array. Because of the adjustable coupling between the right boundary site and its nearest-neighbor, we can realize either coherent destruction of tunneling to coherent tunneling (CDT-CT) transition or coherent tunneling to coherent destruction of tunneling (CT-CDT) transition, by driving or moving the right boundary site while keeping the left boundary site driven by a periodically oscillating fie…
We propose a family of surface codes with general lattice structures, where the error-tolerances against bit and phase errors can be controlled asymmetrically by changing the underlying lattice geometries. The surface codes on various lattices are found to be efficient in the sense that their threshold values universally approach the quantum Gilbert-Varshamov bound. We find that the error-tolerance of surface codes depends on the connectivity of underlying lattices; the error chains on a lattice of lower connectivity are easier to correct. On the other hand, the loss-tolerance of surface codes…
Quantum Carleman Linearization of the Lattice Boltzmann Equation with Boundary Conditions
本文提出带边界条件的格子Boltzmann方程量子Carleman线性化方法，采用BGK平衡函数与反弹格式，并经矩形棱柱绕流验证其精度及加速潜力。
Holographic analysis of boundary correlation functions for the hyperbolic-lattice Ising model
本文通过全息视角，利用CTMRG方法分析双曲格子伊辛模型的边界自旋关联函数，发现其呈幂律衰减并伴有准周期振荡，且标度维度可由体关联长度与背景曲率解释，同时小键维数足以准确描述长距离行为。
Boundary-induced geometrical frustration in a square Rydberg-atom lattice
该研究在Aquila原子量子处理器上，结合张量网络模拟，探究了方形里德堡原子阵列中由晶格尺寸奇偶性诱导的边界相关几何缺陷，明确了不同失谐下的旋转不变态及缺陷特征，为理解可编程量子物质的全局有序性提供了依据。
We present a 1d lattice model that mimics the boundary of the conventional 2d quantum spin-Hall insulator (QSHI) with $U(1)$ symmetry and time-reversal $T$, satisfying $T^2 = (-1)^F$. Our construction utilizes a local tensor product Hilbert space of finite site dimension with a non-onsite symmetry action. We discuss how several signature properties of the QSHI, such as the fractional charge on $T$-domain walls and Kramers parity switching upon $π$-flux threading, are manifested in our treatment. We also present a 1d Hamiltonian whose ground state realizes the conventional Luttinger-liquid phas…
We present a generic and systematic approach for constructing D-dimensional lattice models with exactly solvable d-dimensional boundary states localized to corners, edges, hinges and surfaces. These solvable models represent a class of "sweet spots" in the space of possible tight-binding models---the exact solutions remain valid for any tight-binding parameters as long as they obey simple locality conditions that are manifest in the underlying lattice structure. Consequently, our models capture the physics of both (higher-order) topological and non-topological phases as well as the transitions…
Controlling energy spectra and skin effect via boundary conditions in non-Hermitian lattices
本文通过调节边界 hopping 振幅，研究非厄米特晶格中能量谱和皮肤效应的控制方法，揭示了边界条件对谱和局域化特性的影响。
Emergent Quantum Mechanics at the Boundary of a Local Classical Lattice Model
本文提出一个局域经典晶格模型，通过在额外空间维度中传播经典比特，使其边界动力学以任意精度涌现出由给定哈密顿量描述的量子力学，并解析估计和数值验证了模型与量子力学的偏差。
The 1D AKLT model is a paradigm of antiferromagnetism, and its ground state exhibits symmetry-protected topological order. On a 2D lattice, the AKLT model has recently gained attention because it too displays symmetry-protected topological order, and its ground state can act as a resource state for measurement-based quantum computation. While the 1D model has been shown to be gapped, it remains an open problem to prove the existence of a spectral gap on the 2D square lattice, which would guarantee the robustness of the resource state. Recently, it has been shown that one can deduce this spectr…
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