On the complexity of Commuting Local Hamiltonians, and tight conditions for Topological Order in such systems
Dorit Aharonov, Lior Eldar
备注：
A revised version with a few changes: Title and abstract changed to highlight the transition from local to global entanglement; A remark about the connection to the quantum PCP conjecture was added in the introduction, Section 5.8.3 revised and a figure added, to correct a small error in the first version
The local Hamiltonian problem plays the equivalent role of SAT in quantum complexity theory. Understanding the complexity of the intermediate case in which the constraints are quantum but all local terms in the Hamiltonian commute, is of importance for conceptual, physical and computational complexity reasons. Bravyi and Vyalyi showed in 2003, using a clever application of the representation theory of C*-algebras, that if the terms in the Hamiltonian are all two-local, the problem is in NP, and the entanglement in the ground states is local. The general case remained open since then. In this paper we extend the results of Bravyi and Vyalyi beyond the two-local case, to the case of three-qubit interactions. We then extend our results even further, and show that NP verification is possible for three-wise interaction between qutrits as well, as long as the interaction graph is embedded on a planar lattice, or more generally, "Nearly Euclidean" (NE). The proofs imply that in all such systems, the entanglement in the ground states is local. These extensions imply an intriguing sharp transition phenomenon in commuting Hamiltonian systems: 3-local NE systems based on qubits and qutrits cannot be used to construct Topological order, as their entanglement is local, whereas for higher dimensional qudits, or for interactions of at least 4 qudits, Topological Order is already possible, via Kitaev's Toric Code construction. We thus conclude that Kitaev's Toric Code construction is optimal for deriving topological order based on commuting Hamiltonians.
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University of Cambridge（剑桥大学）；Universität Tübingen（蒂宾根大学）；Universidad Complutense de Madrid（马德里康普顿斯大学）；Instituto de Ciencias Matemáticas (ICMAT)（数学科学研究所）；Instituto de Física Teórica UAM/CSIC（UAM/CSIC理论物理研究所）
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Max-Planck-Institute of Quantum Optics（马克斯·普朗克量子光学研究所）；Inria, Télécom Paris - LTCI, Institut Polytechnique de Paris（法国国家信息与自动化研究所，巴黎高等电信学校-通信与图像处理实验室，巴黎理工学院）；Instituto de Física Teórica UAM/CSIC（马德里自治大学/西班牙国家研究委员会理论物理研究所）；Fachbereich Mathematik, Eberhard Karls Universität Tübingen（蒂宾根大学数学系）；Department of Applied Mathematics and Theoretical Physics, University of Cambridge（剑桥大学应用数学和理论物理系）；Department of Mathematics, Technische Universität München（慕尼黑工业大学数学系）
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School of Engineering and Applied Sciences, Harvard University（哈佛大学工程与应用科学学院）；Department of Computing and Mathematical Sciences, California Institute of Technology（加州理工学院计算与数学科学系）；Simons Institute for the Theory of Computing, University of California, Berkeley（加州大学伯克利分校西蒙斯计算理论研究所）
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