量子代码汉密尔顿人低能状态的电路下下线 (Circuit lower bounds for low-energy states of quantum code Hamiltonians)

from arxiv, Version 2 updates funding sources in the acknowledgments and slightly adjusts the title and introduction. No technical results were modified between Version 1 and Version 2. Version 3 fixes an asymptotic constant in the main theorem and improves the coefficient of epsilon from 1/3 to 1 with logarithmic factors. Version 4 updates the introduction and has no technical result modifications

The No Low-energy Trivial States (NLTS) conjecture of Freedman and Hastings (Quantum Information and Computation, 2014) -- which posits the existence of a local Hamiltonian with a super-constant circuit lower bound on the complexity of all low-energy states -- identifies a fundamental obstacle to the resolution of the quantum PCP conjecture. In this work, we provide new techniques, based on entropic and local indistinguishability arguments, that prove circuit lower bounds for all the low-energy states of local Hamiltonians arising from quantum error-correcting codes. For local Hamiltonians arising from nearly linear-rate and polynomial-distance LDPC stabilizer codes, we prove super-constant circuit lower bounds for the complexity of all states of energy $o(n)$ (which can be viewed as an almost linear NLTS theorem). Such codes are known to exist and are not necessarily locally-testable, a property previously suspected to be essential for the NLTS conjecture. Curiously, such codes can also be constructed on a two-dimensional lattice, showing that low-depth states cannot accurately approximate the ground-energy in physically relevant systems.

翻译：Freedman 和 Hastings (量子信息与计算,2014年) 的无低能三维国家(NLTS) 预测 Freedman 和 Hastings (量子信息与计算,2014年) -- -- 其中假设存在一个本地的汉密尔顿人,其超级通量电路比所有低能州复杂程度低的超通量电路约束所有低能国家的复杂性 -- -- 发现了解决五氯苯酚量子预测的根本障碍。在这项工作中,我们根据昆士兰和地方不可分性论点提供新技术,这些新技术证明当地汉密尔顿人所有低能州由于量子误差校准代码而处于较低的通量范围。对于来自近线性速和多线距离LDPC稳定器的本地汉密尔密尔顿人来说,我们证明所有能源州复杂性超通量电路程较低(美元(可视为几乎线性NLTS 理论) 。此类代码已知存在,而且不一定可在当地进行测试,而以前怀疑这种财产对于NLTS 的测算至关重要。值得怀疑,对于量的州来说,对于近二维基地能系统来说,这种代码也可以在物理深度上构造上构建。