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, Previous versions contain revisions to introduction and non-technical content. Prior to version 3, asymptotic constants were worse and the dependency in Theorem 1 was epsilon^{1/3}. Version 5 introduced the lower bound for linear-distance LDPC stabilizer codes which was not in previous versions as well as additional technical appendices

The No Low-energy Trivial States (NLTS) conjecture of Freedman and Hastings, 2014 -- which posits the existence of a local Hamiltonian with a super-constant quantum 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 or nearly linear-distance LDPC stabilizer codes, we prove super-constant circuit lower bounds for the complexity of all states of energy o(n). 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 even in physically relevant systems.

翻译：Freedman 和 Hastings 的低能三维国家(NLTS) 预测,2014年, Freedman 和 Hastings 的低能三维国家(NLTS), 假设当地有一个汉密尔顿人的存在, 其超常量量子电路在所有低能国家的复杂度上限制较低), 确定了解决五氯苯酚量子猜想的根本障碍。在这项工作中,我们提供了基于昆虫和本地不可分性论点的新技术, 证明由量子错误校正代码产生的当地汉密尔顿人所有低能国家的电路线性下限。对于由近线性率或近线性距离LDPC 稳定码产生的本地汉密尔顿人来说, 我们证明超量量子电路线线下线线线线是所有能源状态复杂性的较低界限。这些代码已知存在,但不一定是本地测试的, 一种原先被怀疑是NLTS 诱导所必不可少的财产。奇怪的是, 这种代码也可以用二维的拉蒂来构建,,, 表明低深度状态甚至连在与物理有关的系统中也无法准确估计地面能源。