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.

中文翻译：

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量子代码哈密顿量低能态的电路下界

Freedman 和 Hastings 的 No Low-energy Trivial States (NLTS) 猜想（Quantum Information and Computation，2014 年）——它假定存在一个局部哈密顿量，其具有所有低能态复杂性的超常数电路下界-- 确定了解决量子 PCP 猜想的根本障碍。在这项工作中，我们提供了基于熵和局部不可区分性论证的新技术，这些技术证明了由量子纠错码引起的局部哈密顿量的所有低能态的电路下界。对于由近线性速率和多项式距离 LDPC 稳定器代码产生的局部哈密顿量，我们证明了所有能量状态复杂性的超常数电路下界 $o(n)$（可以看作是一个几乎线性的 NLTS 定理）。已知此类代码存在且不一定可在本地进行测试，这是先前怀疑对 NLTS 猜想至关重要的属性。奇怪的是，这样的代码也可以构建在二维晶格上，表明低深度状态不能准确地近似物理相关系统中的地面能量。