In an important recent development, Anshu, Breuckmann, and Nirkhe [3] resolved positively the so-called No Low-Energy Trivial State (NLTS) conjecture by Freedman and Hastings. The conjecture postulated the existence of linear-size local Hamiltonians on n qubit systems for which no near-ground state can be prepared by a shallow (sublogarithmic depth) circuit. The construction in [3] is based on recently developed good quantum codes. Earlier results in this direction included the constructions of the so-called Combinatorial NLTS – a weaker version of NLTS – where a state is defined to have low energy if it violates at most a vanishing fraction of the Hamiltonian terms [2]. These constructions were also based on codes.
In this paper we provide a "non-code" construction of a class of Hamiltonians satisfying the Combinatorial NLTS. The construction is inspired by one in [2], but our proof uses the complex solution space geometry of random K-SAT instead of properties of codes. Specifically, it is known that above a certain clause-to-variables density the set of satisfying assignments of random K-SAT exhibits an overlap gap property, which implies that it can be partitioned into exponentially many clusters each constituting at most an exponentially small fraction of the total set of satisfying solutions. We establish a certain robust version of this clustering property for the space of near-satisfying assignments and show that for our constructed Hamiltonians every combinatorial near-ground state induces a near-uniform distribution supported by this set. Standard arguments then are used to show that such distributions cannot be prepared by quantum circuits with depth o(log n). Since the clustering property is exhibited by many random structures, including proper coloring and maximum cut, we anticipate that our approach is extendable to these models as well.

中文翻译：

---

来自 Overlap Gap 属性的组合 NLTS


在最近的一个重要进展中，Anshu、Breuckmann 和 Nirkhe [3] 积极解决了 Freedman 和 Hastings 提出的所谓的无低能琐碎态 （NLTS） 猜想。该猜想假设 n 个量子比特系统上存在线性大小的局部哈密顿量，而浅层（亚对数深度）电路无法为其准备近基状态。[3] 中的构造基于最近开发的良好量子代码。这个方向的早期结果包括所谓的组合 NLTS 的构造——NLTS 的弱版本——其中，如果一个状态最多违反哈密顿项的消失部分，则定义为具有低能量 [2]。这些结构也是基于代码的。

在本文中，我们提供了一类满足组合 NLTS 的哈密顿量的 “非代码” 结构。该结构的灵感来自 [2] 中的一个，但我们的证明使用了随机 K-SAT 的复杂解空间几何而不是代码的属性。具体来说，众所周知，在某个子句到变量密度之上，随机 K-SAT 的满足分配集表现出重叠间隙特性，这意味着它可以被划分为指数级多个集群，每个集群最多构成满足解总集的指数级小部分。我们为近乎令人满意的赋值空间建立了这个聚类属性的某个稳健版本，并表明对于我们构建的哈密顿量，每个组合近基状态都会诱导由该集合支持的近乎均匀的分布。然后，使用标准参数来表明这种分布不能由深度为 o（log n） 的量子电路准备。由于聚类特性由许多随机结构表现出来，包括适当的着色和最大切割，我们预计我们的方法也可以扩展到这些模型。