可检测引理与量子能隙放大
The Detectability Lemma and Quantum Gap Amplification
摘要：
约束满足问题的量子模拟是局部哈密顿量的总和-每个局部哈密顿量指定一个局部约束，其违反有助于给定量子态的能量。形式化哈密尔顿算子的基础（最小）能量与违反约束的最小数量之间的直观联系是有问题的，因为当哈密尔顿算子中的项不对易时，被违反的约束的数量没有很好地定义。本文证明的可检测性引理恰恰提供了这样一个定量的联系。我们应用引理，以获得一个基本的原始在经典复杂性的量子模拟：放大概率随机游走扩展图。在局部哈密顿量的相互作用图是展开子的条件下，它成立。我们的证明是基于一个新的结构强加在希尔伯特空间，我们称之为$XY$分解，这使得减少从量子非对易的情况下，对易的情况下（许多经典的论点通过）。这些结果可能会产生一些有趣的影响。首先，证明PCP定理的量子模拟是量子复杂性理论中最重要的挑战之一。我们的量子能隙放大引理可以看作是Dinur PCP证明中三个主要步骤中第一个步骤的量子模拟。量子隙放大也可以与光谱隙放大相关，并且特别地，与绝热计算的容错相关。最后，检测引理，和$XY$分解提供了一个处理的局部哈密顿和他们的基态的结构。
英文摘要：
The quantum analogue of a constraint satisfaction problem is a sum of local Hamiltonians - each local Hamiltonian specifies a local constraint whose violation contributes to the energy of the given quantum state. Formalizing the intuitive connection between the ground (minimal) energy of the Hamiltonian and the minimum number of violated constraints is problematic, since the number of constraints being violated is not well defined when the terms in the Hamiltonian do not commute. The detectability lemma proved in this paper provides precisely such a quantitative connection. We apply the lemma to derive a quantum analogue of a basic primitive in classical complexity: amplification of probabilities by random walks on expander graphs. It holds under the restriction that the interaction graph of the local Hamiltonian is an expander. Our proofs are based on a novel structure imposed on the Hilbert space that we call the $XY$ decomposition, which enables a reduction from the quantum non-commuting case to the commuting case (where many classical arguments go through). The results may have several interesting implications. First, proving a quantum analogue to the PCP theorem is one of the most important challenges in quantum complexity theory. Our quantum gap amplification lemma may be viewed as the quantum analogue of the first of the three main steps in Dinur's PCP proof. Quantum gap amplification may also be related to spectral gap amplification, and in particular, to fault tolerance of adiabatic computation. Finally, the detectability lemma, and the $XY$ decomposition provide a handle on the structure of local Hamiltonians and their ground states.
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