The quantum PCP conjecture asks whether it is QMA-hard to distinguish between high- and low-energy Hamiltonians even when the gap between "high" and "low" energy is large (constant). A natural proof strategy is gap amplification: start from the fact that high- and low-energy Hamiltonians are hard to distinguish if the gap is small (inverse polynomial) and amplify the Hamiltonians to increase the energy gap while preserving hardness. Such a gap amplification procedure is at the heart of Dinur's proof of the classical PCP theorem. In this work, following Dinur's model, we introduce a new quantum gap amplification procedure for Hamiltonians which uses random walks on expander graphs to derandomise (subsample the terms of) the tensor product amplification of a Hamiltonian. Curiously, our analysis relies on a new technique inspired by quantum de Finetti theorems, which have previously been used to rule out certain approaches to the quantum PCP conjecture.

翻译：量子PCP猜想探讨了即使"高"能与"低"能之间的间隙较大（常数级）时，区分高能与低能哈密顿量是否仍属于QMA困难问题。一种自然的证明策略是间隙放大：从"小间隙（逆多项式级）时高能与低能哈密顿量难以区分"这一事实出发，通过放大哈密顿量来增加能隙同时保持问题难度。此类间隙放大程序是Dinur证明经典PCP定理的核心方法。本工作中，我们遵循Dinur的模型，提出了一种新的量子哈密顿量间隙放大程序，该程序利用扩展图上的随机游走来对哈密顿量的张量积放大进行去随机化（即对哈密顿量项进行子采样）。值得注意的是，我们的分析依赖于一种受量子德菲内蒂定理启发的新技术，而该定理此前曾被用于排除解决量子PCP猜想的某些途径。