We provide first evidence that under certain conditions, 1/2-spin fermions
may naturally behave like a Grover search, looking for topological defects in a
material. The theoretical framework is that of discrete-time quantum walks
(QW), i.e. local unitary matrices that drive the evolution of a single particle
on the lattice. Some QW are well-known to recover the $(2+1)$--dimensional
Dirac equation in continuum limit, i.e. the free propagation of the 1/2-spin
fermion. We study two such Dirac QW, one on the square grid and the other on a
triangular grid reminiscent of graphene-like materials. The numerical
simulations show that the walker localises around the defects in $O(\sqrt{N})$
steps with probability $O(1/\log{N})$, in line with previous QW search on the
grid. The main advantage brought by those of this paper is that they could be
implemented as `naturally occurring' freely propagating particles over a
surface featuring topological---without the need for a specific oracle step.
From a quantum computing perspective, however, this hints at novel applications
of QW search : instead of using them to look for `good' solutions within the
configuration space of a problem, we could use them to look for topological
properties of the entire configuration space.

本研究提供了第一份证据表明，在特定条件下，1/2 自旋费米子可以自然地表现出 Grover 搜索，在材料中寻找拓扑缺陷。作者还研究了两个 Dirac 量子行走模型，并展示了与先前的量子行走搜索一致的步数复杂度。

Grover 搜索作为自然现象

The Grover search as a naturally occurring phenomenon

Machine learning is used to approximate density functionals. For the model
problem of the kinetic energy of non-interacting fermions in 1d, mean absolute
errors below 1 kcal/mol on test densities similar to the training set are
reached with fewer than 100 training densities. A predictor identifies if a
test density is within the interpolation region. Via principal component
analysis, a projected functional derivative finds highly accurate
self-consistent densities. Challenges for application of our method to real
electronic structure problems are discussed.

运用机器学习方法逼近密度泛函，在 1D 非相互作用费米子运动能量模型中，仅需少于 100 个训练密度即可达到与测试密度相似的 1kcal /mol 以下的平均绝对误差，利用主成分分析，还可找到高精度自洽密度。文章探讨了将该方法应用于真实电子结构问题面临的挑战。