The boundary operator is a linear operator that acts on a collection of high-dimensional binary points (simplices) and maps them to their boundaries. This boundary map is one of the key components in numerous applications, including differential equations, machine learning, computational geometry, machine vision and control systems. We consider the problem of representing the full boundary operator on a quantum computer. We first prove that the boundary operator has a special structure in the form of a complete sum of fermionic creation and annihilation operators. We then use the fact that these operators pairwise anticommute to produce an $\mathcal{O}(n)$-depth circuit that exactly implements the boundary operator without any Trotterization or Taylor series approximation errors. Having fewer errors reduces the number of shots required to obtain desired accuracies.
翻译:边界操作员是一个线性操作员,负责收集高维二元点(简易)并将其绘制到边界线性操作员。该边界地图是多种应用中的关键组成部分之一,包括差异方程、机器学习、计算几何、机器视觉和控制系统。我们考虑在量子计算机上代表全部边界操作员的问题。我们首先证明边界操作员有一个特殊结构,其形式是完全组合的发酵和毁灭操作员。然后我们用这些操作员配对反comute来制作一个$\mathcal{O}(n)$-深度电路,该电路完全在没有任何Trotter化或Taylor系列近似错误的情况下执行边界操作员。差少一些差错就会减少获得所需加速度所需的射击次数。