Some quantum algorithms have "quantum speedups": improved time complexity as compared with the best-known classical algorithms for solving the same tasks. Can we understand what fuels these speedups from an entropic perspective? Information theory gives us a multitude of metrics we might choose from to measure how fundamentally 'quantum' is the behavior of a quantum computer running an algorithm. The entanglement entropies for subsystems of a quantum state can be analyzed for subsystems of qubits in a quantum computer throughout the running of an algorithm. Here, a framework for making this entropic analysis is constructed, and performed on a selection of quantum circuits implementing known fast quantum algorithms and subroutines: Grover search, the quantum Fourier transform, and phase estimation. Our results are largely unsatisfactory: known entropy inequalities do not suffice to identify the presence or absence of quantum speedups. Although we know our algorithms must have quantum "magic", the Ingleton inequality, which holds for all entropies of subsystems of stabilizer states, is not violated in any of our examples. In some cases, however, monogamy of mutual information, which is obeyed for product states but violated for highly entangled bipartite states such as the $GHZ$ states, fails at some point in the course of our quantum circuits.

翻译：某些量子算法具有"量子加速"：相较于解决相同任务的最优经典算法，其时间复杂度有所改进。我们能否从熵的视角理解这些加速的驱动机制？信息论为我们提供了多种度量指标，可用于衡量量子计算机运行算法时行为的本质"量子性"。我们可以在算法运行过程中，分析量子计算机中量子比特子系统的量子态纠缠熵。本文构建了进行此类熵分析的框架，并对一系列实现已知快速量子算法与子程序的量子电路进行了分析：Grover搜索、量子傅里叶变换以及相位估计。我们的结果总体而言不尽如人意：已知的熵不等式不足以识别量子加速的存在与否。尽管我们确信这些算法必然具有量子"魔力"，但适用于稳定子态所有子系统熵的Ingleton不等式，在我们所有示例中均未被违反。然而在某些情况下，乘积态满足而高纠缠二分态（如$GHZ$态）会违反的互信息单调性，在我们的量子电路运行过程中的某些节点出现了失效现象。