QREM的多重分形动力学
Multifractal Dynamics of the QREM
摘要：
本文数值研究了在量子随机能量模型下，系统尺寸为$n\leq 20$量子自旋时的布居转移协议及其与量子计算的关系。我们专注于能量匹配问题，即找到多个近似解的组合优化问题时，提供一个已知的近似解作为输入的一部分。我们研究的离域过程引起的人口转移协议通过观察饱和的香农熵的时间演化的波函数作为衡量其蔓延的系统。在饱和状态下的熵值与系统体积的比例确定了模型的三个已知动力学阶段。在非遍历的扩展阶段，我们观察到人口转移完成所需的时间遵循长尾分布。我们设计了两个统计数据来量化协议如何有效和均匀地填充目标能量壳。我们发现，如果选择的横向场参数$Γ$接近安德森转变的临界点的模型，人口转移是最有效的。为了评估人口转移作为量子算法的使用，我们与随机搜索进行了比较。我们检测到一个“黑箱”的优势，有利于PT，但当考虑到人口转移和随机搜索的运行时间，我们没有看到强烈的迹象表明，在系统的大小，我们的数值方法可以访问的加速。我们讨论了这些结果和人口转移对NISQ设备的影响。
英文摘要：
We study numerically the population transfer protocol on the Quantum Random Energy Model and its relation to quantum computing, for system sizes of $n\leq 20$ quantum spins. We focus on the energy matching problem, i.e. finding multiple approximate solutions to a combinatorial optimization problem when a known approximate solution is provided as part of the input. We study the delocalization process induced by the population transfer protocol by observing the saturation of the Shannon entropy of the time-evolved wavefunction as a measure of its spread over the system. The scaling of the value of this entropy at saturation with the volume of the system identifies the three known dynamical phases of the model. In the non-ergodic extended phase, we observe that the time necessary for the population transfer to complete follows a long-tailed distribution. We devise two statistics to quantify how effectively and uniformly the protocol populates the target energy shell. We find that population transfer is most effective if the transverse-field parameter $Γ$ is chosen close to the critical point of the Anderson transition of the model. In order to assess the use of population transfer as a quantum algorithm we perform a comparison with random search. We detect a "black box" advantage in favour of PT, but when the running times of population transfer and random search are taken into consideration we do not see strong indications of a speedup at the system sizes that are accessible to our numerical methods. We discuss these results and the impact of population transfer on NISQ devices.
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