非稳定子性的更快计算
Faster computation of nonstabilizerness
AI总结：
本文提出基于列生成方法和稳定子保真度快速子程序的更高效数值算法，显著降低计算稳定子幅度的时间与空间复杂度，使Haar随机纯态在最多10量子比特下可计算。
摘要：
非稳定子性的刻画因其在门合成和经典模拟中的应用而富有成效。特别是，被称为稳定子幅度的资源单调量是使用基于秩的模拟器估计模拟成本的有用工具，而基于秩的模拟器是Clifford+$T$电路最先进的模拟器之一。在这项工作中，我们提出了更快的数值算法来计算稳定子幅度。我们的算法利用列生成方法，该方法迭代地更新用于计算的纯稳定子态子集。该子集是基于所有稳定子态与目标态之间的重叠来选择的。为了更新该子集，我们利用了一个新提出的用于计算稳定子保真度的子程序，该子程序（i）相对于稳定子态的数量实现了线性时间复杂度，（ii）通过就地计算超指数地降低了空间复杂度，并且（iii）剪枝了计算中不必要的态。因此，我们的算法可以计算多达$n=9$个量子比特的Haar随机纯态的稳定子保真度和稳定子幅度，而朴素方法需要305 EiB的内存。我们进一步表明，当目标态向量为实数时，我们的算法运行得更快。我们证明了与一般情况相比，问题规模减少了$\mathcal{O}(2^n)$，这使得$n=10$个量子比特的情况可以计算。
英文摘要：
The characterization of nonstabilizerness is fruitful due to its application in gate synthesis and classical simulation. In particular, the resource monotone called the stabilizer extent is a useful tool to estimate the simulation cost using rank-based simulators, one of the state-of-the-art simulators of Clifford+$T$ circuits. In this work, we propose faster numerical algorithms to compute the stabilizer extent. Our algorithm utilizes the Column Generation method, which iteratively updates the subset of pure stabilizer states used for calculation. This subset is selected based on the overlaps between all stabilizer states and a target state. In order to update the subset, we make use of a newly proposed subroutine for calculating the stabilizer fidelity that (i) achieves linear time complexity with respect to the number of stabilizer states, (ii) super-exponentially reduces the space complexity by in-place calculation, and (iii) prunes unnecessary states for the computation. As a result, our algorithm can compute the stabilizer fidelity and the stabilizer extent for Haar random pure states up to $n=9$ qubits, which naively requires a memory of 305 EiB. We further show that our algorithm runs faster when the target state vector is real. We prove that the problem size is reduced by $\mathcal{O}(2^n)$ compared to the general cases, which makes it computable for the case of $n=10$ qubits.
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