arXiv 2303.070322023-03quant-phphysics.app-phphysics.comp-phphysics.optics

利用量子计算机的高效光传播算法

Efficient Light Propagation Algorithm using Quantum Computers

Chanaprom Cholsuk, Siavash Davani, Lorcan O. Conlon, Tobias Vogl, Falk Eilenberger

备注：: 8 pages, 7 figures
发表信息：: Phys. Scr. 99, 045110 (2024)
DOI：: 10.1088/1402-4896/ad2d4c

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AI总结：

本文提出量子光束传播方法（QBPM），将经典需 O(N log N) 步的光传播计算降至 O((log N)^2) 个单受控相位门，并在一维和二维系统中验证了指数级加速优势。

摘要：

量子算法有潜力突破计算困难问题的边界。现代光学中的基石之一是光束传播算法，它有助于计算具有特定色散关系的波如何在时间和空间中传播。该算法通过傅里叶变换、与传递函数相乘以及随后的逆变换来求解波动传播方程。该传递函数由相应的色散关系确定，而色散关系通常可以展开为多项式。在自由空间中的傍轴波传播或皮秒脉冲传播情形下，这种展开可以在二次项之后截断。波动传播的经典求解需要 $\mathcal{O}(N log N)$ 个计算步骤，其中 $N$ 是波函数被离散化的点数。在此，我们证明该传播可以作为量子算法执行，仅需 $\mathcal{O}((log{}N)^2)$ 个单受控相位门，这表明计算复杂度呈指数级降低。我们在本文中展示了这种量子光束传播方法（QBPM），并针对双缝实验和高斯光束传播在一维和二维系统中执行了此类传播。我们强调了选择合适可观测量的重要性，以便在面对量子测量过程的统计特性时保持量子优势，因为这种统计特性会导致经典解中不存在的采样误差。

英文摘要：

Quantum algorithms can potentially overcome the boundary of computationally hard problems. One of the cornerstones in modern optics is the beam propagation algorithm, facilitating the calculation of how waves with a particular dispersion relation propagate in time and space. This algorithm solves the wave propagation equation by Fourier transformation, multiplication with a transfer function, and subsequent back transformation. This transfer function is determined from the respective dispersion relation, which can often be expanded as a polynomial. In the case of paraxial wave propagation in free space or picosecond pulse propagation, this expansion can be truncated after the quadratic term. The classical solution to the wave propagation requires $\mathcal{O}(N log N)$ computation steps, where $N$ is the number of points into which the wave function is discretized. Here, we show that the propagation can be performed as a quantum algorithm with $\mathcal{O}((log{}N)^2)$ single-controlled phase gates, indicating exponentially reduced computational complexity. We herein demonstrate this quantum beam propagation method (QBPM) and perform such propagation in both one- and two-dimensional systems for the double-slit experiment and Gaussian beam propagation. We highlight the importance of the selection of suitable observables to retain the quantum advantage in the face of the statistical nature of the quantum measurement process, which leads to sampling errors that do not exist in classical solutions.

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