基于霍克尼（Hockney）方法求解自由场条件下泊松方程的量子算法
A Quantum Algorithm for Solving the Poisson Equation for Free Field Conditions via the Hockney Method
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中文总结 AI 辅助
本研究提出一种基于QFT和Hockney方法的量子算法，可求解自由场及周期边界条件下的泊松方程，对比两种对角矩阵相乘实现方案的性能，验证了算法功能并分析了资源需求。
中文摘要 AI 辅助
针对常见的泊松方程问题，本研究提出了一种基于量子傅里叶变换（QFT）的量子算法，可求解周期边界条件及自由场条件下的泊松方程，其中自由场条件通过霍克尼（Hockney）方法实现。除QFT和用于振幅编码的初始化过程外，该算法仅使用一种将态矢量与对角矩阵相乘的过程（针对振幅编码）。对于该相乘过程，本文考虑了两种替代实现方案：第一种是线性组合单位（LCU）方法的变体，第二种是多控制旋转门序列，该序列将相乘值分解为绝对值和复相位因子。通过将一维和二维测试例的态矢量模拟结果与解析解进行比较，验证了该算法的功能。对于所考虑的测试例，发现LCU版本获得所需辅助量子比特子空间的成功概率约为多控制旋转门序列版本的两倍；但LCU版本所需的辅助量子比特数量最多等于振幅编码中存储泊松方程离散源项的量子比特数，而多控制旋转门序列版本仅需要1个辅助量子比特。此外，两种变体的成功概率计算表明，对于特定问题，随着分辨率提高，成功概率会收敛。关于量子算法所需的计算资源，结论是：尽管QFT比其经典对应过程更高效，但算法中其他必要步骤的当前实现降低了运行时效率。
英文摘要
For the often encountered problem of the Poisson equation, this work presents a quantum algorithm solving it based on the quantum Fourier transform (QFT) for periodic boundary conditions as well as free field conditions, where the latter is realized via the Hockney method. Besides the QFT and an initialization procedure for amplitude encoding, the algorithm just uses a procedure for multiplying the state vector by a diagonal matrix w.r.t. amplitude encoding. For the latter, two alternative implementations are considered here. The first variant is a version of the LCU method and the second is a sequence of multi-controlled rotation gates that represents a factoring of the multiplied values into absolute values and complex phase factors. The functionality of the algorithm is verified via comparing the results obtained from state vector simulations for one- and two-dimensional test examples with their analytical solutions. For the considered test examples, it is found that the success probability for obtaining the desired ancilla qubit subspace in the LCU version is a factor of around two higher than that for the sequence of multi-controlled rotation gates. However, the LCU version requires a number of ancilla qubits up to the number of qubits that is set to store the discretized source term of the Poisson equation in amplitude encoding, whereas the sequence of multi-controlled rotation gates demands only one ancilla qubit. Computations of the success probabilities for both variants furthermore indicate that the success probability converges for a specific problem with increasing resolution. Concerning the required computational resources for the quantum algorithm, the conclusion is drawn that while the QFT is a more efficient procedure than its classical counterpart, the current implementations of the other necessary steps in the algorithm diminish the efficiency w.r.t. the runtime.
