多元函数的量子Fourier分析及其在一类Schrödinger型偏微分方程中的应用
Quantum Fourier analysis for multivariate functions and applications to a class of Schrödinger-type partial differential equations
摘要：
在这项工作中，我们开发了一个高效的表示功能和微分算子的基础上傅立叶分析。使用这种表示，我们创建了一个变分混合量子算法来解决静态，薛定谔型，哈密顿偏微分方程（PDE），使用空间有效的变分电路，包括问题的对称性，全局和基于梯度的优化。我们使用该算法通过计算三个偏微分方程中的基态来对表示技术的性能进行基准测试，即，一维量子谐振子，以及transmon和flux量子比特，研究它们在理想和近期量子计算机中的表现。利用这里开发的傅立叶方法，我们仅使用三到四个量子比特就获得了$10^{-4}-10^{-5}$阶的低不可逆性，证明了量子计算机中的信息压缩。在实际的计算机中，实际的性能受到噪声和成本函数评估误差的限制，但它们也可以通过误差缓解技术得到改善。
英文摘要：
In this work, we develop a highly efficient representation of functions and differential operators based on Fourier analysis. Using this representation, we create a variational hybrid quantum algorithm to solve static, Schrödinger-type, Hamiltonian partial differential equations (PDEs), using space-efficient variational circuits, including the symmetries of the problem, and global and gradient-based optimizers. We use this algorithm to benchmark the performance of the representation techniques by means of the computation of the ground state in three PDEs, i.e., the one-dimensional quantum harmonic oscillator, and the transmon and flux qubits, studying how they would perform in ideal and near-term quantum computers. With the Fourier methods developed here, we obtain low infidelities of order $10^{-4}-10^{-5}$ using only three to four qubits, demonstrating the high compression of information in a quantum computer. Practical fidelities are limited by the noise and the errors of the evaluation of the cost function in real computers, but they can also be improved through error mitigation techniques.
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