The ability to efficiently load functions on quantum computers with high fidelity is essential for many quantum algorithms, including those for solving partial differential equations and Monte Carlo estimation. In this work, we introduce the Fourier series loader (FSL) method for preparing quantum states that exactly encode multi-dimensional Fourier series using linear-depth quantum circuits. Specifically, the FSL method prepares a (Dn)-qubit state encoding the 2 ^Dn -point uniform discretization of a D-dimensional function specified by a D-dimensional Fourier series. A free parameter, m, which must be less than n, determines the number of Fourier coefficients, 2D(m+1) , used to represent the function. The FSL method uses a quantum circuit of depth at most 2(n−2)+⌈log2(n−m)⌉+2D(m+1)+2−2D(m+1) , which is linear in the number of Fourier coefficients, and linear in the number of qubits (Dn) despite the fact that the loaded function’s discretization is over exponentially many (2 ^Dn ) points. The FSL circuit consists of at most Dn+2D(m+1)+1−1 single-qubit and Dn(n+1)/2+2D(m+1)+1−3D(m+1)−2 two-qubit gates; we present a classical compilation algorithm with runtime O(23D(m+1)) to determine the FSL circuit for a given Fourier series. The FSL method allows for the highly accurate loading of complex-valued functions that are well-approximated by a Fourier series with finitely many terms. We report results from noiseless quantum circuit simulations, illustrating the capability of the FSL method to load various continuous 1D functions, and a discontinuous 1D function, on 20 qubits with infidelities of less than 10⁻⁶ and 10⁻³, respectively. We also demonstrate the practicality of the FSL method for near-term quantum computers by presenting experiments performed on the Quantinuum H1-1 and H1-2 trapped-ion quantum computers: we loaded a complex-valued function on 3 qubits with a fidelity of over 95% , as well as various 1D real-valued functions on up to 6 qubits with classical fidelities ≈99%, and a 2D function on 10 qubits with a classical fidelity ≈94%.

中文翻译：

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用于加载任意函数的傅立叶近似的线性深度量子电路


在量子计算机上以高保真度高效加载函数的能力对于许多量子算法至关重要，包括求解偏微分方程和蒙特卡罗估计的算法。在这项工作中，我们介绍了傅里叶级数加载器（FSL）方法，用于准备使用线性深度量子电路精确编码多维傅里叶级数的量子态。具体地，FSL方法准备对由D维傅立叶级数指定的D维函数的2 ^Dn 点均匀离散化进行编码的(Dn)-量子位状态。自由参数 m（必须小于 n）决定用于表示函数的傅里叶系数 2D(m+1) 的数量。 FSL方法使用深度最多为2(n−2)+⌈log2(n−m)⌉+2D(m+1)+2−2D(m+1)的量子电路，其数量与傅立叶系数，并且与量子比特数 (Dn) 呈线性关系，尽管加载函数的离散化超过了指数级多个 (2 ^Dn ) 点。 FSL电路最多由Dn+2D(m+1)+1−1个单量子位和Dn(n+1)/2+2D(m+1)+1−3D(m+1)−2两个量子位组成-量子位门；我们提出了一种运行时间为 O(23D(m+1)) 的经典编译算法来确定给定傅立叶级数的 FSL 电路。 FSL 方法可以高精度加载复值函数，这些函数可以通过有限多项的傅里叶级数很好地逼近。我们报告了无噪声量子电路模拟的结果，说明了 FSL 方法在 20 个量子位上加载各种连续一维函数和不连续一维函数的能力，不保真度小于 10 ⁻⁶ 和 10 ⁻³