Unitary transformations are the fundamental building blocks of gates and operations in quantum information processing, allowing the complete manipulation of quantum systems in a coherent manner. In the case of photons, optical elements that can perform unitary transformations are readily available only for some degrees of freedom, e.g., wave plates for polarization. However, for high-dimensional states encoded in the transverse spatial modes of light, performing arbitrary unitary transformations remains a challenging task for both theoretical proposals and actual implementations. Following the idea of multi-plane light conversion, we show that it is possible to perform a broad variety of unitary operations at high quality by using only a few phase modulation planes. More importantly, we experimentally implement several high-dimensional quantum gates for up to five-dimensional states encoded in the full-field mode structure of photons. In particular, we realize cyclic and quantum Fourier transformations, known as Pauli $ \hat X $-gates and Hadamard $ \hat H $-gates, respectively, with an average visibility of more than 90%. In addition, we demonstrate near-perfect “unitarity” by means of quantum process tomography, unveiling a process purity of 99%. Last, we demonstrate the benefit of the two independent spatial degrees of freedom, i.e., azimuthal and radial, and implement a two-qubit controlled-NOT quantum operation on a single photon. Thus, our demonstrations open up new paths to implement high-dimensional quantum operations, which can be applied to various tasks in quantum communication, computation, and sensing schemes.

中文翻译：

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使用光子全场空间模式的高维量子门

ary变换是量子信息处理中门和运算的基本构建块，允许以连贯的方式完整地操作量子系统。在光子的情况下，可以执行单一变换的光学元件仅在某些自由度下才可用，例如，用于偏振的波片。但是，对于以光的横向空间模式编码的高维状态，执行任意proposals变换对于理论建议和实际实现而言仍然是一项艰巨的任务。遵循多平面光转换的想法，我们表明仅使用几个相位调制平面就可以以高质量执行多种单一操作。更重要的是，我们实验性地实现了几个高维量子门，用于以光子全场模式结构编码的多达五维状态。特别是，我们实现了称为Pauli的循环和量子傅立叶变换$ \ hat X $ -gates和Hadamard $ \ hat H $ -gates的平均可见性分别超过90％。此外，我们通过量子过程层析成像技术展示了近乎完美的“统一性”，揭示了99％的过程纯度。最后，我们展示了两个独立的空间自由度（即方位角和径向）的好处，并在单个光子上实现了两个量子位控制的NOT量子操作。因此，我们的演示开辟了实现高维量子运算的新途径，可以将其应用于量子通信，计算和传感方案中的各种任务。