# Optical Realization of Wave-Based Analog Computing with Metamaterials
## 1 Introduction  
Exploring novel approaches to improve the computational capacity and efficiency is a goal that humans have been continuously pursuing. Early computers are constructed mechanically [1] or electronically [2; 3; 4] on the principle of analog, aiming to perform mathematical operations. Despite the impressive success achieved in the fields of weather prediction, aerospace and nuclear industry, these machines faced significant obstacles of slow response and large size [5]. In the 20 th century, digital computing emerged with a rapid development of semiconductor technology and large-scale integrated circuits. It began to gradually substitute for their conventional analogue counterparts on the strength of easy programmability, high speed and flexibility. However, in solving specialized computational tasks, such as imaging-processing and edge detection, digital computers are often inefficient and hindered by high-power consumptions. As [PERSON]'s law is approaching its physical limitations, the long-abandoned analog approach as an alternative paradigm has attracted renewed attention for its potential abilities to overcome these shortcomings [6; 7; 8]. Analog optical computing offers unique advantages of real-time, power-efficient, high-throughput imaging processing abilities originating from the wave-based nature [7; 9; 10; 11]. Compared with standard digital processes, all-optical analog approaches do not refer to photoelectric converters but directly manipulate optical signals both temporally and spatially [12]. In the temporal domain, high-speed pulse waveform modulation enables numerous applications, including analog computing [13; 14; 15; 16; 17; 18; 19], differential equations solving [19; 20; 21; 22; 23], optical memory [24; 25], photonic neural networks [26; 27] and complex nonlinear system simulation [28; 29; 30]. In the spatial domain, the input functions are indicated by the spatial wavefronts which will be mathematically transformed with pre-designed optical systems. Therefore, this computing platform has intrinsic parallel characteristic, showing great potential for accelerating the processing of megalo-capacity datasets and images [31; 32; 33]. In addition to classical mathematical operations, researchers have also extended the concept of analog computing to quantum algorithms that promise an exponential speedup that is far beyond classical ones in solving problems of large integer factorization ([PERSON]'s algorithm) [34] and combinatorial optimization ([PERSON]'s algorithm) [35; 36]. Some fundamental properties of quantum computing, including superposition principle and interference phenomena, are the essence of wave nature, which are not exclusive to quantum mechanical but are common to classical waves. By encoding quantum bit (qubits) into different degrees of freedom for the electromagnetic field (e.g., frequency, polarization, orbital angular momentum, space and time bins), many quantum computations can be efficiently simulated in optics [37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51; 52; 53; 54; 55; 56; 57; 58; 59; 60; 61; 62; 63; 64; 65; 66; 67; 68]. However, traditional analog optical computing requires bulky optical components, such as diffractive lenses and spatial filters, which are inconvenient for miniatur and integration of modern ultracompact optics. Empowered by recent development of nanofabrication technologies, metamaterial [69; 70; 71; 72; 73; 74; 75; 76; 77; 78; 79] or its two-dimensional counterpart, metasurface [80; 81; 82; 83; 84; 85; 86; 87; 88; 89], is able to tailor subwavelength building blocks on the scales of micro- or nanometers, providing unprecedented flexibility to arbitrarily control the electromagnetic waves that are unattainable in the nature. Owning to their powerful wave manipulation abilities and deeply subwavelength characteristics, those meta-structures can significantly reduce the complexity and shrink the size of computing systems, making it possible to realize chip-level all-optical information processing systems [100; 101].