# Optical Realization of Wave-Based Analog Computing with Metamaterials
## 3 Other Emerging Approaches
### Quantum Computing with Metamaterials
The designed metamaterial consisted of four cascade subblocks (an oracle subblock \(U_{m}\), two Fourier transform subblocks \(F\), a phase plate subblock \(I-2|0\rangle\langle 0|\)) corresponding to the operators of oracle, Walsh-Hadamard transform and inversion-about-average (IAA) operation in [PERSON]'s search algorithm (see Figure 13a). In this scheme, the quantum states are directly mapped onto classical optical fields. For instance, the incident electric field amplitude \"\(E(y)\)\" is the analogue of quantum probability amplitude, transversal coordinate \"\(y\)\" is used to label the item of the database and the maximum number of the database is depending on the full width at half-maximum of the beam \"D\" (listed in Table 1). When light occurs on the metamaterial, each functional subblock it passed is equivalent to performing a quantum operation; therefore, each roundtrip represents one iteration of the search algorithm. After multiple iterations in the metamaterial, the marked item is found by measuring the field distribution on the output plane (see Figure 13b). Recently, [PERSON] et al. [155] verified that Deutsch-Jozsa (DJ) algorithm could be simulated by using a similar strategy (see Figure 13c,d). In Table 1, we compare the general protocol of performing [PERSON]'s search algorithm in quantum and classical realm to clarify the differences between the two strategies. As the spatial freedom of the optical field (cbit) is adopted to simulate the qubit, the classical wave-based architecture shows a simple and enlightening way to the field of signal processing. It should be mentioned that despite that it has been proved that entanglement is not necessary for the efficiency of some quantum algorithms [156], a lack of entanglement will essentially limit the physical resources for these classical analogies, leading to an exponential growth of the width of the beam as the numbers of qubits increase. In addition, the size of the database is also limited by the spatial resolution, diffraction effect and paraxial approximation of the optical system. With the rapid development of the field of artificial electromagnetism, we can expect more novel functionalities on simulating quantum behaviors, such as three-dimensional all-optical search and data classification. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{**Quantum**} & \multicolumn{1}{c}{**Classical**} \\ \hline Items in the database \(\ket{i}\) & “\(y^{\prime}\) \\ Probability amplitude of the equivalent quantum state & “\(E(y)^{\prime}\) \\ The maximum number of the database \(N\) & “\(D^{\prime}\) \\ \hline \(U_{m}\) & \(I-2\ket{s}/\ket{s}\) & \(\exp(i2\pi\sqrt{\epsilon_{m}(y)}d_{m}/\lambda_{0})\) \\ \(H^{\otimes n}\) & \(F\) \\ \(IAA\) & \(I-2\ket{0}\bra{0}\) & \(\exp(i2\pi\sqrt{\epsilon_{0}(y)}d_{0}/\lambda_{0})\) \\ \(H^{\otimes n}\) & \(F\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The general protocol of performing [PERSON]’s search algorithm with metamaterial in quantum and classical realm. Figure 13: (**a**) Schematic of the general protocol of simulating quantum search algorithm with metamaterials. (**b**) Experimental results of the time traces (top) and the field intensity (bottom) of the output with different iterations. Adapted with permission from [154], [PERSON], 2017. (**c**) Simulation results of metamaterial-based DJ algorithm analogy with constant, simple balance and complex balance functions, respectively. (**d**) The photograph (top) and experimental results of three samples for simulating DJ algorithm. Adapted with permission from [155] © The Optical Society.