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Journal of Physics Communications

Implementation of quantum permutation algorithm with classical light

Shihao Zhang, Pengyun Li, Bo Wang, Qiang Zeng, Xiangdong Zhang

Published Jan 4, 2019• 7 citations
Publisher

Abstract

We report the experimental implementation of quantum permutation algorithm using polarization and orbital angular momentum of the classical optical beam. The easy-handling optical setup to realize all eight cyclic permutation transformations for an input four-dimensional system has been constructed. The two-to-one speed-up ratio to determine the parity of each permutation has been demonstrated. Moreover, we have theoretically discussed the extension to the case with eight elements, and the limitations on the generalization of our proposal to higher-dimensional cases. Our scheme exhibits outstanding stability and demonstrates that optical quantum computation is possible using classical states of light.

DOI
10.1088/2399-6528/aafc1c

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We report the experimental implementation of quantum permutation algorithm using polarization and orbital angular momentum of the classical optical beam. The easy-handling optical setup to realize all eight cyclic permutation transformations for an input four-dimensional system has been constructed. The two-to-one speed-up ratio to determine the parity of each permutation has been demonstrated. Moreover, we have theoretically discussed the extension to the case with eight elements, and the limitations on the generalization of our proposal to higher-dimensional cases. Our scheme exhibits outstanding stability and demonstrates that optical quantum computation is possible using classical states of light.
Quantum algorithms provide a more efficient way to solve computational tasks than classical algorithms. We experimentally realize quantum permutation algorithm using light's orbital angular momentum degree of freedom. By exploiting the spatial mode of photons, our scheme provides a more elegant way to understand the principle of quantum permutation algorithm and shows that the high dimension characteristic of light's orbital angular momentum may be useful in quantum algorithms. Our scheme can be extended to higher dimension by introducing more spatial modes and it paves the way to trace the source of quantum speedup.
The spatial modes of photons are one realization of a QuDit, a quantum system that is described in a D -dimensional Hilbert space. In order to perform quantum information tasks with QuDits, a general class of D -dimensional unitary transformations is needed. Among these, cyclic transformations are an important special case required in many high-dimensional quantum communication protocols. In this paper, we experimentally demonstrate a cyclic transformation in the high-dimensional space of photonic orbital angular momentum (OAM). Using simple linear optical components, we show a successful four-fold cyclic transformation of OAM modes. Interestingly, our experimental setup was found by a computer algorithm. In addition to the four-cyclic transformation, the algorithm also found extensions to higher-dimensional cycles in a hybrid space of OAM and polarization. Besides being useful for quantum cryptography with QuDits, cyclic transformations are key for the experimental production of high-dimensional maximally entangled Bell-states.
The quantum permutation algorithm provides computational speed-up over classical algorithms for determining the parity of a given cyclic permutation. For its nnn-qubit implementations, the number of required quantum gates scales quadratically with nnn due to the quantum Fourier transforms included. We show here for the nnn-qubit case that the algorithm can be simplified so that it requires only O(n)O(n)O(n) quantum gates, which theoretically reduces the complexity of the implementation. To test our results experimentally, we utilize IBM's 5-qubit quantum processor to realize the algorithm by using the original and simplified recipes for the 2-qubit case. It turns out that the latter results in a significantly higher success probability which allows us to verify the algorithm more precisely than the previous experimental realizations. We also verify the algorithm for the first time for the 3-qubit case with a considerable success probability by taking the advantage of our simplified scheme.
We report an experiment to demonstrate a quantum permutation determining algorithm with linear optical system. By employing photon's polarization and spatial mode, we realize the quantum ququart states and all the essential permutation transformations. The quantum permutation determining algorithm displays the speedup of quantum algorithm by determining the parity of the permutation in only one step of evaluation compared with two for classical algorithm. This experiment is accomplished in single photon level and the method exhibits universality in high-dimensional quantum computation.
High-dimensional quantum computing has emerged as a recent alternative to conventional qubit-based approaches, offering advantages such as an increased information density and enhanced robustness against noise. In this work, we explore the use of the orbital angular momentum (OAM) of light as a degree of freedom in which high-dimensional quantum gates can be implemented. We present a theoretical and experimental study of structured light beams carrying OAM, with emphasis on Laguerre–Gauss modes, as a natural basis for high-dimensional quantum computing. We then describe the generation of OAM beams using computer-generated holograms and spatial light modulators, detailing practical methods for encoding arbitrary phase profiles. Building on this framework, we analyze the implementation of high-dimensional quantum logic gates, focusing on the generalized Pauli Z_d and X_d operations. Optical setups based on Dove prisms, Sagnac interferometers, spiral phase plates, and parity sorters are proposed and theoretically validated. Finally, we report experimental results demonstrating the generation of Laguerre–Gauss beams, the transformation properties of Dove prisms, and the operation of a parity sorter, which constitutes a building block for realizing high-dimensional quantum gates. These results establish a solid foundation for future implementations of high-dimensional quantum information processing using the orbital angular momentum of light.
Classical optics can be used to efficiently implement certain quantum information processing tasks with a high degree of control, for example, one-dimensional quantum walks through the space of orbital angular momentum of light directed by its polarization. To explore the potential of quantum information processing with classical light, we here suggest a method to realize ddd-dimensional quantum walks with classical optics---an important step towards robust implementation of certain quantum algorithms. In this scheme, different degrees of freedom of light, such as frequency, orbital angular momentum, and time bins, represent different directions for the walker while the coin to decide which direction the walker takes is realized by employing the polarization combined with different light paths.