Shanghai Jiaotong University implements a Haar random unitary matrix based on quantum random walking for the first time
Recently, the research group of Jin Xianmin and Tang Hao from the School of Physics and Astronomy of Shanghai Jiao Tong University published a paper entitled "Generating Haar-uniform Randomness using Stochastic Quantum Walks on a Photonic Chip" in the authoritative journal of physics "Physical Review Letters" [PRL 128, 050503 (2022)]. In recent years, quantum random walks have been proposed to construct random unitary matrices conforming to the Haar measure. This work establishes the mapping of this theoretical scheme in a three-dimensional optical quantum chip system, and is the first experimental realization of Haar random unitary matrices based on quantum random walks.
Random operations of quantum systems play an important role in quantum information processing. Especially as various studies on Bose sampling have demonstrated the superiority of quantum computing, the Haar random unitary matrices required for these studies have received more and more attention. The Haar measure is not only a theoretical tool to study randomness, but also a practical module for building quantum protocols or algorithms, widely used in Bose sampling, quantum cryptography, quantum process tomography, entanglement generation, fidelity estimation, etc. .
Therefore, how to construct a random unitary matrix conforming to the Haar measure in quantum physics hardware has become a research point that has attracted much attention in the current quantum computing community. Some experimental schemes that have been reported so far, some are based on general quantum circuits, which require the use of a large number of quantum gates; some are based on programmable optical networks using Reck or Clements configurations, which require the use of photonic beam splitters and interferometers with the square of the number of modes. level increased. Therefore, these implementations are relatively complex. Inspired by quantum control and open quantum system theory, Professor L. Banchi from the University of Florence, Italy, deduced that the Haar random unitary matrix can be realized more efficiently based on time-continuous quantum random walks. was tested for a uniform distribution, but this theoretical scheme has never been experimentally verified.
This work explores the experimental realization of the above theoretical scheme based on the research group's many years of experiments in realizing various quantum walks based on three-dimensional optical quantum chips. This work improves the original theoretical scheme, and proposes a quantum random walk theoretical model based on the random perturbation of the waveguide transmission constant. Therefore, in the experiment, the experimental parameters in the preparation process of femtosecond laser direct writing are controlled by classical random numbers, so that the perturbation value Δβ of the waveguide transmission constant of different intensities is introduced into the array, which will be on the diagonal of the Hamiltonian. term introduces a perturbation value to inject photons into such a large-scale three-dimensional optical waveguide array for quantum random walks.
Fig. 1 Schematic diagram of the waveguide array on a three-dimensional photonic chip The colors of different depths represent the perturbation value Δβ of the waveguide transmission constant with different intensities.
In the experiment, up to 17 groups of 5 × 5 three-dimensional optical waveguide arrays with different Δβ random perturbations were prepared, each group including 8 samples with different evolution lengths from 1 to 8 cm. The histograms of Δβ random disturbance values in all these hundreds of samples satisfy the uniform distribution of 0-0.4/mm, and maintain a constant Δβ random disturbance value in each segment with Δz of 2 mm. The photon evolution distribution image of quantum random walk is obtained experimentally, and the average evolution distribution of 17 groups under the same evolution length is obtained. Compared with the completely uniform distribution, the norm of the difference matrix between the two distributions is obtained, because it conforms to the expectation of the evolution distribution in the Haar measure. will be evenly distributed. Experiments show that the evolution of about 8 cm gradually conforms to the Haar measure (the norm tends to zero), which is consistent with the theoretical simulation results. For pure quantum walks without the perturbation of transport constants, the norm cannot be reduced, which reflects the unique advantage of quantum random walks in constructing Haar random unitary matrices.
Fig. 2 The experimental result of realizing Haar random unitary matrix by quantum walk
On this basis, this work also explores the factors affecting the convergence time and convergence limit through supplementary experiments and numerical simulations. According to the research, the increase of △β amplitude will speed up the speed of norm convergence, while the increase of the number of sample groups will improve the accuracy of convergence (that is, to stabilize at a lower norm), and the parameter △z that reflects the frequency of random disturbance changes Also affects the convergence speed. In the existing photonic quantum chip technology, by selecting appropriate experimental parameters, a random unitary matrix conforming to the Haar measure can be fully realized.
Figure 3 Exploration of factors affecting the convergence limit
This work verifies that the expectation of a large number of quantum random walks on a three-dimensional optical quantum chip conforms to the Haar measure through abundant experimental work. After this verification, in future applications, it is only necessary to prepare a quantum random walk with a set of specific experimental parameters each time, and it can be identified as a Haar random unitary matrix, which can be applied to a series of quantum information processing modules such as Bose sampling middle. Large-scale three-dimensional photonic quantum chips and flexible introduction of random disturbances have brought a more efficient and innovative approach to the construction of Haar random unitary matrices.
The first author of the paper is Tang Hao, associate researcher at the School of Physics and Astronomy, Shanghai Jiao Tong University, and the corresponding author of the paper is Professor Jin Xianmin, School of Physics and Astronomy, Shanghai Jiao Tong University. The collaborators include Professor L. Banchi of the University of Florence, Italy, Professor M.S. Kim of Imperial College London, etc. The research work has been funded by the National Key R&D Program, the National Natural Science Foundation of China, the Shanghai Municipal Science and Technology Commission and the Shanghai Municipal Education Commission.
Link:
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.128.050503?ft=1
