Generating Haar-uniform Randomness using Stochastic Quantum Walks on a Photonic Chip Hao Tang, 1, 2 Leonardo Banchi, 3, 4 Tian-Yu Wang, 1, 2 Xiao-Wen Shang, 1, 2 Xi Tan, 1, 2 Wen-Hao Zhou, 1, 2 Zhen Feng, 1, 2 Anurag Pal, 1, 2 Hang Li, 1, 2 Cheng-Qiu Hu, 1, 2 M.S. Kim, 5, 6 and Xian-Min Jin 1,2, ∗ 1 Center for Integrated Quantum Information Technologies (IQIT), School of Physics and Astronomy and State Key Laboratory of Advanced Optical Communication Systems and Networks, Shanghai Jiao Tong University, Shanghai 200240, China 2 CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China 3 Department of Physics and Astronomy, University of Florence, via G. Sansone 1, I-50019 Sesto Fiorentino (FI), Italy 4 INFN Sezione di Firenze, via G. Sansone 1, I-50019 Sesto Fiorentino (FI), Italy 5 QOLS, Blackett Laboratory, Imperial College London, London SW7 2AZ, UK. 6 Korea Institute of Advanced Study, Seoul 02455, South Korea. As random operations for quantum systems are intensively used in various quantum information tasks, a trustworthy measure of the randomness in quantum operations is highly demanded. The Haar measure of ran- domness is a useful tool with wide applications such as boson sampling. Recently, a theoretical protocol was proposed to combine quantum control theory and driven stochastic quantum walks to generate Haar-uniform random operations. This opens up a promising route to converting classical randomness to quantum random- ness. Here, we implement a two-dimensional stochastic quantum walk on the integrated photonic chip and demonstrate that the average of all distribution profiles converges to the even distribution when the evolution length increases, suggesting the 1-pad Haar-uniform randomness. We further show that our two-dimensional array outperforms the one-dimensional array of the same number of waveguide for the speed of convergence. Our work demonstrates a scalable and robust way to generate Haar-uniform randomness that can provide useful building blocks to boost future quantum information techniques. Random operations for quantum systems[1] play an im- portant role for a large variety of tasks in quantum infor- mation processing. Especially, as various studies on bo- son sampling[2–4, 6, 7, 7] have emerged in recent years to demonstrate quantum computational supremacy[8, 9], the Haar random unitary matrices[10] required for these studies have drawn ever increasing attention. The Haar measure of randomness is now investigated as more than a theoretical tool, but also as a useful building block for quantum proto- cols or algorithms. It has wide applications covering boson sampling[2–4, 6, 7], quantum cryptography[11, 12], quan- tum process tomography[13], entanglement generation[14], fidelity estimation[15] etc, which, therefore, motivated a se- ries of experimental schemes on implementing random or pseudorandom quantum operations[16–18]. So far, these ex- perimental schemes decompose a random unitary matrix ei- ther by using a large number of quantum gates[16–18], or using photonic beamsplitters and interferometers[19, 20] via Reck/Clements decomposition method[21, 22], both of con- siderably high complexity in implementation. An alternative approach to generate Haar uniform random operations has recently been proposed[9] using what we call a stochastic quantum walk. The rationale is based on quan- tum control theory, which allows for a coherent driving of permanently coupled quantum systems via classical control pulses and stochastic pulses. Instead of using quantum cir- cuits or programmable photonic networks with beam splitters and phase shifters, in this alternative approach, random op- erations can be implemented via permanently coupled pho- tonic waveguides by applying stochastic modulations. This scheme, which effectively implements a stochastic version of a continuous-time quantum walk[24], could be scalable and beneficial for practical quantum experiments including larger- scale boson sampling. However, up to now, this scheme has never been demonstrated in experiments. Photonic lattice is an ideal physical platform to implement continuous-time quantum walk. A large evolution space in the photonic lattice allowing for real spatial two-dimensional quantum walks has been recently demonstrated[3, 26]. While this physical system is suitable for coherent and pure quantum walk, the environmental decoherence term can also be inten- tionally introduced by lattice manipulation. The key process is to introduce classical randomness to the propagation constant along different segments of each waveguide, which causes the randomness in the diagonal part of the Hamiltonian matrix. Therefore, a different kind of evolution, namely, the stochastic quantum walk, has been successfully demonstrated in the pho- tonic lattice to simulate various open quantum systems[6, 28]. In this work, we experimentally demonstrate an instance of Haar-uniform randomness using stochastic quantum walks on the integrated photonic chips. We prepare different samples with different random settings of the propagation constant and detunings, and then measure the light intensity distribution af- ter the evolution inside each chip. The different samples cre- ated according to the above procedure yields different unitary evolutions that, in the ideal case, should represent indepen- dent samples from the Haar distribution. We show that the average of all distribution profiles converges to the even distri- bution when the evolution length increases, suggesting the 1- pad Haar-uniform randomness. We further show that our two- arXiv:2112.06549v1 [quant-ph] 13 Dec 2021