Generating Haar-Uniform Randomness Using Stochastic Quantum Walks on a Photonic Chip

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 randomness 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 randomness. 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 Letter demonstrates a scalable and robust way to generate Haar-uniform randomness that can provide useful building blocks to boost future quantum information techniques.

Figure 1

Generating Haar-uniform randomness using stochastic quantum walks. (a) Illustration of averaging many stochastic quantum walks of a certain evolution time to reach the Haar measure. The red columns represent the distribution of $I/N$ that appear in Eq. (2). (b) Schematic diagram of introducing random delta beta detunings to implement stochastic quantum walks on the photonic chip. The color bar shows the detuning strength of the propagation constant, where “max” corresponds to the given $\mathrm{\Delta }\beta$ amplitude. Photons are injected into one waveguide, and the evolution in the lattice corresponds to a unitary operation.

Figure 2

Experimental results for stochastic quantum walks. The photonic evolution patterns of different evolution lengths for one random setting of photonic lattice. The corresponding evolution length of each graph is marked beside the graph. The mask illustrates how we read the figure data to get the probability distribution, with details explained in Supplemental Material Note 5 [30].

Figure 3

Convergence to the Haar measure. (a) The heat maps show all the elements of the matrix $M$, the diagonal elements of ${\mathbb{E}}_i[U_i\rho U_i]-I/N$, for different evolution lengths $z$ as shown above each heat map. The average for each $z$ is estimated from 17 random settings with the same evolution length. Note that a few elements have a value above 0.033 or below $-0.033$. They are represented in the heat map using the color for 0.033 or $-0.033$, respectively. (b) The L2 norm $\parallel M\parallel$ for samples of different evolution lengths. QSW and QW stand for stochastic quantum walk and quantum walk, respectively. The theoretical results are obtained by setting $\mathrm{\Delta }z$ of 2 mm and averaging 17 random settings, which are consistent with the experiment. The shading area shows a possible range of theoretical results considering a 15% fluctuation of the $\mathrm{\Delta }\beta$ amplitude; that is, the upper bound and lower bound of the shading area correspond to a $\mathrm{\Delta }\beta$ amplitude of 0.34 and $0.46{\mathrm{mm}}^{-1}$, respectively. The error bar for the experimental results is the standard error of the mean, with details explained in Supplemental Material Note 6 [30].

Figure 4

Compare the performance in one- and 2D array. (a) Schematic diagram of the 1D photonic lattice of 25 waveguides with random tunings of the propagation constant. (b) The calculated $L2$ norm $\parallel M\parallel$ for 1D and 2D arrays. For 2D array, we get two sets of norm values, each by averaging six random settings separately.