Experimental realization of self-guided quantum process tomography

Characterization of quantum processes is a preliminary step necessary in the development of quantum technology. The conventional method uses standard quantum process tomography, which requires $d^2$ input states and $d^4$ quantum measurements for a $d$-dimensional Hilbert space. These experimental requirements are compounded by the complexity of processing the collected data, which can take several orders of magnitude longer than the experiment itself. In this paper we propose an alternative self-guided algorithm for quantum process tomography, tuned for the task of finding an unknown unitary process. Our algorithm is a fully automated and adaptive process characterization technique. The advantages of our algorithm are the following: it has an inherent robustness to both statistical and technical noise; it requires less space and time since there is no postprocessing of the data; it requires only a single input state and measurement; and it provides on-the-fly diagnostic information while the experiment is running. Numerical results show our algorithm achieves the same $1/n$ scaling as standard quantum process tomography when $n$ uses of the unknown process are used. We also present experimental results wherein the algorithm and its advantages are realized for the task of finding an element of SU(2).

Figure 1

Numerical results without noise. The solid lines are the median performance of SGQPT over 100 randomly (according to the Haar measure) chosen SU(2) operators. The shadow regions represent interquartile error. We use up to ${10}^4$ iterations in the simulation and the scaling behavior of infidelity is fitted with respect to the number of iterations. The parameters in the algorithm are optimized to achieve the best-fitting scaling behavior, specified in the figure. In each iteration, two controls are added and both are measured $N$ times.

Figure 2

Numerical results with three levels of random noise. Noise is added to the rotation angles of the three wave plates in the implementation of the learning control $V$ by choosing a random value in the uniform interval $(-\epsilon ,\epsilon )$, where $\epsilon$ is specified in the legend. Parameters in our algorithm are numerically optimized to achieve the best scaling behavior of the median performance with respect to the number of iterations. The optimized parameters are $\alpha =0.94$ and $\gamma =0.21$. In each iteration, the measurements are repeated 1000 times for both controls. The shadow regions represent interquartile error.

Figure 3

Experimental setup. The system and ancilla qubits are encoded on the polarization and the path degree of freedom of the photon, respectively. The learning controls $V_k$ are automatically realized by a python program to learn the unknown evolution $U$ in a black box. Bell measurements on the system and ancilla qubits are implemented by a three-step quantum walk.

Figure 4

Experimental results without noise. We only perform 50 iterations of learning and $N=10$ and 100 repeated measurements for each iteration due to limited time. Parameters $\alpha =0.85$ and $\gamma =0.06$ are optimized to achieve the best median performance over 20 randomly chosen SU(2) operators after 50 iterations of learning. The horizontal lines are simulated median infidelities of QPT with the same total number of photons used in SGQPT with 50 iterations, respectively. The simulation results of quantum process tomography are performed with six eigenstates of three Pauli matrices as probe states and performing three Pauli measurements for each probe state.

Figure 5

Experimental results under three levels of random experimental errors. The lines with dots show the median infidelities of 20 randomly chosen SU(2) unitary processes. The shaded regions represent the interquartile range of infidelities. The horizontal lines are simulated data, which represent median infidelities of QPT with the same levels of error for the same 20 processes above. The noise level means the uncertainty of rotation angle of each electric motor stage in degrees. The total number of photons used in each tomography for each chosen unitary process is ${10}^4$.