A. I. Lvovsky

A prerequisite to the successful development of quantum computers and simulators is precise understanding of physical processes occurring therein, which can be achieved by measuring the quantum states they produce. However, the resources required for traditional quantum-state estimation scale exponentially with the system size, highlighting the need for alternative approaches. Here we demonstrate an efficient method for reconstruction of significantly entangled multi-qubit quantum states. Using a variational version of the matrix product state ansatz, we perform the tomography (in the pure-state approximation) of quantum states produced in a 20-qubit trapped-ion Ising-type quantum simulator, using the data acquired in only 27 bases with 1000 measurements in each basis. We observe superior state reconstruction quality and faster convergence compared to the methods based on neural network quantum state representations: restricted Boltzmann machines and feedforward neural networks with autoregressive architecture. Our results pave the way towards efficient experimental characterization of complex states produced by the quench dynamics of many-body quantum systems.

We propose a refined iterative likelihood-maximization algorithm for reconstructing a quantum state from a set of tomographic measurements. The algorithm is characterized by a very high convergence rate and features a simple adaptive procedure that ensures likelihood increase in every iteration and convergence to the maximum-likelihood state. We apply the algorithm to homodyne tomography of optical states and quantum tomography of entangled spin states of trapped ions and investigate its convergence properties.