Fast Reconstruction of High-qubit Quantum States via Low Rate Measurements

Abstract

Quantum state tomography is a fundamental technique for quantum technology, with many applications in quantum control and quantum communication. Due to the exponential complexity of the resources required for QST, people are looking for approaches that identify quantum states with less efforts and faster speed. In this Letter, we provide a tailored efficient method for reconstructing mixed quantum states up to $12$ (or even more) qubits from an incomplete set of observables subject to noises. Our method is applicable to any pure state $\rho$, and can be extended to many states of interest in quantum information tasks such as $W$ state, GHZ state and cluster states that are matrix product operators of low dimensions. The method applies the quantum density matrix constraints to a quantum compressive sensing optimization problem, and exploits a modified Quantum Alternating Direction Multiplier Method (Quantum-ADMM) to accelerate the convergence. Our algorithm takes $8,35, 226$ seconds to reconstruct arbitrary superposition state density matrices of $10,11,12$ qubits with acceptable fidelity respectively, using less than $1 \%$ measurements of expectation on a normal desktop, which is the fastest realization to date. We further discuss applications of this method using experimental data of mixed states obtained in an ion trap experiment up to $8$ qubits.