Title: Fast Reconstruction of High-qubit Quantum States via Low Rate Measurements,
Abstract: Due to the exponential complexity of the resources required by quantum state
tomography (QST), people are interested in approaches towards identifying
quantum states which require less effort and time. In this paper, we provide a
tailored and 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 or nearly pure state $\rho$, and
can be extended to many states of interest in quantum information processing,
such as multi-particle entangled $W$ 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 $8,35$
and $226$ seconds respectively to reconstruct superposition state density
matrices of $10,11,12$ qubits with acceptable fidelity, using less than $1 \%$
of measurements of expectation. To our knowledge it is the fastest realization
that people can achieve using a normal desktop. We further discuss applications
of this method using experimental data of mixed states obtained in an ion trap
experiment of up to $8$ qubits.