  Recently, recovering an unknown signal from quadratic measurements has gained
popularity because it includes many interesting applications as special cases
such as phase retrieval, fusion frame phase retrieval, and positive
operator-valued measure. In this paper, by employing the least squares approach
to reconstruct the signal, we establish the non-asymptotic statistical property
showing that the gap between the estimator and the true signal is vanished in
the noiseless case and is bounded in the noisy case by an error rate of
$O(\sqrt{p\log(1+2n)/n})$, where $n$ and $p$ are the number of measurements and
the dimension of the signal, respectively. We develop a gradient regularized
Newton method (GRNM) to solve the least squares problem and prove that it
converges to a unique local minimum at a superlinear rate under certain mild
conditions. In addition to the deterministic results, GRNM can reconstruct the
true signal exactly for the noiseless case and achieve the above error rate
with a high probability for the noisy case. Numerical experiments demonstrate
the GRNM performs nicely in terms of high order of recovery accuracy, faster
computational speed, and strong recovery capability.
