  In this paper, we tackle the compressive phase retrieval problem in the
presence of noise. The noisy compressive phase retrieval problem is to recover
a $K$-sparse complex signal $s \in \mathbb{C}^n$, from a set of $m$ noisy
quadratic measurements: $ y_i=| a_i^H s |^2+w_i$, where $a_i^H\in\mathbb{C}^n$
is the $i$th row of the measurement matrix $A\in\mathbb{C}^{m\times n}$, and
$w_i$ is the additive noise to the $i$th measurement. We consider the regime
where $K=\beta n^\delta$, with constants $\beta>0$ and $\delta\in(0,1)$. We use
the architecture of PhaseCode algorithm, and robustify it using two schemes:
the almost-linear scheme and the sublinear scheme. We prove that with high
probability, the almost-linear scheme recovers $s$ with sample complexity
$\Theta(K \log(n))$ and computational complexity $\Theta(n \log(n))$, and the
sublinear scheme recovers $s$ with sample complexity $\Theta(K\log^3(n))$ and
computational complexity $\Theta(K\log^3(n))$. To the best of our knowledge,
this is the first scheme that achieves sublinear computational complexity for
compressive phase retrieval problem. Finally, we provide simulation results
that support our theoretical contributions.
