ered.

Moore and Russell [@MR01] analyzed both the discrete and the continuous quantum walk on a hypercube. Kendon and Tregenna [@KT03] performed a numerical analysis of the effect of decoherence in the discrete case. In this article, we extend the continuous case with the model of decoherence described above. In particular, we show that up to a certain rate of decoherence, both linear instantaneous mixing times and linear instantaneous hitting times still occur. Beyond the threshold, however, the walk behaves like the classical walk on the hypercube, exhibiting $\Theta(n \log n)$ mixing times. As the rate of decoherence grows, mixing is retarded by the quantum Zeno effect.

Results
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Consider the continuous quantum walk on the $n$-dimensional hypercube with energy $k$ and decoherence rate $p$, starting from the initial wave function $\Psi_0 = \vert 0 \rangle ^{\otimes n}$, corresponding to the corner with Hamming weight zero. We prove the following theorems about this walk.

When $p < 4k$, the walk has instantaneous mixing times at $$t_{mix} = \frac {n (2\pi c - \arccos(p^2/8k^2-1))}{\sqrt{16k^2 -  p^2}}$$ for all $c \in \mathbb{Z}$, $c > 0$. At these times, the total variation distance between the walk distribution and the uniform distribution is zero.

This result is an extension of the results in [@MR01], and an improvement over the classical random walk mixing time of $\Theta(n \log n)$. Note that the mixing times decay with $p$ and disappear altogether when $p \geq 4k$. Further, for large $p$, we will see that the walk is retarded by the quantum Zeno effect.

When $p < 4k$, the walk has approximate instantaneous hitting times to the opposite corner $(1, \dots , 1)$ at times $$t_{hit} = \frac{2 \pi n (2c + 1)}{\sqrt{16k^2 - p^2}}$$ for all $c \in \mathbb{Z}$, $c \geq 0$. However, the probability of measuring an exact hit decays exponentially in $c$; the probability is $$P_{hit} = \left[\frac{1}{2} + \frac{1}{2}e^{-\frac{p \pi (2c + 1)}
    {\sqrt{16k^2 -  p^2}}}\right]^n\enspace.$$ In particular, w