 Solution

The chain is a random walk on the finite abelian group $(\mathbb{Z}/n\mathbb{Z}{)}^d$, so its stationary distribution is uniform and its characters diagonalize the transition operator. For one coordinate and $\theta =2\pi k/n$, the eigenvalue isUniformly in $p\in [0,1]$,for an absolute $c>0$. Hence the one-coordinate chi-squared distance after $t\ge n^2$ is at most $C{e}^{-ct/n^2}$. The coordinates evolve independently, so the product formula for chi-squared distance givesThe chi-squared divergence bound on total variation distance now yieldsuniformly in $p_1,\dots ,p_d$.

For $d=1$, the first nonconstant character has eigenvalue modulus $1-O(n^{-2})$, uniformly in $p$. Testing against its real or imaginary part gives a fixed positive total-variation distance until time $c{n}^2$. Thus