Two-bit parity path and sparse normalization

Substituting $n=2$ gives $w_1=w_2=\sqrt{2}\mathrm{/}2$. For $b\in \{0,1\}$, edge 1 joins $\mathrm{\mid }0,b⟩$ to $\mathrm{\mid }1,b\oplus x_1⟩$, and edge 2 joins $\mathrm{\mid }1,b⟩$ to $\mathrm{\mid }2,b\oplus x_2⟩$. Starting with $b=0$, these two edges produce the four paths in the table; each endpoint bit is $x_1\oplus x_2$. Each internal path vertex has degree two and each endpoint degree one.

Therefore $d_H=2$ and $h_{\max }=1\mathrm{/}\sqrt{2}$, so [F2] gives ${\alpha }_{\mathrm{s}\mathrm{p}}=d_H{h}_{\max }=\sqrt{2}$. This exact-weight example illustrates the path underlying [F3], whose lower bound instead uses rounded weights: at $n=2$ its grid exponent is $b=5$, so its two weights are $23\mathrm{/}32$ and its sparse normalization is $23\mathrm{/}16$. Thus [F3] does not identify this exact matrix with its rounded family or supply a lower bound for arbitrary block-encoding access.

In the parity-free basis, the path matrix is $H_0=\frac{1}{\sqrt{2}}\left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 1\\ 0& 1& 0\end{array}\right),H_0^2=\frac{1}{2}\left(\begin{array}{ccc}1& 0& 1\\ 0& 2& 0\\ 1& 0& 1\end{array}\right)\mathrm{.}$ Its eigenvalues are $1,0,-1$, so spectral functional calculus gives $e^{-i\pi H_0}=I-2H_0^2$, which sends $\mathrm{\mid }0⟩$ to $-\mathrm{\mid }2⟩$. The prefix-parity basis permutation of [F1] restores the endpoint bit $x_1\oplus x_2$; equivalently, [F1] gives $(-i{)}^2=-1$ times the same endpoint.