Sparse Hamiltonian row and value oracles

Fix $N\ge 1$, a sparsity bound $d\ge 1$, an integer $b\ge 1$, and an injective code $\mathrm{e}\mathrm{n}\mathrm{c}:C\to \{0,1{\}}^b$ for a finite set $C\subset \mathbb{C}$ containing $0$ and closed under conjugation. Put $w=\lceil {\log }_{2}N\rceil$ and $v=\lceil {\log }_{2}d\rceil$, identify $[N]$ with the first $N$ strings in $\{0,1{\}}^w$, and regard any remaining strings as invalid labels. A valid $w$-bit index is zero-extended to $w+1$ bits. Let $H=(H_{xy}{)}_{x,y\in [N]}$ be Hermitian, have entries in $C$, and have at most $d$ nonzero entries in every row. For each valid row $x$, list its nonzero column indices in increasing order. Define the total location function $f_H(x,\mathrm{\ell })$ to be the $\mathrm{\ell }$-th listed index for a valid slot $0\le \mathrm{\ell }<d$ when that slot exists, and a distinguished sentinel otherwise. Invalid row labels and invalid slot labels also return the sentinel. Encode the sentinel by a fixed $(w+1)$-bit word outside the valid index codes. Define the total value function $g_H(x,y)$ to return $\mathrm{e}\mathrm{n}\mathrm{c}(H_{xy})$ on valid labels and $\mathrm{e}\mathrm{n}\mathrm{c}(0)$ on invalid labels. All label widths and codes are part of the finite input description.

The row-location oracle and entry-value oracle are the coherent XOR maps $O_{f_H}\mathrm{\mid }x,\mathrm{\ell },z⟩=\mathrm{\mid }x,\mathrm{\ell },z\oplus f_H(x,\mathrm{\ell })⟩$ and $O_{g_H}\mathrm{\mid }x,y,z⟩=\mathrm{\mid }x,y,z\oplus g_H(x,y)⟩$, where the output target has respectively $w+1$ and $b$ bits. The slot register has $v$ bits; all unused slot strings are invalid and use the sentinel. These are total Boolean-function oracles on every bitstring basis input. They are instances of Coherent XOR function oracle, whose justified-by supplier proves unitarity and involutivity for arbitrary total functions. Each named call and any separately granted controlled or inverse call is charged under Circuit-family and computation cost model. This exact-code model describes the decoded matrix $H$; approximating a different target matrix requires a separate bound on the matrix difference. Row and value access alone does not supply a term list or term-evolution circuits.