Continuous-time walk resource accounting

Suppose a declared access model assigns query/gate/description-bit cost triples $C_{\mathrm{i}\mathrm{n}}=(q_{\mathrm{i}\mathrm{n}},g_{\mathrm{i}\mathrm{n}},b_{\mathrm{i}\mathrm{n}})$, $C_H=(q_H,g_H,b_H)$, and $C_{\mathrm{o}\mathrm{u}\mathrm{t}}=(q_{\mathrm{o}\mathrm{u}\mathrm{t}},g_{\mathrm{o}\mathrm{u}\mathrm{t}},b_{\mathrm{o}\mathrm{u}\mathrm{t}})$ to the initial-state preparation, Hamiltonian evolution or simulator, and final measurement/readout of a continuous-time walk. If the model charges serial composition componentwise, the composed cost is $C_{\mathrm{i}\mathrm{n}}+C_H+C_{\mathrm{o}\mathrm{u}\mathrm{t}}=(q_{\mathrm{i}\mathrm{n}}+q_H+q_{\mathrm{o}\mathrm{u}\mathrm{t}},g_{\mathrm{i}\mathrm{n}}+g_H+g_{\mathrm{o}\mathrm{u}\mathrm{t}},b_{\mathrm{i}\mathrm{n}}+b_H+b_{\mathrm{o}\mathrm{u}\mathrm{t}}),$ plus separately declared classical processing. The number of measured output bits is reported separately.

For the exact $n$-dimensional hypercube transfer, the initial state $\mathrm{\mid }{0}^n⟩$ requires no preparation gates. On the local-operation branch of Hypercube continuous-time state transfer, the evolution factors into $n$ one-qubit operations $U_X(\pi \mathrm{/}2)=e^{-i\pi X\mathrm{/}2}$; this branch uses $n$ local evolution calls and $n$ gates when those operations are primitive in the declared gate model. A different finite gate set must charge its supplied synthesis cost. On the whole-operation branch, the procedure makes one call to the supplied $U_{A_n}(\pi \mathrm{/}2)$ and its query, gate, and description-bit costs are those declared by the access contract; the factorization identity alone does not imply $n$ such calls or gates. Either branch reads out $n$ bits.

For complete-graph search on arbitrary $N\ge 2$, the total includes the supplied preparation cost $C_s=(q_s,g_s,b_s)$, the declared ideal-or-simulated Hamiltonian-evolution cost, and the final readout cost. When $N=2^n$ with standard binary encoding, $A_s=H^{\otimes n}$ is verified to prepare the uniform state; it has zero oracle queries and $n$ Hadamard gates when $H$ is primitive, and otherwise its declared synthesis costs are charged. The readout produces $n$ bits. For a general $m$-bit encoding it produces $m$ bits before any separately charged classical decoding. Simulator precision, oracle access, and description bits remain in their declared coordinates; a matrix alone gives no cost bound.