SDP maximally entangled reference-state access

For an integer matrix dimension $d\ge 1$, define the maximally entangled reference vector $\mathrm{\mid }{\mathrm{\Phi }}_d⟩=d^{-1\mathrm{/}2}{\sum }_{i=0}^{d-1}\mathrm{\mid }i⟩\mathrm{\mid }i⟩$ on two ordered $d$-dimensional registers. The access contract supplies a finite preparation unitary $V_{\mathrm{\Phi }}$, its inverse, width/ancillas, input encoding, and costs $C_{\mathrm{\Phi }}$, satisfying $V_{\mathrm{\Phi }}(\mathrm{\mid }0⟩\mathrm{\mid }0⟩\mathrm{\mid }{0}^a⟩)=\mathrm{\mid }{\mathrm{\Phi }}_d⟩\mathrm{\mid }{0}^a⟩$ exactly in the declared encoding. Its action on other inputs is any supplied unitary completion; the inverse is that of the complete unitary. When $d=2^q$, a concrete preparation applies $H$ to each of the $q$ first-register qubits and then CNOT from each first-register qubit to its matching second-register qubit, using exactly $q$ Hadamards and $q$ CNOTs on $2q$ wires. For other $d$, $V_{\mathrm{\Phi }}$ and its cost are supplied inputs satisfying the same preparation equation; unused encoded labels have zero output amplitude, and no padding is allowed unless its effect on traces is accounted for.