Manuscript En08-B@1 · En08-B.tex · section · 23cac52a

Quantum registers and gate resources

Resource ledger and effective specification › Quantum registers and gate resources · lines 987–1034

For the constructed route, let $N$ be its unpadded tick count, $m$ the number of nonzero increments, $q = m + 1$ the number of distinct route sites, and $w \in {0, 1}$ its terminal padding count. The total number of ticks is $n = N + w$ .

All $q$ registers exist at time zero and are retained until the terminal output reduction. Thus the global accounted peak is $q$ , even though only two registers participate in a nontrivial gate at a given tick.

Quantity	Exact statement or uniform bound
Logical payload dimension	$D$ , fixed independently of $j$ .
Supplied payload registers	One, at the source.
Fresh auxiliary registers	$q - 1 = m$ , each prepared once in $σ$ .
Total initialized sites	$q = m + 1 \leq N + 1$ .
Global retained-register peak	$q$ .
Terminal quantum output	One $D$ -dimensional destination register.
Non-output registers ignored at the end	$q - 1$ , all blank in the ideal protocol.
Two-site SWAP count	$m \leq N$ .
Tick count	$n = N + w$ , with $w \leq 1$ for null targets.
Largest active block	Two registers; block Hilbert dimension $D^{2}$ .
Physical interaction radius	$a_{j} = ℓ / j$ .
Ideal logical channel error	Zero.
Null-target spatial error	At most $κ ℓ / (2 j^{2})$ .
Null-target arrival delay	At most $B_{j} (L) = O (j^{- 1})$ .
Uniform ticks and sites	At most $H_{j} (L) \leq 2 L j / ℓ + 3$ .

The total quantum Hilbert dimension is $D^{q}$ . A fixed logical dimension is therefore not a constant global-memory bound. If a finite reference of dimension $r_{R}$ is included in a global Hilbert-space count, that dimension becomes $r_{R} D^{q}$ . The reference is not initialized or operated on by the transport protocol.

The ledger concerns quantum register ownership and circuit resources. It assigns no heat, work, hardware fabrication cost, or Hamiltonian energy bound to a SWAP or to blank-state preparation.