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

Lemma: Uniform lattice routing

Explicit routing and complete quantum transport › A deterministic rounding convention · lines 512–587 · proof included or linked

En08-B@1#lem:routing, En08-B@1#eq:route-count, En08-B@1#eq:lattice-delay, En08-B@1#eq:uniform-ticks, En08-B@1#eq:route-points

Let $j > κ$ and $b \in Λ_{j}$ . Set $r = ∥ b ∥_{2}$ . For $b = 0$ , set $N_{j} (b) = 0$ . For $b \neq = 0$ , set

N_{j} (b) = ⌈ \frac{r}{h _{j} ( j - κ )} ⌉ .

There is an explicit route

p_{0} = 0, p_{1}, \dots, p_{N} = b, N = N_{j} (b),

whose increments belong to $h_{j} S_{j}$ . Its time

τ_{j} (b) = N_{j} (b) δ_{j}

satisfies

0 \leq τ_{j} (b) - \frac{r}{c} \leq \frac{κ r}{c ( j - κ )} + δ_{j} .

If $r \leq L$ and $j \geq 2 κ$ , then

N_{j} (b) \leq \frac{2 L j}{ℓ} + 1.

Proof. The zero case uses the single point $0$ , no gates, and zero time.

For $b \neq = 0$ , let $N$ be given by (eq:route-count) and define

p_{k} = Q_{j} (\frac{k}{N} b), 0 \leq k \leq N .

Because rounding fixes lattice points, $p_{0} = 0$ and $p_{N} = b$ . Write

e_{k} = p_{k} - \frac{k}{N} b, ∥ e_{k} ∥_{2} \leq ε_{j} .

Then

∥ p_{k} - p_{k - 1} ∥_{2} \leq \frac{r}{N} + 2 ε_{j} \leq h_{j} (j - κ) + κ h_{j} = h_{j} j = a_{j} .

Both points are lattice points, so every increment is in $h_{j} S_{j}$ . An increment may be zero; it is a permitted wait.

The triangle inequality along the route gives $r \leq N a_{j} = c N δ_{j}$ , proving the lower bound. Also,

N \leq \frac{r}{h _{j} ( j - κ )} + 1,

N δ_{j} \leq \frac{r j}{c ( j - κ )} + δ_{j} .

Subtracting $r / c$ proves (eq:lattice-delay). Finally, $j - κ \geq j /2$ gives

N \leq \frac{r j ^{2}}{ℓ ( j - κ )} + 1 \leq \frac{2 L j}{ℓ} + 1.

Displayed formulas in this passage

En08-B@1#eq:route-count · line 518
$N_{j} (b) = ⌈ \frac{r}{h _{j} ( j - κ )} ⌉ .$
En08-B@1#eq:lattice-delay · line 533
$0 \leq τ_{j} (b) - \frac{r}{c} \leq \frac{κ r}{c ( j - κ )} + δ_{j} .$
En08-B@1#eq:uniform-ticks · line 540
$N_{j} (b) \leq \frac{2 L j}{ℓ} + 1.$
En08-B@1#eq:route-points · line 550
$p_{k} = Q_{j} (\frac{k}{N} b), 0 \leq k \leq N .$
unlabeled · line 524
$p_{0} = 0, p_{1}, \dots, p_{N} = b, N = N_{j} (b),$
unlabeled · line 529
$τ_{j} (b) = N_{j} (b) δ_{j}$
unlabeled · line 557
$e_{k} = p_{k} - \frac{k}{N} b, ∥ e_{k} ∥_{2} \leq ε_{j} .$
unlabeled · line 561
$∥ p_{k} - p_{k - 1} ∥_{2} \leq \frac{r}{N} + 2 ε_{j} \leq h_{j} (j - κ) + κ h_{j} = h_{j} j = a_{j} .$
unlabeled · line 573
$N \leq \frac{r}{h _{j} ( j - κ )} + 1,$
unlabeled · line 577
$N δ_{j} \leq \frac{r j}{c ( j - κ )} + δ_{j} .$
unlabeled · line 583
$N \leq \frac{r j ^{2}}{ℓ ( j - κ )} + 1 \leq \frac{2 L j}{ℓ} + 1.$