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

Proposition: A vanishing per-gate error need not vanish globally

Implementation errors and their boundaries › Vanishing local errors can accumulate · lines 1225–1274 · proof included or linked

En08-B@1#prop:accumulation

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Fix a nonzero null target of distance $r > 0$ and a number $λ > 0$ . There are local gate implementations with per-gate half-diamond error at most $λ / j$ , for all sufficiently large $j$ , whose logical channels do not converge to $id_{D}$ .

Proof. Let

R_{D} (Z) = Tr (Z) \frac{I _{D}}{D}, P_{p} = (1 - p) id_{D} + p R_{D} .

At each nonzero route step, apply the ideal SWAP and then $P_{p_{j}}$ to the arriving payload register, where $p_{j} = λ / j$ for sufficiently large $j$ . This is a channel on the same permitted two-site block. Its distance from the ideal SWAP channel is at most $p_{j}$ .

All other registers remain blank. If the route has $m_{j}$ SWAPs, the logical channel is

P_{p_{j}}^{m_{j}} = (1 - p_{j})^{m_{j}} id_{D} + (1 - (1 - p_{j})^{m_{j}}) R_{D} .

Theorem thm:main gives $m_{j} / j \to r / ℓ$ , so

(1 - λ / j)^{m_{j}} ⟶ e^{- λ r / ℓ} < 1.

To see a positive complete-channel discrepancy, apply the channel to one half of

∣ Ω_{D} ⟩ = \frac{1}{D} a = 0 \sum^{D - 1} ∣ a ⟩ ∣ a ⟩ .

The replacement component has overlap $D^{- 2}$ with $∣ Ω_{D} ⟩$ . Testing the projector $∣ Ω_{D} ⟩ ⟨ Ω_{D} ∣$ therefore gives

d_{⋄} (P_{p_{j}}^{m_{j}}, id_{D}) \geq (1 - (1 - p_{j})^{m_{j}}) (1 - D^{- 2}),

whose limit is positive.

Displayed formulas in this passage

unlabeled · line 1236
$R_{D} (Z) = Tr (Z) \frac{I _{D}}{D}, P_{p} = (1 - p) id_{D} + p R_{D} .$
unlabeled · line 1249
$P_{p_{j}}^{m_{j}} = (1 - p_{j})^{m_{j}} id_{D} + (1 - (1 - p_{j})^{m_{j}}) R_{D} .$
unlabeled · line 1255
$(1 - λ / j)^{m_{j}} ⟶ e^{- λ r / ℓ} < 1.$
unlabeled · line 1261
$∣ Ω_{D} ⟩ = \frac{1}{D} a = 0 \sum^{D - 1} ∣ a ⟩ ∣ a ⟩ .$
unlabeled · line 1268
$d_{⋄} (P_{p_{j}}^{m_{j}}, id_{D}) \geq (1 - (1 - p_{j})^{m_{j}}) (1 - D^{- 2}),$

Nearby in this section

section: Vanishing local errors can accumulate