Manuscript Oai212-A@1 · Oai212-A.tex · section · 79323841

Bridge 1 supplement: a standard random-circuit comparison

Bridge 1 supplement: a standard random-circuit comparison · lines 749–768

In particular, an occupied site can become empty with positive probability. This is an explicit obstruction to an exact finite- $q$ identification with a monotone FPP infected set.

For completeness, support is an OTOC observable after a declared probe average. For a Pauli $P$ on $V$ , let

D_{x} (P, B) = \frac{1}{4 q ^{∣ V ∣}} Tr ([P, B_{x}]^{†} [P, B_{x}]),

and average $B$ uniformly over nonidentity local Paulis. Then

\frac{2 ( q ^{2} - 1 )}{q ^{2}} E_{B} D_{x} (P, B) = 1_{{x \in supp P}} .

Indeed, exactly $q^{2} /2$ local Paulis anticommute with a fixed nonidentity local Pauli; identity is not among them.

Displayed formulas in this passage

unlabeled · line 757
$D_{x} (P, B) = \frac{1}{4 q ^{∣ V ∣}} Tr ([P, B_{x}]^{†} [P, B_{x}]),$
unlabeled · line 762
$\frac{2 ( q ^{2} - 1 )}{q ^{2}} E_{B} D_{x} (P, B) = 1_{{x \in supp P}} .$

Bridge 1 supplement: a standard random-circuit comparison

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