Let $A, B \subseteq P_n$, 2 finite sets of k-local commuting Pauli operators from the Pauli group $P_n$.
Can we always a finite depth unitary $U$ such that $U^ \dagger AU=B$?
2026-03-26 12:52:40.1774529560
mapping of local Pauli operators
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Elements of A commute with each other and the same goes for elements in B.
K-local means that each operator in the set acts on at most k qubits when the total number of qubits is n.
Finite depth unitary in the sense of finite depth quantun circuit.
I have already proved the answer is NO when we restrict ourselves to finite depth circuits with spacially local gates on surface codes. The general case seems much more challenging to prove