Algorithm to Verify Local Equivalence of Stabilizer States

Adam Burchardt; Jarn de Jong; Lina Vandr\'e

Algorithm to Verify Local Equivalence of Stabilizer States

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arxiv 2410.03961 v2 pith:TJSB6OFU submitted 2024-10-04 quant-ph

Adam Burchardt , Jarn de Jong , Lina Vandré This is my paper

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keywords stateslocalstabilizeralgorithmequivalencegraphverifyapproach

verification ladder T0 review T1 audit T2 compute T3 formal

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abstract

We present an algorithm for verifying the local unitary (LU) equivalence of graph and stabilizer states. Our approach reduces the problem to solving a system of linear equations in modular arithmetic. Furthermore, we demonstrate that any LU transformation between two graph states takes a specific form, naturally generalizing the class of local Clifford (LC) transformations. Lastly, using existing libraries, we verify that for up to $n=11$, the number of LU and LC orbits of stabilizer states is identical.

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Local Equivalences of Graph States
quant-ph 2025-11 conditional novelty 8.0 of 10

Graph states are LU-equivalent if and only if they are linked by r-local complementations for some integer r; LU-equivalence is decidable in quasi-polynomial time, and LU=LC holds on at most 19 qubits.

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