{"ID":2844605,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2511.19439","arxiv_id":"2511.19439","title":"Polynomial Algorithms for Simultaneous Unitary Similarity and Equivalence","abstract":"We present an algorithm to solve the Simultaneous Unitary Similarity(S.U.S) problem which is to check if there exists a Similarity transformation determined by a Unitary $U$ s.t $UA_lU^*=B_l$, $l \\in \\{1,...,p\\}$, where $A_l$ and $B_l$ are $nxn$ complex matrices. We observe that the problem is simplest when $U$ is diagonal, where we see that the `paths' in the graph defined by non-zero elements of $A_l$ and $B_l$ determine the solution. Inspired by this we generalize this to the case when $U$ is block-diagonal to identify a form refered to as the `Solution-form' using `paths' determined by non-zero sub-matrices of $A_l,B_l$ which are non-zero multiples of Unitary. When not in Solution form we find an equivalent problem to solve by diagonalizing a Hermitian or a Normal matrix related to the sub-matrices. The problem is solved in a maximum of $n$ steps. The same idea can be extended to solve the Simultaneous Unitary Equivalence (S$.$U$.$Eq) problem where we solve for $U,V$ in $UA_lV^*=B_l$, $A_l,B_l$ being $mxn$ Complex rectangular matrices. Here we work with the 'paths' in the related bi-graph to define the Solution-form. The algorithms have a complexity of $O(pn^4)$. This work finds application in Quantum Evolution, Quantum gate design and Simulation. The salient features of each step of the algorithm can be retained as Canonical features to classify a given collection of complex matrices up to Unitary Similarity.","short_abstract":"We present an algorithm to solve the Simultaneous Unitary Similarity(S.U.S) problem which is to check if there exists a Similarity transformation determined by a Unitary $U$ s.t $UA_lU^*=B_l$, $l \\in \\{1,...,p\\}$, where $A_l$ and $B_l$ are $nxn$ complex matrices. We observe that the problem is simplest when $U$ is diag...","url_abs":"https://arxiv.org/abs/2511.19439","url_pdf":"https://arxiv.org/pdf/2511.19439v1","authors":"[\"Harikrishna VJ\",\"Vittal Rao\",\"Ramakrishnan K. R\"]","published":"2025-11-08T09:24:59Z","proceeding":"math.RA","tasks":"[\"math.RA\",\"cs.DS\",\"quant-ph\"]","methods":"[]","has_code":false}