Polynomial Algorithms for Simultaneous Unitary Similarity and Equivalence

TL;DR

The paper tackles the problem of determining when a unitary transformation can simultaneously map multiple matrix pairs via S.U.S or S.U.Eq. It introduces a path-based graph framework and a Solution form that, together with an Equivalent-problem mechanism (via diagonalization of submatrices), progressively reduces the problem to diagonal U blocks. The authors prove that this approach yields a polynomial-time algorithm with complexity $O(pn^4)$ and extend the framework to the bi-graph setting for S.U.Eq. They provide implementation details, discuss numerical stability (noting eigen-gap sensitivity), and propose canonical features to classify matrix collections under Unitary Similarity, with potential applications in quantum evolution and gate design.

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.

Figures (2)

• Figure 1: Pictorial depiction of S.U.S's Solution form with paths and partition of blocks in $U$
• Figure 2: Pictorial depiction of S.U.Eq's Solution-form with paths and partition of blocks in $U$ and $V$