Problem StatementThe problem is to calculate the coefficients $A_{j_1\cdots j_n}$, of a square matrix A with size $N$ by $N$ of complex double elements, whose weighted sum with $N^2$ irreducible matrices $[j_1]\otimes \cdots \otimes [j_n]$ is equal to $A$. $N=2^n$, where $n$ is a natural number. $[j]$ are the Pauli matrices for $j=(1,2,3)$ or identity for $j=0$. $\otimes$ is the Kronecker product. InputSquare matrix $A$ with size (2^n,2^n), where $n$ is a natural number.OutputWeighted coefficients $A_{j_1\cdots j_n}$, where $j=(0,1,2,3)$.Details that might be relevant in identifying Missing ApplicationI wrote a program that computes the full decomposition of a $N\times N$ matrix into the sum of weighted gamma matrices and their multi-linears in $\mathcal{O}(N^2\log{}N)$ steps.$$A = \sum_{k=0}^{n}\sum_{j=1}^{\begin{pmatrix}2n+1 \\ k\end{pmatrix}}A_{k\{j\}}\sigma^{k\{j\}}$$$A$ is the matrix to decompose. $N=2^n$, where $n$ is a natural number. $A_{k\{j\}}$ are the weighted elements to be computed. $\sigma^{k\{j\}}=\gamma^{j_{1}}\gamma^{j_{2}}\gamma^{j_{3}}\cdots$ are the multi-linears equal to the product of $k$ unique gamma matrices. $k\{j\}$ is the $j$th set of uniquely chosen $k$ indices. For example in $N=4$ and $k = 2$: $k\{j\}$ is pick the $j$th element from the list $(01,02,03,04,12,13,14,23,24,34)$, where none of elements share the same indices. Another representation of$$A = \sum_{j_1=0}^{3}\cdots \sum_{j_n=0}^{3} \[j_1]\otimes \cdots \otimes[j_n] \A_{j_1 \cdots j_n}$$where $\otimes$ is the Kronecker product, $[1]$, $[2]$, $[3]$, are the Pauli matrices, $[0]$ is identity, $A_{j_1 \cdots j_n}$ are the weighted elements to be computed in a different basis, and $(j_1 \cdots j_n)$ can be treated as a number in base 4. One can show that multipliting two matrices $[j_1]\otimes \cdots \otimes[j_n]$ and $[l_1]\otimes \cdots \otimes[l_n]$ to get $[g_1]\otimes \cdots \otimes[g_n]$ is equivalent to $(j_1 \cdots j_n)$ xor $(l_1 \cdots l_n)$ = $(g_1 \cdots g_n)$ but missing the sign.There exists a simple map between $A_{k\{j\}} \longleftrightarrow A_{j_1\cdots j_n}$.Questions(1) Does this algorithm already exist?(2) Does it have an application?
What can I do with this algorithm?
algorithms;reference request;matrices