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I have a non-unitary and non-hermitian matrix of $16\times16$. I have used singular value decomposition to break that matrix into three matrices i.e. $A = UDV$, Where $U$ and $V$ are unitary but still non-hermitian, while D is the diagonal matrix. Now, I want to implement this matrix on a quantum circuit in qiskit in form of universal set of gates.

Here, I have came across the term Shannon decomposition and then quantum Shannon decomposition. Currently, I am unable to grasp the concept and cannot find the basic mathematics in matrix form that governs the decomposition process. I know it must be simple but stuck here.

If there are relevant books on the topic as research papers make it complex. Also, if there is a qiskit code that can do it?

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Given that the matrix that you are trying to implement is non-unitary, the quantum Shannon decomposition (originally introduced in this paper by Shende, Bullock and Markov) is not a useful approach. To implement a non-unitary operation on a quantum computer, one must embed it in a larger unitary operation in a process that goes by the name of block encoding, as discussed in this previous question. For a more in-depth pedagogical introduction, I would recommend Lin Lin's lecture notes (Chapter 6, in particular). Finally, since your matrix is not even Hermitian, you should consider quantum singular value transformation (see Chapter 8 of the aforementioned lecture notes or this PRX Quantum tutorial).

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  • $\begingroup$ Thankyou for the help, it really cleared my concepts on the topic from the Lin Lin's lecture notes. Additionally, I used Singular value decomposition in Mathematica to decompose the matrices into unitary and diagonal matrix. Further, I used the code from quantum-decomp to form the quantum circuit in qiskit. Currently, I want to test this circuit, if it gives correct results or not. I will share if it works. $\endgroup$ Commented Jan 30 at 4:48

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