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I have a 32 X 32 matrix show in the picture below and I want to make a circuit for this matrix. Also in the picture below to the matrix picture I have written the each column in bra-ket notation to get the idea of operators required in the circuit by seeing the qubits.

32X32 matrix

enter image description here

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If I'm reading your matrix correctly, it can be written as $$ \cos(\theta)I+i\sin(\theta)Z\otimes X^{\otimes 4}. $$ So, if I were you, I'd start by performing a rotation $R_z(2\theta)$ on the first qubit, creating $$ (\cos(\theta)I+i\sin(\theta)Z)\otimes I, $$ and then apply four controlled-nots, controlled off each of the other 4 qubits, and targetting the first qubit. That creates $$ \cos(\theta)I+i\sin(\theta)Z^{\otimes 5}. $$ You can then convert the $Z$s into $X$s using Hadamards. Overall, the circuit looks something like: enter image description here

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  • $\begingroup$ thanks @DaftWullie for answering my question if you don't mind can you please tell me what should I read for having a good command over matrix decomposition and circuit construction. $\endgroup$
    – zircon
    Commented Feb 19, 2020 at 14:18
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    $\begingroup$ I would start with chapter 4 of Nielsen & Chuang, although, really, it's just about practice. $\endgroup$
    – DaftWullie
    Commented Feb 19, 2020 at 14:23
  • $\begingroup$ I think in the first equation [cos(θ) + i sin(θ) Z⊗X^⊗4].There is minus not the plus. So the correct equation is [cos(θ) - i sin(θ)Z⊗^X⊗4]. I understand this mistake happen due to the poor quality picture. Now Is there any change in the first operator or just in the value of theta to change the sign? @DaftWullie $\endgroup$
    – zircon
    Commented Feb 20, 2020 at 10:55
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    $\begingroup$ @vardhannegi You're probably right about the sign. As you say, just flip the sign of the rotation angle, $\theta\mapsto-\theta$. $\endgroup$
    – DaftWullie
    Commented Feb 20, 2020 at 11:33

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