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


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. $\endgroup$ – quantum_boy Feb 19 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 Feb 19 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 $\endgroup$ – quantum_boy Feb 20 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 Feb 20 at 11:33

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