This article serves as a reference for myself. I also hope it can be helpful for anyone: self-studying quantum computation wants to follow academic updates in qc needs career advice learn the theory study resources list maintained by poig on GitHub contains learning material for all levels has links to online textbooks in pdf Simons Institute’s YouTube Channel doing interdisciplinary studies including qc regularly updates presentations on cutting edge topics project & research opportunity Unitary Fund provides opportunities to work on a quantum open source project, QuTiP mentored by the core developers for a 3-month period. Quantum Open Source Foundation(qosf) has monthly challenge on their GitHub and also holds Quantum Computing Mentorship Program once or twice every year.

Personal solution to the book Quantum Computation and Quantum Information by Michael A. Nielsen & Isaac L. Chuang, chapter 4. Currently under construction. Obviously won’t solve them all, but I’ll try my best. Open to valuable critics and better solutions. S4.2 Single qubit operations ex 4.2 method 1: use taylor expansion to check left and right are equal. method 2: Since the matrix $A$ is involutory, it is non-defective and can be eigen-decomposed with $\pm 1$ eigenvalues. $$ A = P^{-1}SP $$ in which $S$ is a signature matrix (diagonal matrix with $\pm 1$ on diagonal). Then $$ \exp(iAx)= P^{-1} \exp(iSx) P $$ For the part in the middle, $\exp(\pm ix)=\cos(x)\pm i\sin(x)$, $\exp(iSx)=\cos(x)I+i\sin(x)S$