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>these matrices must be their own inverse (in mathematical terms, Unitary and Hermitian)

Not necessarily. The only requirement on quantum operators is that they must be unitary. Only observables have to be Hermitian (so they have real eigenvalues).

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Neither unitary or hermitian means that operators are there own inverse. Unitary here means that the inverse is the same as the complex transpose - which means that there is an inverse and it is easy to find. This is important as quantum circuits must be composed of unitaries, i.e. the circuit is reversable. Hermitian operators have no particular special property with respect to inverses.

Quantum gates are unitary because of conservation of probability (the out state must be remain normalized to 1). If it's also hermitian (equal to conjugate transpose) then it's its own inverse.

Thanks for the correction! I thought all quantum operators were their own inverse, but I guess I was wrong.

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