Why a Unitary Matrix in Quantum Cryptography?

Question

I've read that in Quantum circuits the matrix that encodes the transformation has to be unitary. I understand that it must preserve the norm of the resulting qubit, but what is the intuition behind it?

Answer 1

Short (unhelpful?) answer: the transformation are unitary, because quantum physics says all transformations on quantum states are unitary. So your question boils down to “Why are unitaries the only allowed transformations?”.

For two reasons:

1. Quantum mechanics restricts the allowed transformation on quantum states to be linear, transforming the vector $|ψ\rangle$ to $V|ψ\rangle$, where V is a matrix. I have no obvious intuition on why the transformation have to be linear$^\dagger$, except that
   1. it gives an elegant and simple mathematical theory;
   2. attempts at constructing nonlinear quantum mechanics have all lead to problematic consequences, e.g. in causality.
2. Once you accept the linearity of the transformation $V$, the unitarity comes from the conservation of the norm of $|φ\rangle$, since this norm corresponds to the sum of probabilities of the different measurement results, which is therefore $1$. We have then, for all $|φ\rangle$, \begin{align*} \langle φ|φ \rangle &=1 & \langle φ|V^\dagger V|φ\rangle=1. \end{align*} Since it is true for all $|φ\rangle$, it implies that the eigenvalues of the Hermitian operator $V^\dagger V$ are all $1$, that is $ V^\dagger V=\mathbb I$: in words, $V$ is unitary. Another (maybe mor intuitive) explanation of this is that unitaries are the only matrices preserving the the scalar products $\langle φ|ψ\rangle$, which measures the distinguisability of states $|φ\rangle$ and $|\psi\rangle$.

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$\dagger$: if anyone knows an intuitive explanation of the linearity of QM, I’d love to read it!

Answer 2

I am not a physicist, but I did take a quantum mechanics elective many decades ago.

The following question and its answers from Physics Stack Exchange illustrate the significance of unitary evolution of states in Quantum Mechanics.

Intutitively, until we make a measurement (at the very end of the computation) and the wavefunction collapses, all the other quantum evolutions are unitary. See especially the references to Wigner's and Kadison's theorems.

Quantum circuits are governed by Quantum Mechanics.