According to Wigner’s theorem, every symmetry operation must be represented in quantum mechanics by an unitary or an anti-unitary operator. To see this, we can see that given any two states $|\psi\rangle$ and $|\psi'\rangle$, you would like to preserve
$$|\langle\psi|\psi\rangle'|^2=|\langle O\psi|O\psi \rangle'|^2 \tag{1}$$
under some transformation $O$. If $O$ is unitary, that works. Yet, we can verify that an anti-unitary $A$ operator such that
$$\langle A\psi| A\psi'\rangle= \langle\psi|\psi'\rangle^* ,\tag{2}$$ works too; where ${}^*$ is the complex conjugate. Note that I cannot write it as $\color{red}{\langle \psi| A^\dagger A|\psi\rangle'}$ as $A$ does not behave as usual unitary operators, it is only defined on kets $|A\psi\rangle=A|\psi\rangle$ not bras.
I was wondering if one could make an anti-unitary gate? Would that have any influence on what can be achieved with a quantum computer?
Example
I was thinking on some kind of time reversal gate $T$. The time reversal operator is the standard example of anti-unitarity.
As a qubit is a spin-1/2 like system, for a single qubit we need to perform $$T=iYK\tag{3}$$ where $K$ is the complex conjugate. If you have an state $$|\psi\rangle=a|0\rangle+b|1\rangle\tag{4}$$ then $$K|\psi\rangle=a^*|0\rangle+b^*|1\rangle.\tag{5}$$ I do not think you can build the operator $K$ from the usual quantum gates without knowing the state of the qubit beforehand. The gate has to be tailored to the state of the qubit.
Experiments
I know now of one preprint Arrow of Time and its Reversal on IBM Quantum Computer (2018) where the authors claim to have achieved this on a IBM backend. But I do not know if their time reversal algorithm is the same as making an anti-unitary gate.
