Are anti-unitary gates possible?

Question

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.

Answer 1

No, you can't perform anti-unitary gates. Or rather: if they are possible, then faster than light communication is possible.

For example, here is a circuit that remotely detects whether or not a conjugation gate was performed:

This works because conjugating sqrt(X) is equivalent to inverting it. The conjugation between the X^(1/2) and the X^(-1/2) makes them add up to an X instead of cancelling to an I. So they act as conjugation detectors.

(It might be a bit silly to associate a conjugation with a specific qubit, as it is a global effect. Perhaps it is coming along for the ride as part of another operation, such as the "Universal Not".)

Of course, this diagram assumes a particular notion of simultaneity (indicated by the dashed vertical line). But that means antiunitary operations would reveal a preferred rest frame to the universe! In fact it's even worse than that. Antiunitary operations also break rotational symmetry! The X rotations act as conjugation detectors, as do Z rotations, but Y rotations don't, so you could find the preferred Y axis of the universe. (Well... okay, technically the exact thing you expect to break, and exactly how you expect it to break, will depend on what your favorite interpretation of quantum mechanics is.)