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Are quantum controlled non-unitary operations possible?

For instance can I define a controlled-reset where I reset a target qubit to $|0 \rangle$ if a controlled qubit is in the $|1\rangle$ state? How about if the control is in a superposition?

If the example above is problematic I'd be interested in other examples of non-unitary operations for which this makes sense.

If such a notion does make sense, how might I think about such an operation mathematically?

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  • $\begingroup$ You can check the article "Non-Unitary Quantum Circuits" by Terashima and Ueda, 2003. The short answer is yes it is possible. However, it is inefficient due to the gate cost in implementation of Repeat until success method. $\endgroup$ Commented Mar 29 at 7:29

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You are asking -- and I quote -- if you can

define a controlled-reset where I reset a target qubit to |0⟩ if a controlled qubit is in the |1⟩ state

Which is nothing but classical logic.

Instead, in case you admit the qubits to be in a superposition, then you are asking to apply, as you said, a non-unitary operation. This is something meaningful only in some theories that go beyond quantum computation, which instead works with unitaries.

The closest thing I can think to what you ask for is to use Fredkin gate -- a.k.a. a controlled swap -- with an auxiliary state on the ground state. But this can still entangle your target qubit with the other qubits.

enter image description here

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  • $\begingroup$ Thanks for your response. Just to clarify… does the |i> in your CSWAP diagram refer a single input qubit in an arbitrary state? $\endgroup$
    – Callum
    Commented May 19, 2023 at 22:00
  • $\begingroup$ No, it is because I used quirk (algassert.com/quirk) to write it and I'm forced to chose each state. $\endgroup$ Commented May 19, 2023 at 23:39
  • $\begingroup$ Okay. Perhaps my example with the controlled reset wasn't all that clear. I know that you can do non unitary operations on a quantum computer by conditioning gates on a measurement outcome. Can you do a quantum controlled non-unitary op in this manner? Or is there some no-go result that disallows this? $\endgroup$
    – Callum
    Commented May 20, 2023 at 9:45
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    $\begingroup$ Conditioning gates on a measurement outcome doesn't mean "performing a non-unitary". Regarding your question, as far as I know, non-unitaries are physically meaningful only to describe correlation of a system with the environment, which should be intrinsically not drivable. $\endgroup$ Commented May 20, 2023 at 10:11

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