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A quantum gate is usually defined as a unitary transformation, like the definition found in "Mathematics of Quantum Mechanics" by Scherer. According to this definition, can we consider a quantum gate synonymous with a unitary gate?

However, this definition seems to exclude measurements, which are non-unitary operations, from being considered as quantum gates. How can we address this shortcoming? Are measurements the only non-unitary operations to consider? What are some arguments for and against including measurement operations as quantum gates?

Would it be appropriate to modify the definition of a quantum gate to something like:

A quantum gate is any unitary transformation coupled with the measurement gate.

I have come across a similar question, where the general consensus seems to be that "quantum gates are unitary". The intent of my question is to clarify whether it would be inherently wrong to consider the measurement operator as a quantum gate, and whether there would be any merit in considering this proposed definition.

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  • $\begingroup$ If you want an expanded definition that includes feasible non-unitary operations, people generally talk about "quantum channels" or "CPTP maps," which include unitaries, measurements, and combinations of unitaries and measurements. $\endgroup$ Commented May 27 at 16:01
  • $\begingroup$ I am familiar with quantum channels. My question specifically relates to quantum gates and an appropriate definition for them: what counts as a quantum gate and what does not. $\endgroup$
    – Josh
    Commented May 27 at 16:09
  • $\begingroup$ My point is that "unitary transformations plus measurements" gives you all possible quantum channels. So "unitary transformations plus measurements" already has a name, and calling that collection "quantum gates" would be redundant. $\endgroup$ Commented May 27 at 16:12
  • $\begingroup$ but then when we say "measurement gate", what do we mean by "gate"? are we being informal? or do we mean a gate that is not a quantum gate, since it is not unitary? $\endgroup$
    – Josh
    Commented May 27 at 16:18

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Sometimes people mean unitary-only. Sometimes they mean generically anything you might do to the qubits (like a measurement gate or a reset gate or a dynamical decoupling gate or a leakage removal gate).

Personally I prefer the wider definition, since in error correction the way we plan to implement most unitary effects is with measurements. It's weird to switch from calling your CNOT gate a "gate" to calling it "not a gate" when you start explaining that you did it with lattice surgery with classically-tracked feedback.

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I wouldn't say it's coupled with the measurement gate. I think they are inherently different:

A quantum gate is equivalent to a Unitary matrix. Quantum gates create together a quantum circuit (which, in turn, can also be thought of as a Unitary matrix). The measurement operator is equivalent to a projection and, therefore, of course, non-unitary.

There are other manipulations you can do on a qubit, like the reset operator (practically a measurement), which is also non-unitary, and therefore I wouldn't define it as a quantum gate.

I very much agree with the Wikipedia definition (https://en.wikipedia.org/wiki/Quantum_logic_gate), focusing on the fact that the gates are building blocks of a quantum circuits. With regards to your question, they state the gates are reversible (and, therefore, not including measurements).

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  • $\begingroup$ If measurement is not a quantum gate, then what is meant by "measurement gate"? Why do we use the word "gate", when it is not a gate? $\endgroup$
    – Josh
    Commented May 29 at 9:45
  • $\begingroup$ That's the definition I think is correct. Measurement is a quantum operation that, unlike quantum gates, collapses the state. Because of its non-unitary nature, it is inherently different, and therefore I believe we should treat it as different $\endgroup$
    – Nati Erez
    Commented May 30 at 11:15

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