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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.

Josh
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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.

Craig Gidney
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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).

Nati Erez
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