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The standard way to implement a reversible XOR gate is by means of a controlled-NOT gate or CNOT; this is the "standard quantum XOR operation". Physics.Stackexchange

Is there a "standard quantum XNOR operation"?

The XNOR gate (sometimes ENOR, EXNOR or NXOR and pronounced as Exclusive NOR) is a digital logic gate whose function is the logical complement of the exclusive OR (XOR) gate. Wikipedia

Alternatively, what is the logical complement of the CNOT gate?

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There is no "standard" method to implement XNOR, but it can be logically obtained by attaching a NOT gate (often called an X gate in quantum computing) to a logical XOR (which you know is implemented using CNOT). The X gate is applied to the target qubit of the CNOT.

To answer your question more directly, there is no standard "quantum gate" that is equivalent to XNOR. The best way to implement XNOR in a quantum circuit is with a CNOT and an X on the second qubit.

The reason why {CNOT,X} can give you a logical XNOR was explained in this answer to your own question 3.5 months ago.

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  • $\begingroup$ $\text{cNOT}\cdot(\mathbb{1}\otimes X)$? What would the circuit look like (not sure which qubit would get the x)? Also, would it be accurate to call this 'gate' not CNOT? $\endgroup$
    – user820789
    Commented Oct 1, 2018 at 1:22
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    $\begingroup$ Yes that is correct! :) $\endgroup$
    – user1271772 No more free time
    Commented Oct 1, 2018 at 1:22
  • $\begingroup$ I would like to update this question to more accurately ask the question I haven't figured out how to succinctly articulate in a meaningful way & am wondering if perhaps you could advise, as you have already taken time to provide an answer. I had asked in the comments of the initial XNOR question about SWAP. Upon further reflection however, I'm not sure that is correct. I have also learned since that time that XOR == "Symmetric Difference" & XNOR == "Logical Biconditional." "Quantum Symmetric Difference" & "Quantum Logical Biconditional" both yield less than 5 results in Google.. $\endgroup$
    – user820789
    Commented Oct 6, 2018 at 3:37
  • $\begingroup$ I also have been looking into quantum, vector & fuzzy logics (De Morgan's laws seem to keep popping up a lot as well). The concepts of equality & equivalence also are recurrent. As you seem very versed in maths & quantum, perhaps you can understand what it is I am after, &/or identify the gaps in my current thinking & make a suggestion on what area(s) of math/quantum to focus on (has been on set/type theory + complexity classes; very interested in optical computing as well) in my continuing education. Thank you again for taking the time to provide so many great answers; I appreciate it! $\endgroup$
    – user820789
    Commented Oct 6, 2018 at 3:55

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