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I have been told that one of the great keys that unlock quantum computing's potential is the reversibility of quantum logic gates as for classical gates there's some loss of information, but I cannot grasp this concept. Mathematically I see why the quantum logic gate is reversible, it is a mere unitary operator but on the classical one I don't see where the information is lost, could someone clarify it?

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Mathematically I see why the quantum logic gate is reversible, it is a mere unitary operator but on the classical one I don't see where the information is lost, could someone clarify it?

For illustration, let's take the classical XOR gate.

Say you know that the output or end result of a certain XOR operation is 1. Now what could have been the possible values for A and B? Either A = 0 & B = 1 or A=1 & B = 0. That is, you cannot unambiguously reconstruct its two inputs from its single output i.e. the information about the initial state of A and B is lost after the XOR logic operation. This is exactly what they mean when they mean by "irreversibility of classical logic gates".

Interestingly, the reversible quantum version of the XOR gate is the CNOT gate.

However, there are exceptions like the classical NOT gate.

For the NOT gate, if you're given that the output state 1, then you can unambiguously say the initial state of A was 0. Similarly, if you're given NOT A is 0, you can unambiguously say that A was 1. No information is lost here during the logic operation.

It's possible to perform all classical logic operations using only reversible gates. But in those cases, often it becomes necessary to use ancilla bits. For instance, the reversible Toffoli gate gate can implement all classical logic functions. The Toffoli gate has a quantum version too. Basically, all reversible classical logic operations can be directly mapped to quantum gate operations, which are unitary and reversible. In case you're wondering why quantum gates are unitary, read this answer.

I have been told that one of the great keys that unlock quantum computing's potential is the reversibility of quantum logic gates as for classical gates there's some loss of information, but I cannot grasp this concept.

That's true. Read up the Wikipedia articles on reversible computing and Landauer's principle. Note that there's two kinds of reversibilities: logical reversibility and physical reversibility. A universal Turing machine can be made both physically and logically reversible. Logical and physical reversibility together would enable running efficient algorithms and energy-efficient computation in quantum computers.

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