Quantum Circuits Must Be Reversible

A closed quantum circuit cannot throw information away. Its gates are unitary operators, and unitary operators have inverses.

This is the basic reason quantum circuits are reversible.

Unitary means invertible

A closed-system quantum gate is represented by a unitary operator . By definition,

Therefore

So every valid closed-system quantum gate can be undone. If a state evolves as

then applying recovers the original state:

Irreversible classical functions lose information

Many classical functions are not reversible. For example,

loses information if only the sum is kept.

The output could have come from many inputs:

So the map

cannot be inverted in general. The output does not contain enough information to recover the input.

That kind of collapse is not allowed for a unitary gate, because a unitary gate must have a unique way back.

Reversible embeddings keep the input

To use a classical function inside a quantum circuit, one usually embeds it into a reversible transformation.

For a classical function , the standard pattern is

where is bitwise XOR.

The important point is that is preserved. The function value is written into a separate target register by modifying .

This transformation is reversible because applying it twice recovers the original target:

The input was kept all along, so the second application uses the same again.