Addition modulo two reduces binary addition to its remainder modulo two. Written as ⊕, it maps the Boolean negation 1 − x to 1 ⊕ x and provides a common operation for reversible computing. section 6.2.1
Applied to a quantum basis state, the operation connects classical Boolean functions with quantum gates: the function 1 − x becomes an X-gate transformation, while preserving an input bit enables reversible extensions such as (x, y ⊕ x). section 6.2.1
Negation and reversible extension
For x = 0 or 1, compute 1 − x as 1 ⊕ x; the corresponding quantum operation is X. To preserve recoverability, extend a function with an auxiliary bit: F(x, y) = (x, y ⊕ x). section 6.2.1
