## 2.5 The square root of NOT

Now that we have poked our heads into the quantum world, let us see how quantum interference challenges conventional logic.
Consider a following task: design a logic gate that operates on a single bit and such that when it is followed by another, identical, logic gate the output is always the negation of the input.
Let us call this logic gate **the square root of \texttt{NOT}**, or

A simple check, such as an attempt to construct a truth table, should persuade you that there is no such operation in logic.
It may seem reasonable to argue that since there is no such operation in logic, ^{32}

We could also step through the circuit diagram and follow the evolution of the state vector:

Or, if you prefer to work with column vectors and matrices, you can write the two consecutive application of *right to left*)^{33}, where each

One way or another, quantum theory explains the behaviour of ^{34} that corroborate this theory, logicians are now entitled to propose a new logical operation

There are infinitely many unitary operations that act as the square root of

\texttt{NOT} .↩︎Just remember that circuits diagrams are read from

*left to right*, and vector and matrix operations go from*right to left*.↩︎We discuss this in more detail in [Appendix:

**!!to-do!!**Physics against logic, via beamsplitters].↩︎