## Quantum Country exercise 16

This is one in a series of posts working through the exercises in the Quantum Country online introduction to quantum computing and related topics. The exercises in the original document are not numbered; I have added my own numbers for convenience in referring to them.

Exercise 16. Show that the inverse of the Toffoli gate is the Toffoli gate itself.

Answer: With input states $\vert x \rangle$, $\vert y \rangle$, and $\vert z \rangle$ the output states of the Toffoli gate are $\vert x \rangle$, $\vert y \rangle$, and $\vert z \oplus \left( x \wedge y \right) \rangle$.

If those output states are used as the input states to a second Toffoli gate, the first two output states will again be $\vert x \rangle$ and $\vert y \rangle$. The third output state will be $\vert \left[ z \oplus \left( x \wedge y \right) \right] \oplus \left( x \wedge y \right) \rangle$. We have

$\left[ z \oplus \left( x \wedge y \right) \right] \oplus \left( x \wedge y \right) = z \oplus \left[ \left( x \wedge y \right) \oplus \left( x \wedge y \right) \right]$

The expression $x \wedge y$ will have either the value 0 or the value 1. If it has the value 0 then we have

$z \oplus \left[ \left( x \wedge y \right) \oplus \left( x \wedge y \right) \right] = z \oplus \left( 0 \oplus 0 \right) = z \oplus 0 = z$

If it has the value 1 then we have

$z \oplus \left[ \left( x \wedge y \right) \oplus \left( x \wedge y \right) \right] = z \oplus \left( 1 \oplus 1 \right) = z \oplus 0 = z$

We therefore have

$\left[ z \oplus \left( x \wedge y \right) \right] \oplus \left( x \wedge y \right) = z$

so that the third output state from the second Toffoli gate is simply $\vert z \rangle$. Since the first and second output states from the second Toffoli gate were $\vert x \rangle$ and $\vert y \rangle$, the output states of the second Toffoli gate are the same as the input states of the first Toffoli gate. The Toffoli gate therefore acts as its own inverse.

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