# Notes - Quantum Information HT24, Unitary transformations and reversible processes

### Flashcards

What are the reversible gates for a classical bit?

- Identity
- NOT

What are the reversible gates for a qubit?

Every unitary matrix $U$, up to global phase.

Suppose a quantum system is in state $ \vert \psi \rangle$ and we apply the reversible process described by the unitary matrix $U$. What is the state of the system at the end of the process?

Suppose you have an orthonormal basis $\{ \vert \psi _ k \rangle : k = 0, \cdots, d-1 \}$ and a set of real numbers $\{\gamma _ k \in \mathbb R : k = 0, \cdots, d-1\}$. How can you construct a unitary matrix, and what is in fact true about this expression?

In fact, for a suitable choice of phases and ONB, any unitary matrix can be written this way.

How could you immediately tell that the matrix

\[M = i|+\rangle\langle0| - i|-\rangle\langle 1 |\]
is unitary without expanding $M^\dagger M$?

It’s of the form

\[U = \sum^{d-1}_{k = 0} |\phi_k \rangle \langle \psi_k |\]where $\{ \vert \psi _ k \rangle : k = 0, \cdots, d-1 \}$ and $\{ \vert \phi _ k \rangle : k = 0, \cdots, d-1 \}$ are orthonormal bases.

Suppose you have two orthonormal bases

\[\\{|\psi_k \rangle : k = 0, \cdots, d-1 \\}\]
and

\[\\{|\phi_k \rangle : k = 0, \cdots, d-1 \\}\]
How can you construct a unitary matrix, and what is in fact true about this expression?

In fact, for a suitable choice of ONB, any unitary matrix can be written this way.

How can we use reversible processes to reduce a measurement in an arbitrary orthnormal basis $\{ \vert \phi _ k \rangle : k = 0, \cdots, d-1 \}$ into a measurement in the computational basis? State the explicit unitary matrix used.

- Apply the reversible transformation corresponding to $U = \sum _ {k = 0}^{d-1} \vert k\rangle \langle \phi _ k \vert $
- Perform a measurement in the computational basis
- Apply the inverse of $U$, which is $U^\dagger$.

Formally if the system is initially in the state $\psi$, then we have

\[\begin{aligned} p_n &= |\langle n | U | \psi \rangle|^2 \\\\ &= |\langle \phi_n | \psi\rangle|^2 \end{aligned}\]Suppose you are asked to check if a matrix $T$ is a unitary matrix, where $T$ is defined in terms of its action on $ \vert x\rangle \otimes \vert y\rangle \otimes \vert z\rangle$ is unitary, e.g. like for $\mathtt{TOFFOLI}$

\[\mathtt{TOFFOLI}|x\rangle \otimes |y\rangle \otimes |z\rangle = |x\rangle \otimes |y\rangle \otimes |z + \mathtt{AND}(x, y)\rangle\]
How can you do this?

Use the fact that $U^\dagger U = I$ iff

\[(\langle x| \otimes \langle y| \otimes \langle z|) U^\dagger U (|x'\rangle \otimes |y'\rangle \otimes |z'\rangle) = \delta_{xx'} \delta_{yy'}\delta_{zz'}\]