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
\[M = i \vert +\rangle\langle0 \vert - i \vert -\rangle\langle 1 \vert\]
is unitary without expanding $M^\dagger M$?
It’s of the form
\[U = \sum^{d-1} _ {k = 0} \vert \phi _ k \rangle \langle \psi _ k \vert\]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
\[\\{ \vert \psi _ k \rangle : k = 0, \cdots, d-1 \\}\]
and
\[\\{ \vert \phi _ k \rangle : k = 0, \cdots, d-1 \\}\]
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
\[\begin{aligned} p _ n &= \vert \langle n \vert U \vert \psi \rangle \vert ^2 \\\\ &= \vert \langle \phi _ n \vert \psi\rangle \vert ^2 \end{aligned}\]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 $\mathtt{TOFFOLI}$
\[\mathtt{TOFFOLI} \vert x\rangle \otimes \vert y\rangle \otimes \vert z\rangle = \vert x\rangle \otimes \vert y\rangle \otimes \vert z + \mathtt{AND}(x, y)\rangle\]
How can you do this?
Use the fact that $U^\dagger U = I$ iff
\[(\langle x \vert \otimes \langle y \vert \otimes \langle z \vert ) U^\dagger U ( \vert x'\rangle \otimes \vert y'\rangle \otimes \vert z'\rangle) = \delta _ {xx'} \delta _ {yy'}\delta _ {zz'}\]