Understanding Hamiltonian's of the Single Qubit Gates and Toffoli gate

Question

As a most general shape, we can write our unitary(unitary here single qubit gates and toffoli gates) in that shape:

$U = \exp({iHt})$

H is the hamiltonian. However single qubit gates does not reqire entanglement. We need just one qubit to apply single qubit gate whereas toffoli gate requires entanglement. Toffoli gate can be written single and two qubit(CNOT) gates. So toffoli gates requires at least two body interaction(hope I am correct until now). And of course there are some quantum optimum pulse techniques to implement toffoli gate at one shot.

My question is how can we prove if a unitary requires 2 body interaction to implement it or not?

For instance when we write the equation $U = exp(iHt)$ what is the next step to prove that? And how can we write/prove it for toffoli gate?

What is the proof differences between toffoli gate and single qubit gate when we try to write to most general shape of the hamiltonian?

Here also another point I am concern is that, I wonder the hamiltonain structure to implement toffoli gate with single shot. I do not want to use single and two qubit gates to implement toffoli gate but I want to implement a direct toffoli gate. For this, what is the general structure of toffoli gate? Maybe this is just a quantum optimum control problem?

Answer 1

It is true that every unitary can be written as $U=e^{i H t}$ where $H$ is a Hermitian matrix.

Also, if $H$ is a 2-local Hamiltonian acting on $n$ qubits, then $U=e^{i H t}$ is a $2^n \times 2^n$ unitary matrix that may entangle pairs of qubits. The amount of entanglement in each pair depends on the parameter $t$ and the structure of the Hamiltonian $H$.

Let's look at a simple example, let $H = t (X \otimes X \otimes I \otimes \cdots \otimes I)$ acting on $n$ qubits and $t \in \mathbb{R}$. Then we have $$e^{-iH/2} = RXX_{1,2}(t).$$ So, we get a 2-qubit gate $RXX_{1,2}(t)$ that can entangle qubits 1 and 2. The resulting gate introduces "interaction" between the first two qubits.

This example hints that if a Hamiltonian is 2-local then the resulting unitary matrix can entangle pairs of qubits. This implies that the unitary matrix must be implemented with 2-qubit gates. Clearly, if $H$ is a 1-local Hamiltonian, then it will generate a bunch of 1-qubit gates.

Now, let's look at the Toffoli gate. A straightforward calculation of eigenvectors and eigenvalues of the Toffoli gate allows finding the generating Hamiltonian. Note that the ket $(|110\rangle -|111\rangle)/\sqrt{2}$ is the eigenvector of the Toffoli gate with the respective eigenvalue $e^{\pi i}$. The rest of the eigenvectors have eigenvalues $e^{i 0}$. This means the generating Hamiltonian has eigenvalues that are all zero except one eigenvalue which is $-2\pi$. Therefore, the generating Hamiltonian of the Toffoli gate is as follows: $$H = -2\pi \frac{|110\rangle -|111\rangle}{\sqrt{2}} \frac{\langle 110| - \langle 111|)}{\sqrt{2}} = -\pi (|110\rangle -|111\rangle)(\langle 110| - \langle 111|).$$ We can rewrite the above in a matrix form: \begin{equation} H = \begin{pmatrix} O & O & O & O\\ O & O & O & O\\ O & O & O & O\\ O & O & O & \pi(X - I)\\ \end{pmatrix} \end{equation} such that $O$ is a $2\times 2$ zero matrix, $X$ and $I$ are the usual $2\times 2$ Pauli operators. It is straighforwad to verify that what follows holds: $$e^{iH/2} = \textrm{Toffoli}.$$