Solving Quantum Control Theory Problem with Toffoli Gate Evolution

[Kreizhn]

For the sake of simplicity, I'm dropping unnecessary constants like $\hbar$. I'm working on a problem in quantum control theory, and trying to find a test case to synthesize the ternary Toffoli gate. Unfortunately, I cannot seem to find a basis that admits a possible solution.

In order to remove the control theoretical aspect from this problem and make it only quantum mechanical, if (X,Y,Z) are the two-level Pauli spin matrices, I have the following basis vectors

$$B= \left\{ I \otimes I \otimes X, I \otimes I \otimes Y, I \otimes X \otimes I, I \otimes Y \otimes I, X \otimes I \otimes I, Y \otimes I \otimes I, Z \otimes Z \otimes Z \right\}$$

and I'm trying to create the Toffoli gate

$$T=\begin{pmatrix} 1 \\ & 1 \\ && 1 \\ &&& 1 \\ &&&&1 \\ &&&&&1 \\ &&&&&& 0 & 1 \\ &&&&&& 1 & 0 \end{pmatrix}$$

Now according to numerical calculations, the basis set spans a 72-dimensional Lie group(which seems wrong since it should only span a 64 dimensional space). In any case, the problem seems to be that there's no way to express the Toffoli gate as a discrete time evolution of the identity operator using elements of the basis set, that is

$$T = e^{i H_n t_n} e^{i H_{n-1} t_{n-1}} \cdots e^{i H_1 t_1} I$$

where $t_i > 0, H_i \in B$ for all i=1,...,n.

So to sum up, I have numerically computed a seemingly inconsistent dimension of the Lie Group, and in any case it seems that despite having a suitable quorum to construct the Toffoli gate, I am unable to. Any thoughts?

 

[Tomsk]

Hi, you're right that you can't write the gate in terms of Hamiltonians acting on each qubit (because it creates entangled states), but you should be able to write the whole thing as e^{iHt}. Also you may have an extra 8 dimensions from the I you have in the equation
$$T = e^{i H_n t_n} e^{i H_{n-1} t_{n-1}} \cdots e^{i H_1 t_1} I$$
I don't think you need the I at the end because each e^{iH_i t_i} is a 2x2 matrix.

I think the Hamiltonian is $$|11\rangle\langle 11|\otimes (\sigma_x - I)$$