# Is there a matrix exponential $e^{iA}$ gate in IBM Quantum Experience?

Is there a gate that can perform the matrix exponential operation

$$e^{iA}|\Psi\rangle$$

in IBM quantum experience API?

What is the name and symbol for this type of gate (or some other gates that can perform operations like the matrix exponential)?

I need a 2nd opinion that can confirm whether or not

$$e^{i\begin{bmatrix} 8 & 6+i \\ 6-i & -1\end{bmatrix}} = \begin{bmatrix} -0.06558 -0.63357i & 0.38542 -0.66763i \\ 0.14805 -0.75654i & -0.46568 + 0.43456i \end{bmatrix}$$

is correct?

• $A$ is a Hamiltonian - an arbitrary Hermitian matrix that is probably local (or at least sparse). $A$ is a black-box that can be used to represent or encode many varied problems. To understand "this type of gate" I recommend you study Hamiltonian simulation. The W'dia article is not bad - but there they call your matrix $A$ the Hamiltonian matrix $H$. There are also some decent lectures about Hamiltonian simulation on YouTube; this forum certainly has a number of other pointers, too. Commented Jun 20, 2023 at 22:11
• This may or may not be a helpful observation but RX, RY, RZ, RZZ, RXX are all exponentials of Pauli matrices (or their tensor products) with some phase angle $\theta$. The RZZ gate is equal to $e^{-i\frac{\theta}{2}Z \otimes Z}$ for instance. I don't know if more general $e^{iA}$ gates are available in IBM quantum experience. Commented Jun 20, 2023 at 22:37
• @MarkSpinelli thank you. Presently what I need is a second opinion/calculation to check my implementation of $e^{i\begin{bmatrix} 8 & 6+i \\ 6-i & -1\end{bmatrix}} = \begin{bmatrix} -0.06558 -0.63357i & 0.38542 -0.66763i \\ 0.14805 -0.75654i & -0.46568 + 0.43456i \end{bmatrix}$ if anyone has other software that can compute the complex matrix exponential, please? Commented Jun 20, 2023 at 23:21
• Actually I found a site that can do the calculation emathhelp.net/en/calculators/linear-algebra/… . The answer seems correct. Commented Jun 20, 2023 at 23:31
• Sure, your matrix is hermitian, and you probably correctly calculated it’s corresponding unitary (the matrix exponential). What you’d like to do now, is rewrite that matrix exponential in terms of other, well-used quantum gates such as $X,Y,Z$, CNOT, etc. For that, you’ll have to study Hamiltonian simulation. Or, you could ask a direct question like “How can I simulate the following $2\times 2$ Hamiltonian?” Commented Jun 21, 2023 at 0:36