How to implement a exponential of a hamiltonian, but non-unitary, matrix in QISKIT?

Question

I need a way to implement exponential of a matrix so that I can create a gate that is analogous to rotation using that matrix, similar to how rotation in the $x$ axis uses the Pauli-$X$ gate. This is one of the matrices that I want to implement the exponential.

$A = \frac{1}{2\sqrt{2}}\begin{pmatrix} 0 & 1 & 1 & 0\\ 1 & 0 & 0 & 1\\ 1 & 0 & 0 & 1\\ 0 & 1 & 1 & 0 \end{pmatrix}$

I want to to make a circuit that computes $e^{i \phi A}$

Answer 1

Hamiltonian needs to be Hermitian not unitary. Matrix $A$ is Hermitian so it can be exponentiated using the classes in qiskit's opflow.

Note that, your matrix can be written as $A = \frac{1}{2\sqrt{2}}(I \otimes X + X \otimes I)$

So the following circuit computes $e^{-i \phi A}$

from qiskit.opflow import I, X
from qiskit.opflow import PauliTrotterEvolution, Suzuki
from qiskit.circuit import Parameter
from numpy import sqrt

_const = 1 / (2 * sqrt(2))
A = _const * (I ^ X) + _const * (X ^ I)

phi = Parameter('ϕ')

evolution_op = (phi * A).exp_i() # exp(-iϕA)

trotterized_op = PauliTrotterEvolution(trotter_mode = Suzuki(order = 1)).convert(evolution_op)
circ = trotterized_op.to_circuit()
circ.draw('mpl')

Answer 2

Or, if you have the actual matrix representation you can do

from qiskit.circuit import QuantumCircuit

phi = # some numerical value
A = # your matrix

circuit = QuantumCircuit(2)
circuit.hamiltonian(A, time=phi, qubits=range(2))

Though this will give an inefficient decomposition into gates as it takes the exact matrix exponential and synthesis that into a circuit. If you know the Pauli decomposition of your matrix and can use @Egretta.Thula's solution, that'll be much more efficient.