Need help regarding building a custom VQE

Question

I want to write code for a custom Variational Quantum Eigensolver (VQE) capable of computing eigenvalue(s) for non-Hermitian systems, based on this paper. I have formulated a cost function (provided below, which may need correction), and I wish to test this custom VQE on a non-Hermitian matrix, for example, $$H = \begin{bmatrix}1 & -2 \\ 3 & 4 \end{bmatrix}\,.$$ I attempted to do this using VQE from qiskit_algorithm but encountered two issues. Firstly, it doesn't seem to allow the cost_function as an input, which it used to do earlier, I believe. Secondly, I tried to convert the matrix into an Operator through SparsePauliOp, but that also doesn't seem to work. Though I used qiskit, I don't mind suggestions or guidance in other libraries.

def cost_function(params):
    # Apply the ansatz
    U = ansatz.assign_parameters(params)
    # n is number of qubits (len(H))
    # Compute the quantum process snapshot
    QPS = qiskit.QuantumCircuit(n + 1, n)
    QPS.h(0)
    QPS.append(U, range(1, n + 1))
    QPS.append(qiskit.quantum_info.Operator(A), range(1, n + 1))
    QPS.append(U.inverse(), range(1, n + 1))
    QPS.measure(range(1, n + 1), range(n))
    
    # Execute the circuit
    result = quantum_instance.execute(QPS).result()
    counts = result.get_counts()
    
    # Compute the expectation value
    expectation = 0
    for key, value in counts.items():
        # Convert the key to a binary string
        key = format(int(key), '0' + str(n) + 'b')
        
        # Compute the probability
        prob = value / shots
        
        # Compute the phase
        phase = 0
        for i in range(n):
            phase += int(key[i]) * 2 * np.pi / (2 ** n)
        
        # Add the contribution to the expectation value
        expectation += prob * np.exp(1j * phase)
    
    # Return the absolute value of the expectation value
    return numpy.abs(expectation)

Answer 1

1. Instead of using the VQE class which is rather a black box, use Estimator and define the VQE cycle step by step. Check this tutorial
2. SparsePauliOp.from_operator(Operator(H)) should make it. Both SparsePauliOp and Operator are imported from qiskit.quantum_info