Hello!

I am trying to implement VQLS to solve linear systems of equations of the form Ax=b.

I know that VQLS finds an |x\rangle such that A|x\rangle \propto |b\rangle.

I am using the code from Variational Quantum Linear Solver | PennyLane Demos, but I changed the vector b from the uniform superposition to an unormalized vector b = [1,3,4,2,0.3,2,3,6].

Consequently, the only thing that I changed is the U_b() function so that it uses amplitude encoding to encode this vector:

```
def U_b():
"""Unitary matrix rotating the ground state to the problem vector |b> = U_b |0>."""
qml.AmplitudeEmbedding(features = vector, wires = range(n_qubits), normalize = True)
```

I also changed the Hadamard test function so that it calculates the adjoints of U_b() and CA():

```
def local_hadamard_test(weights, l=None, lp=None, j=None, part=None):
# First Hadamard gate applied to the ancillary qubit.
qml.Hadamard(wires=ancilla_idx)
# For estimating the imaginary part of the coefficient "mu", we must add a "-i"
# phase gate.
if part == "Im" or part == "im":
qml.PhaseShift(-np.pi / 2, wires=ancilla_idx)
# Variational circuit generating a guess for the solution vector |x>
variational_block(weights)
# Controlled application of the unitary component A_l of the problem matrix A.
CA(l)
# Adjoint of the unitary U_b associated to the problem vector |b>.
# In this specific example Adjoint(U_b) = U_b.
qml.adjoint(U_b)()
# Controlled Z operator at position j. If j = -1, apply the identity.
if j != -1:
qml.CZ(wires=[ancilla_idx, j])
# Unitary U_b associated to the problem vector |b>.
U_b()
# Controlled application of Adjoint(A_lp).
# In this specific example Adjoint(A_lp) = A_lp.
qml.adjoint(CA)(lp)
# Second Hadamard gate applied to the ancillary qubit.
qml.Hadamard(wires=ancilla_idx)
# Expectation value of Z for the ancillary qubit.
return qml.expval(qml.PauliZ(wires=ancilla_idx))
```

I have two questions:

1 - Can my implementation correctly solve A|x\rangle \propto |b\rangle or am I missing some steps?

2 - How can I “find” x from the output of VQLS |x\rangle? In the sense, how can I find the true x of Ax=b given the output of the VQLS?

Thank you very much in advance!