Why can't Grover's algorithm apparently solve this simple situation?

Question

I'm utilizing Grover's algorithm to solve a straightforward problem. It presents eight options, but only four are correct (the 1's in the S column)

$$\begin{array}{c|c|c|c} q_0& q_1\ & q_2\ &S \\ \hline 0 & 0 & 0 &1 \\ \hline 0 & 0 & 1 &1 \\ \hline 0 & 1 & 0 &1 \\ \hline 0 & 1 & 1 &0 \\ \hline 1 & 0 & 0 &0 \\ \hline 1 & 0 & 1 &0 \\ \hline 1 & 1 & 0 &0 \\ \hline 1 & 1 & 1 &1 \\ \hline \end{array}$$

Why can't Grover's algorithm apparently solve this simple situation?

The oracle produce the state (the order is $|q_2 q_1 q_o \rangle$):

$-\frac{\sqrt{2}}{4} |000\rangle+\frac{\sqrt{2}}{4} |001\rangle- \frac{\sqrt{2}}{4} |010\rangle+\frac{\sqrt{2}}{4} |011\rangle- \frac{\sqrt{2}}{4} |100\rangle+\frac{\sqrt{2}}{4} |101\rangle+\frac{\sqrt{2}}{4} |110\rangle- \frac{\sqrt{2}}{4} |111\rangle$

where the solutions clearly are $|000 \rangle$ , $|010 \rangle$ , $|100 \rangle$ and $|111 \rangle$, but the code did not find any.

Here is the code

arquivo='002.dimac'
with open(arquivo, 'r') as f:
    dimacs = f.read()
print(dimacs)  # let's check the file is as promised

       
# steps 2 & 3 of Grover's algorithm
from qiskit import QuantumCircuit
from qiskit.circuit.library import GroverOperator
from qiskit.circuit.library import PhaseOracle
oracle = PhaseOracle.from_dimacs_file(arquivo)
grover_operator = GroverOperator(oracle)


qc=QuantumCircuit(3)
qc.h([0,1,2])
qc = qc.compose(grover_operator)

qc.measure_all()
qc.draw()

# Simulate the circuit
from qiskit import Aer, transpile
sim = Aer.get_backend('aer_simulator')
t_qc = transpile(qc, sim)
counts = sim.run(t_qc,shots=1024).result().get_counts()
  
print(counts)
   

Output:

                c example DIMACS-CNF 3-SAT
                p cnf 3 4
                -1 2 3 0
                1 -2 -3 0
                -1 -2 3 0
                -1 2 -3 0

                ┌───┐┌────┐ ░ ┌─┐      
           q_0: ┤ H ├┤0   ├─░─┤M├──────
                ├───┤│    │ ░ └╥┘┌─┐   
           q_1: ┤ H ├┤1 Q ├─░──╫─┤M├───
                ├───┤│    │ ░  ║ └╥┘┌─┐
           q_2: ┤ H ├┤2   ├─░──╫──╫─┤M├
                └───┘└────┘ ░  ║  ║ └╥┘
        meas: 3/═══════════════╩══╩══╩═
                               0  1  2 

        {'001': 122, '110': 119, '101': 125, '100': 110, '111': 138, '011': 139, '010': 126, '000': 145}
```

Answer 1

In this case exactly 50% of the basis states in the search space of the problem are correct answers, so any number of Grover iterations don't improve the probability of getting the answer when measuring the state - it remains 50%.

The easiest way to see this (at least it was for me) is by using a visualization of the system state during Grover's search as a unit circle, with horizontal axis being the superposition of all non-solution basis states and the vertical axis - the superposition of all solution basis states (see https://github.com/microsoft/QuantumKatas/blob/main/tutorials/ExploringGroversAlgorithm/VisualizingGroversAlgorithm.ipynb for the visuals). If 50% of the basis states in the search space are solutions, the system starts in the state that is tilted at an angle $45^{\circ}$ to the horizontal axis, and each Grover iteration rotates the state by $90^{\circ}$, so the state always remains exactly halfway between the vertical and the horizontal axes and always has 50% probability of measuring a state that is an answer.