grovers-solution.ipynb

Kernel: Unknown Kernel

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These are solutions to exercises from the grovers.ipynb notebook.

# EXERCISE 1 SOLUTION
# Define U_9 so that U_9|1001> = -|1001> but fixes 
# all other computational basis states |x>

# Set the number of qubits for the computational basis states
num_qubits = 4

@cudaq.kernel
def U9(qubits: cudaq.qvector, aux_qubit: cudaq.qubit):
    
    x(qubits[1])
    x(qubits[2])
    z.ctrl(qubits, aux_qubit)
    x(qubits[1])
    x(qubits[2])
    
# Create a kernel to test our defintion of U9
@cudaq.kernel
def flip_phase_of_9_only(num_qubits : int):
    qubits = cudaq.qvector(num_qubits)
    aux_qubit = cudaq.qubit()
    # Place the aux_qubit in the minus state
    x(aux_qubit)
    h(aux_qubit)
    
    # Initialize the the qubits in a state of equal superposition
    equal_superposition(qubits)
    
    # Apply the U9 kernel
    U9(qubits, aux_qubit)

# Get the state of the flip_phase_of95_only kernel
result_with_aux_qubit = cudaq.get_state(flip_phase_of_9_only,num_qubits)

print('The phases of the computational basis states after an application of U9 are', check_phase(result_with_aux_qubit))

# EXERCISE 2 SOLUTION
# Define a phase oracle that flips the phase of both 1111 and 1001, fixing all other computational basis states
# assuming that the aux_qubit is initialized in the minus state
@cudaq.kernel
def phase_oracle(qubits: cudaq.qvector, aux_qubit: cudaq.qubit):
    U9(qubits, aux_qubit)
    U15(qubits, aux_qubit)

EXERCISE 3 SOLUTION

$$f(000) = f(110) = 1, \quad f(x) = 0 \text{ otherwise}.$$

Applying Grover's algorithm, we note that the total number of states is $N = 2^3 = 8$ and the number of solutions is $t = 2$. The superposition of solutions is given by $$|G\rangle = \frac{1}{\sqrt{2}}(|000\rangle + |110\rangle),$$ while the equal superposition of the remaining states is $$|B\rangle = \frac{1}{\sqrt{8-2}}(|001\rangle + |010\rangle + |100\rangle + |011\rangle + |101\rangle + |111\rangle).$$

As we have $N = 2^3 = 8$, the angle $\theta$ satisfies: $$\theta = \arcsin\left(\sqrt{\frac{t}{N}}\right) = \arcsin\left(\sqrt{\frac{2}{8}}\right) = \arcsin\left(\frac{1}{2}\right) = \frac{\pi}{6}.$$ We solve for $m$: $$(2m+1)\theta = \frac{\pi}{2} \quad \Rightarrow \quad (2m+1)\frac{\pi}{6} = \frac{\pi}{2} \quad \Rightarrow \quad 2m+1 = 3 \quad \Rightarrow \quad m = 1.$$ Thus, a single iteration of the Grover diffusion operator maps $|\xi\rangle$ exactly to $|G\rangle$, ensuring that the subsequent measurement yields a valid solution with $100\%$ probability!

Try writing the code for this. In particular, you will need to create new code for the phase oracle.