A3: Generate State $\frac{1}{\sqrt{2}}\lparen\ket{0}-\ket{2^{n}-1}\rparen$ I

Editorial

First, apply the Hadamard gate to the first qubit.

Next, we transform the state $\ket{1000 \cdots 0}$ to $\ket{1100 \cdots 0}$. To achieve this, apply a controlled-X gate (CNOT gate) with the first qubit as the control bit and the second qubit as the target bit:

Similarly, to transform $\ket{1100 \cdots 0}$ to $\ket{1110 \cdots 0}$, apply a controlled-X gate, with the first qubit as the control bit and the third qubit as the target bit.

By continuing to apply the controlled-X gate until the $n$-th qubit is the target bit, we can prepare the quantum state $\ket{\psi} = \frac{1}{\sqrt{2}} (\ket{00 \cdots 00} + \ket{11 \cdots 11})$.

Finally, apply a Z gate to flip the phase of $\ket{11 \cdots 11}$.

Summarizing these operations, we obtain the following circuit:

Sample Code

Below is a sample program:

from qiskit import QuantumCircuit
 
 
def solve(n: int) -> QuantumCircuit:
    qc = QuantumCircuit(n)
 
    qc.h(0)
 
    for i in range(1, n):
        qc.cx(0, i)
 
    qc.z(0)
 
    return qc