A2: Generate Uniform Superposition State

Editorial

In problem A1, we prepared the "uniform superposition" state on a single qubit.

Now, how can we express the state of $n$ qubits when each qubit is in a uniform superposition?

When measuring one qubit, we observe either of the computational basis states $\ket{0}$ or $\ket{1}$ with equal probabiliy. Therefore, when measuring $n$ qubits in sequence, we can observe any one of the $2^n$ possible computational basis states — from $\ket{0...0}$ to $\ket{1...1}$ — with equal probabilities of $1/2^{n}$.

Consequently, the state $\ket{A}$ of the uniform superposition state of all $2^n$ computational basis states can be expressed as

As a result, we can prepare an $n$-qubit uniform superposition by applying a Hadamard gate to each of the $n$ qubits:

Sample Code

Below is a sample program:

from qiskit import QuantumCircuit
 
 
def solve(n: int) -> QuantumCircuit:
    qc = QuantumCircuit(n)
 
    qc.h(range(n))
 
    return qc

Alternatively, you can use a for-loop to apply $n$ Hadamard gates in-turn.

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