QCoder

Editorial

Define the integer $k$ as the "smallest integer such that $2^k \geq L$ ". Now consider applying the Hadamard gate to the first $k$ qubits.

By doing so, the absolute values of the probability amplitudes $|a_i|$ of the computational basis states $\ket{0},\ \ket{1},\ ...\ \ket{L-1}$ are all equal, and satisfy:

\begin{equation} |a_0| = |a_1| = \cdots = |a_{L-1}| = \frac{1}{\sqrt{2^k}} \end{equation}

Since $2^{k-1} < L \leq 2^k$ , the sum of the observation probabilities satisfies the following inequality:

\begin{align} \sum_{i=0}^{L-1} |a_i|^2 &= L \times \left(\frac{1}{\sqrt{2^k}}\right)^2 \nonumber\\ &= \frac{L}{2^k} > \frac{2^{k-1}}{2^k} = 0.5\\ \end{align}

Note that the "smallest integer $k$ such that $2^k \geq L$ " can be represented as $\lceil\log_2(L)\rceil$ , where $\lceil(\cdot)\rceil$ denotes the ceiling function.

As a result, you can solve this problem by applying the Hadamard gate to the first $\lceil\log_2(L)\rceil$ qubits.

Pay attention to the corner case that occurs when $L = 1$ .

Sample Code

Below is a sample program:

from qiskit import QuantumCircuit
import math
 
 
def solve(n: int, L: int) -> QuantumCircuit:
    qc = QuantumCircuit(n)
 
    if L == 1:
        return qc
 
    k = math.ceil(math.log2(L))
    qc.h(range(k))
 
    return qc

Supplements

Quantum algorithms that amplify the probability amplitude of the desired state are known as Grover's algorithm and amplitude amplification. This problem can also be solved using such algorithms.

C1: Generate Uniform Amplitude Superposition State I

Editorial

Sample Code

Supplements