C2: Generate Uniform Amplitude Superposition State II

Problem Statement

You are given integers $n$ and $L$. Implement the operation of preparing the state $\ket{\psi}$ on a quantum circuit $\mathrm{qc}$ with $n$ qubits such that the states $\ket{0},\ \ket{1},\ ...\ \ket{L-1}$ are observed with equal probabilities, and the sum of these probabilities equals $1$.

The error in the sum of probabilities can be up to $5.0 \times 10^{-3}$.

More Precise Problem Statement

Define the state $\ket{\psi}$ prepared by $\mathrm{qc}$ as

where $a_i$ denotes the probability amplitude of the computational basis state $\ket{i}$.

Implement $\mathrm{qc}$ satisfying following conditions:

• $|a_0| = |a_1| = \cdots = |a_{L-1}|$
• $\sum_{i=0}^{L-1} |a_i|^2 > 1 - 5.0 \times 10^{-3}$

Constraints

• $1 \leq n \leq 10$
• $1 \leq L \leq 2^n$
• The circuit depth must not exceed $1000$.
• Integers must be encoded by little-endian notation, i.e., $\ket{100} = 1 \neq \ket{001}$.
• The submitted code must follow the specified format:

from qiskit import QuantumCircuit
 
 
def solve(n: int, L: int) -> QuantumCircuit:
    qc = QuantumCircuit(n)
    # Write your code here:
 
    return qc

Sample Input

• $n = 3,\ L=3$: Implemented circuit $\mathrm{qc}$ should perform the following transformation.

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