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Why is the fastest time complexity of a quantum computer O(sqrt(n))?

Henry Tom
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⚡ Quantum Brief
A 2025 Stack Exchange discussion examines why quantum algorithms like Grover’s achieve O(√n) time complexity for unstructured search problems, outperforming classical O(n) brute-force methods. Grover’s algorithm demonstrates quantum advantage by quadratically speeding up searches in unsorted databases, requiring roughly √n queries to find a marked item among n possibilities. The √n limit stems from quantum amplitude amplification, where each iteration boosts the target state’s probability by a fixed amount, converging optimally at O(√n) steps. Classical computers must check each entry sequentially, while quantum parallelism enables simultaneous evaluation, though measurement collapses the state, necessitating repeated amplification cycles. Experts note this bound applies to unstructured search; structured problems (e.g., Shor’s algorithm) can achieve exponential speedups, but √n remains the proven limit for generic quantum search tasks.
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