What is the Bernstein-Vazirani algorithm?

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The Bernstein-Vazirani (BV) algorithm is a quantum algorithm designed to determine a hidden string efficiently using a quantum computer. It demonstrates how quantum computers can solve certain problems faster than classical algorithms, even though it is simpler than algorithms like Shor’s or Grover’s.

Problem Statement:

• There is a hidden $n$-bit string $s = s_1 s_2 … s_n$.

• You are given an oracle function $f(x)$ defined as:

  $$f(x) = s \cdot x \ (\text{mod 2})$$

  where $x$ is an $n$-bit input and $\cdot$ is the bitwise dot product (XOR sum).

• Goal: Find the hidden string $s$ using the fewest queries.

Classical Approach:

• Requires $n$ queries in the worst case, since each bit of $s$ must be determined separately.

Quantum Approach (Bernstein-Vazirani Algorithm):

1. Initialize qubits:

   • Prepare $n$ qubits in $|0\rangle$ and one ancilla qubit in $|1\rangle$.

2. Apply Hadamard gates:

   • Put all qubits into a superposition, representing all possible inputs simultaneously.

3. Oracle query:

   • Apply the oracle $U_f$ which encodes the hidden string into the phase of the superposition.

4. Apply Hadamard gates again:

   • This step interferes the amplitudes so that the hidden string $s$ is encoded in the computational basis.

5. Measurement:

   • Measure the $n$ qubits to directly obtain the hidden string $s$ in one query.

Why it’s important:

• Shows a quantum advantage: Only one oracle call versus $n$ classically.

• Illustrates quantum parallelism and interference.

• Serves as a foundation for understanding more advanced quantum algorithms.

✅ In short: The Bernstein-Vazirani algorithm is a quantum method for finding a hidden bit string with a single query, showcasing how quantum superposition and interference can outperform classical approaches.

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Explain amplitude amplification in Grover’s algorithm.

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