Explain Simon’s algorithm.

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🔑 What is Simon’s Algorithm?

Simon’s Algorithm (proposed by Daniel Simon in 1994) is a quantum algorithm that solves a special problem called Simon’s Problem exponentially faster than any classical algorithm.

It was the first quantum algorithm to show an exponential speedup, laying the foundation for Shor’s algorithm.

📌 Simon’s Problem

We are given a black-box function $f: \{0,1\}^n \to \{0,1\}^n$ with the following property:

• There exists a secret string $s \in \{0,1\}^n$.

• The function is 2-to-1:

  • For every $x$, $f(x) = f(x \oplus s)$, where $\oplus$ is bitwise XOR.

  • Meaning: each output corresponds to exactly two inputs that differ by $s$.

👉 Goal: Find the secret string $s$.

⚡ Why is this hard classically?

• A classical algorithm would need about $2^{n/2}$ evaluations to find $s$.

• Simon’s algorithm finds $s$ in about $O(n)$ evaluations using quantum computing → exponential speedup.

⚙️ How Simon’s Algorithm Works

1. Superposition

   • Prepare a uniform superposition of all inputs $|x\rangle$.

2. Oracle Application

   • Apply the black-box function (oracle) → entangles input with output:

     $$\frac{1}{\sqrt{2^n}} \sum_x |x\rangle |f(x)\rangle$$

3. Measurement of Output

   • Measure the second register (output).

   • This collapses the first register into a superposition of two inputs: $|x\rangle$ and $|x \oplus s\rangle$.

4. Hadamard Transform

   • Apply Hadamard gates to the first register.

   • This produces a state that encodes information about $s$.

5. Measurement

   • Measure → gives a random string $y$ such that:

     $$y \cdot s = 0 \ (\text{mod } 2)$$

     (dot product modulo 2).

6. Repeat

   • Repeat steps until enough independent equations are collected.

   • Solve the linear system of equations to recover $s$.

Read More  :

Explain Shor’s algorithm and its importance.

What is the Quantum Fourier Transform (QFT)?

How does Shor’s algorithm impact modern cryptography?

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