Explain Deutsch’s algorithm.

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🔹 The Problem

Suppose we are given a function:

$$f : \{0,1\} \to \{0,1\}$$

There are four possible functions:

1. Constant 0 → $f(0)=0, f(1)=0$

2. Constant 1 → $f(0)=1, f(1)=1$

3. Balanced → $f(0)=0, f(1)=1$

4. Balanced → $f(0)=1, f(1)=0$

👉 The task: Determine if $f$ is constant (same output for both inputs) or balanced (different outputs).

• Classical computer → Needs 2 evaluations (check both inputs 0 and 1).

• Quantum computer (Deutsch’s algorithm) → Solves it with 1 evaluation using superposition + interference.

🔹 Steps of Deutsch’s Algorithm

1. Initialize qubits
   We start with two qubits:

$$|0\rangle \otimes |1\rangle = |0,1\rangle$$

2. Apply Hadamard gates

• First qubit (input): goes into superposition.

• Second qubit (output): also transformed.

$$|0,1\rangle \xrightarrow{H \otimes H} \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle) \otimes \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)$$

3. Apply Oracle $U_f$
   The oracle is a quantum black box that encodes the function:

$$U_f |x,y\rangle = |x, y \oplus f(x)\rangle$$

👉 Due to clever setup of second qubit in $|0\rangle - |1\rangle$, the result is:

$$\frac{1}{\sqrt{2}} \big( (-1)^{f(0)}|0\rangle + (-1)^{f(1)}|1\rangle \big) \otimes \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)$$

Now, the phase of the first qubit encodes the answer.

4. Apply Hadamard again on first qubit
   This interference step reveals if $f$ is constant or balanced:

• If constant → result = $|0\rangle$.

• If balanced → result = $|1\rangle$.

5. Measure first qubit

• Output 0 → Function is constant.

• Output 1 → Function is balanced.

🔹 Why is it important?

• It’s the first quantum algorithm showing how quantum computation can outperform classical.

• Classical: needs 2 evaluations.

• Quantum: solves with 1 evaluation (quadratic speedup for this problem).

• Foundation for more advanced algorithms like Deutsch–Jozsa, Simon’s algorithm, and Shor’s algorithm.

✅ In short:

Deutsch’s algorithm determines if a function$f$$($$x$$)$$f(x)$ is constant or balanced using one quantum evaluation instead of two classical evaluations. It works by leveraging superposition, phase kickback, and interference.

Read More :

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What is a unitary matrix in the context of quantum gates?

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