what is Deutsch–Jozsa in quantum computing. explain with examples

The Deutsch-Jozsa algorithm is a foundational quantum algorithm (proposed in 1992) and was the first to demonstrate that a quantum computer could solve a specific problem exponentially faster than a deterministic classical computer.

It is designed to solve a "black box" problem: determining if a hidden function is constant or balanced using only one query.

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1. The Problem Statement

Imagine you are given a "black box" (often called an Oracle) containing a hidden boolean function $f(x)$ that takes a string of bits (0s and 1s) as input and outputs either a 0 or a 1.

You are promised that the function belongs to one of two categories:

• Constant: The function returns the same output (either always 0 or always 1) for every possible input.

• Balanced: The function returns 0 for exactly half of the inputs and 1 for the other half.

The Goal: Determine if $f(x)$ is constant or balanced.

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2. Examples: Constant vs. Balanced

To understand the difference, imagine a simple function with a 3-bit input. There are $2^3 = 8$ possible input combinations (000 to 111).

Example A: Constant Function

Regardless of what you put in, the output is identical.

• Input 000 $\rightarrow$ Output 1

• Input 001 $\rightarrow$ Output 1

• ...

• Input 111 $\rightarrow$ Output 1

  (This is like a coin that has "Heads" on both sides. No matter how many times you flip it, you get Heads.)

Example B: Balanced Function

The outputs are split 50/50.

• Input 000 $\rightarrow$ Output 0

• Input 001 $\rightarrow$ Output 0

• Input 010 $\rightarrow$ Output 1

• Input 011 $\rightarrow$ Output 1

  (This is like a fair coin. If you flip it enough times, you get half Heads and half Tails.)

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3. Classical vs. Quantum Solution

This is where the quantum advantage becomes clear. Suppose you have $N$ input bits, meaning there are $2^N$ possible inputs.

• Classical Approach:

  To be 100% sure, you might have to check more than half of all possible inputs. In the worst-case scenario (if you keep getting the same answer), you need to check $2^{N-1} + 1$ inputs. For a large number of inputs, this takes a massive amount of time.

• Deutsch-Jozsa Approach:

  A quantum computer can solve this with exactly 1 query, regardless of how large $N$ is.

Feature	Classical Computer	Quantum Computer
Method	Check inputs one by one	Check all inputs at once (Superposition)
Queries Needed	Exponential ($2^{N-1}+1$)	One (1)
Speed	Slow for large inputs	Instant

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4. How It Works (The Quantum Circuit)

The algorithm relies on two key quantum principles: Superposition and Interference.

1. Superposition:

   Instead of inputting a single string (like 000), the algorithm initializes the qubits into a state of superposition. This allows the Oracle to process all $2^N$ inputs simultaneously.

2. Phase Kickback (The Oracle):

   The Oracle doesn't just give an answer; it encodes the answer into the "phase" of the quantum state.

   • If the function is Balanced, it adds a negative phase (a minus sign) to exactly half of the states.

   • If the function is Constant, the phases remain aligned (or all flipped globally, which is physically equivalent).

3. Interference:

   Finally, the algorithm applies a transformation (Hadamard gates) that causes the quantum states to interfere with each other like waves.

   • Constructive Interference: If the function was Constant, all the waves add up to point back to the starting state (all zeros).

   • Destructive Interference: If the function was Balanced, the positive and negative phases cancel each other out, ensuring you never measure the all-zero state.

5. Summary of Results

After running the circuit once, you measure the output:

• Result is 000...0: The function is Constant.

• Result is anything else: The function is Balanced.

Why is this important?

While distinguishing constant vs. balanced functions isn't very useful for daily tasks (like sending emails or browsing the web), Deutsch-Jozsa was a proof-of-concept. It proved that quantum computers are not just faster classical computers, but operate on fundamentally different rules that allow for exponential speedups on specific problems.

Would you like me to walk you through the linear algebra math step-by-step for a simple 1-qubit example?

... Explanation and Step-by-Step Example ...

I selected this video because it provides a clear, step-by-step walkthrough of the math and logic behind the Deutsch-Jozsa algorithm, which complements the high-level overview provided above.