1. Input Domain of the Function

• Deutsch Algorithm: Works on a 1-bit function only. That is, the function is defined as f: {0,1\} → {0,1\}, meaning the input can only be 0 or 1.
• Deutsch-Jozsa Algorithm: Works on n-bit functions. The function can be defined as f : {0,1}ⁿ → {0,1}. Here, the input consists of multiple bits, and the function’s nature is resolved in a much more general and broader input space.

2. Problem Difficulty and Classical Algorithm Requirements

• Deutsch Algorithm: This algorithm only works for 1-bit functions. For classical computers, determining whether f is constant or balanced requires 2 queries evaluating f(0) and f(1).
• Deutsch-Jozsa Algorithm: This algorithm works with multi-bit functions. For classical computers, determining whether f is constant or balanced requires 2ⁿ queries, which means checking all possible inputs. In contrast, the quantum algorithm can determine the nature of the function with just 1 query.

3. Scope

• Deutsch Algorithm: This algorithm can only identify 1-bit functions as either constant or balanced, which limits its applicability significantly.
• Deutsch-Jozsa Algorithm: This algorithm addresses a much broader range of problems because it can solve n – bit functions. The function f is not limited to being constant or balanced but can include a much more complex and extensive class of functions.

4. Oracle Usage

• Deutsch Algorithm: Uses a 1-bit oracle that evaluates the result of f(x) as f(0) and f(1). The oracle operates on only two states.
• Deutsch-Jozsa Algorithm: Uses an n-bit oracle. The oracle evaluates f(x) for 2ⁿ different inputs, effectively working on a much larger dataset.

5. Algorithm Complexity and Applications

• Deutsch Algorithm: Has lower complexity as it only solves 1-bit functions. Consequently, its practical applications are very limited.
• Deutsch-Jozsa Algorithm: Has higher complexity as it works with larger n-bit functions. However, thanks to the advantages of quantum computers, this type of problem, which is infeasible for classical computers, can be solved efficiently with quantum computers.

Conclusion : While the Deutsch Algorithm offers a simple and limited application, the Deutsch-Jozsa Algorithm provides a more powerful and extensive application.