5.2 Deutsch-Jozsa Algorithm

The Deutsch-Jozsa algorithm is a quantum computing powerhouse. It can tell if a function always gives the same output or a balanced mix of outputs. This algorithm does in one step what classical computers might need billions of steps for.

Quantum circuits make this magic happen. Using special gates and measurements, the algorithm creates a superposition of states, applies the function, and then reads the result. It's like solving a puzzle by looking at all the pieces at once.

Understanding the Deutsch-Jozsa Algorithm

Purpose of Deutsch-Jozsa algorithm

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• Determines if boolean function $f: \{0,1\}^n \rightarrow \{0,1\}$ is constant or balanced
• Demonstrates quantum speedup over classical algorithms solves problem with single function evaluation
• Constant function returns same output for all inputs (always 0 or always 1)
• Balanced function returns 0 for half of inputs and 1 for other half (parity function)

Quantum circuit for Deutsch-Jozsa

• Circuit uses n input qubits and 1 auxiliary qubit
• Initial state prepares input qubits in $|0\rangle^{\otimes n}$ and auxiliary qubit in $|1\rangle$
• Hadamard gates applied to all qubits create superposition
• Oracle implements quantum black box representing function f
• Measurement performed on input qubits determines function type

Steps in Deutsch-Jozsa algorithm

1. State preparation creates initial state $|0\rangle^{\otimes n}|1\rangle$
2. Hadamard gates transform state to superposition
3. Oracle encodes function information in phase
4. Hadamard gates applied to input qubits interfere amplitudes
5. Measurement of input qubits reveals function type
   • Constant function: All qubits in $|0\rangle$ state
   • Balanced function: At least one qubit in $|1\rangle$ state

Efficiency vs classical approaches

• Classical algorithm requires worst-case $2^{n-1} + 1$ function evaluations (checking half plus one)
• Quantum algorithm always requires 1 function evaluation
• Exponential speedup achieved: Quantum O(1) vs Classical O($2^n$)
• Quantum algorithm deterministic and always correct
• Probabilistic classical algorithm can be wrong

Applications of Deutsch-Jozsa algorithm

• Solves specific problem instances (constant function f(x) = 0, balanced function f(x) = x_1 ⊕ x_2)
• Implements algorithm steps: construct circuit, prepare state, apply steps, interpret results
• Considers practical aspects: qubit count, circuit depth, error mitigation strategies
• Serves as foundation for more complex quantum algorithms (Shor's, Grover's)
• Demonstrates quantum parallelism and interference principles