Books

Quantum Algorithms

Quantum algorithms leverage quantum phenomena to solve problems more efficiently than classical algorithms.

Deutsch-Jozsa Algorithm

Problem: Determine if a function is constant or balanced.

Quantum Solution:

• Uses qubits
• Solves with 1 query (deterministic)
• Classical requires queries in worst case

Circuit:

1. Apply Hadamard to all qubits
2. Apply oracle
3. Apply Hadamard to first n qubits
4. Measure

Grover's Search Algorithm

Problem: Find marked item in unstructured database of N items.

Quantum Advantage: vs classical.

Algorithm Steps:

1. Initialize uniform superposition
2. Apply Grover operator times
3. Measure

Grover operator = Oracle + Diffusion operator

Shor's Factoring Algorithm

Problem: Factor integer N into its prime factors.

Quantum Advantage: Exponential speedup - best classical is sub-exponential.

Key Components:

1. Quantum Fourier Transform (QFT)
2. Modular exponentiation
3. Period finding

Complexity: quantum operations

Quantum Fourier Transform

QFT is the quantum analogue of discrete Fourier transform:

Used in:

• Shor's algorithm
• Phase estimation
• Hidden subgroup problems

Variational Quantum Algorithms

Hybrid classical-quantum algorithms:

1. VQE (Variational Quantum Eigensolver)
• Find ground state energy of molecules
• Applications in chemistry and drug discovery
2. QAOA (Quantum Approximate Optimization Algorithm)
• Solve combinatorial optimization problems
• Applications in logistics and finance

Quantum Machine Learning

1. Quantum Support Vector Machine (QSVM)
2. Quantum Neural Networks (QNN)
3. Quantum Principal Component Analysis (QPCA)

Algorithm Comparison