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๐Ÿ“ Phase Estimation

Extract eigenvalues with exponential precision using quantum interference

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Quantum Fourier Transform

๐ŸŽฏ Understanding Quantum Phases

Quantum Phase Estimation (QPE) is one of the most important quantum algorithms. It determines the eigenvalue of a unitary operator with exponential precision. If U|ฯˆโŸฉ = e^(2ฯ€iฯ†)|ฯˆโŸฉ, QPE finds ฯ† to n bits of accuracy using only n qubitsโ€”a precision that would require exponentially many measurements classically.

๐Ÿ’Ž Exponential Precision

With n counting qubits, QPE achieves precision of 1/2^n. Just 10 qubits give precision of 1/1024 โ‰ˆ 0.1%, while 20 qubits reach 1 part per million. This exponential scaling is impossible classically.

4 qubits
1/16
6.25% precision
8 qubits
1/256
0.39% precision
10 qubits
1/1024
0.098% precision

๐ŸŒŸ Why It Matters

  • โ€ขFoundation of Shor's: Extracts period from quantum state for integer factorization
  • โ€ขQuantum Chemistry: Finds molecular energy eigenvalues for drug discovery
  • โ€ขMaterials Science: Simulates quantum systems and predicts material properties
  • โ€ขMachine Learning: Enables quantum principal component analysis and classification

๐Ÿ”‘ Key Components

๐ŸŽฏ

Eigenvalue Problem

Find phase ฯ† where U|ฯˆโŸฉ = e^(2ฯ€iฯ†)|ฯˆโŸฉ

๐Ÿ”ข

Counting Register

n qubits that encode phase to n bits of precision

โšก

Controlled-U Gates

Apply U^(2^k) to accumulate phase information

๐ŸŒŠ

Inverse QFT

Extract phase from quantum superposition