Quantum Phase Estimation

Estimate eigenvalues of unitary operators with applications in quantum computing.

Number of bits of precision

int (min: 1)

Target phase to estimate (0-1)

float (min: 0.0)

Noise probability

float (min: 0.0)

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Quantum Explanation

Quantum Phase Estimation

Quantum Phase Estimation (QPE) is an algorithm that determines the eigenvalue of a unitary operator with respect to an eigenvector, providing a binary approximation of the phase.

Key Quantum Concept: QPE leverages quantum parallelism and the quantum Fourier transform to extract phase information exponentially more efficiently than classical methods, making it a critical subroutine in Shor's factoring algorithm and quantum chemistry applications.

Quantum Phase Estimation Algorithm

Quantum Phase Estimation (QPE) is a quantum algorithm that determines the eigenphase (or eigenvalue) of a unitary operator with respect to an eigenvector. It is a fundamental building block for many important quantum algorithms, including Shor's factoring algorithm and the HHL algorithm for solving linear systems.

Classical vs. Quantum: Classical algorithms for estimating eigenvalues typically require multiple applications of the operator and complex numerical methods. QPE leverages quantum parallelism to extract phase information with exponentially fewer resources for certain problems, providing a speed advantage for many quantum algorithms.

How Quantum Phase Estimation Works: Step-by-Step

Initialization: Start with two registers - a phase register of n qubits (initially |0⟩) and an eigenstate register containing an eigenvector |u⟩ of unitary operator U.
Superposition Creation: Apply Hadamard gates to all qubits in the phase register, creating a uniform superposition.
Controlled Operations: Apply a series of controlled-U operations with increasing powers (U, U², U⁴, U⁸, etc.) controlled by each qubit in the phase register.
Inverse Quantum Fourier Transform: Apply the inverse QFT to the phase register, converting the phase information to a computational basis state.
Measurement: Measure the phase register to obtain an n-bit estimate of the phase φ, where e^(2πiφ) is the eigenvalue of U corresponding to |u⟩.

Quantum Phase Estimation Circuit

Quantum circuit implementing phase estimation with n control qubits and an eigenvector register. H gates create superposition, followed by controlled-U operations and inverse QFT.

Key Quantum Principles in Phase Estimation

Quantum Parallelism: The algorithm evaluates U^(2^j) for all j simultaneously through superposition.
Quantum Fourier Transform: The inverse QFT efficiently extracts the phase information encoded in the quantum state.
Phase Kickback: Controlled operations transfer phase information from the eigenstate to the phase register.
Quantum Measurement: Collapses the superposition into an approximate binary representation of the phase.

Applications of Quantum Phase Estimation

QPE serves as a key subroutine in numerous quantum algorithms:

Shor's Algorithm: QPE is used to find the period of a function, which is critical for factoring large numbers.
Quantum Chemistry: Estimating eigenvalues of molecular Hamiltonians to determine ground state energies.
HHL Algorithm: Solving systems of linear equations by estimating eigenvalues of a Hermitian matrix.
Quantum Simulation: Determining the spectrum of quantum systems for physics simulation.

Limitations and Challenges

Eigenstate Preparation: Requires starting with a good approximation of an eigenvector, which can be difficult to prepare.
Circuit Depth: The number of controlled operations scales exponentially with bit precision.
Decoherence: Deep circuits are susceptible to noise and quantum decoherence.
Implementation Complexity: Requires complex controlled unitary operations that may be difficult to implement efficiently.

Beginner's Analogy: Understanding Phase Estimation

Imagine trying to measure the speed of a carousel's rotation. Instead of timing it directly, you stand at a fixed point and count how many times a particular horse passes by in a given time period.

Now, imagine you have multiple observers who each watch for different periods of time: 1 minute, 2 minutes, 4 minutes, and 8 minutes. Each observer counts full rotations and fractions.

QPE works similarly: different qubits in the phase register track different "timescales" of the operator's rotation in the complex plane. The controlled operations apply different powers of the operator (like watching for different time periods), and the inverse QFT combines these observations to determine the precise rotation rate (the phase).

Further Learning Resources

Phase Estimation on Wikipedia

QFT and Phase Estimation Video

Qiskit Learning on QPE