Amplitude Amplification
Amplitude Amplification is a fundamental technique in Quantum Computing that enhances the probability of measuring a desired state in a quantum system. This method is an extension of the Grover's Algorithm and was initially proposed by Lov Grover in 1996.
History and Development
The concept of amplitude amplification emerged from the need to speed up unstructured search problems in quantum computing. Lov Grover introduced his quantum search algorithm, which provided a quadratic speedup over classical algorithms for searching unsorted databases. Subsequently, this technique was generalized into what is now known as amplitude amplification:
- 1996: Grover's algorithm published, laying the groundwork for amplitude amplification.
- 1998: Amplitude amplification was formalized as a more general algorithm applicable to various problems beyond searching.
- Early 2000s: Further theoretical developments and optimizations of the algorithm were made by researchers like Gilles Brassard and Peter Høyer.
Principle
Amplitude amplification works by:
- Amplifying the amplitude of the desired state while reducing the amplitude of other states.
- Using a series of quantum gates to rotate the quantum state in the desired direction.
- Repeating this process to increase the probability of measuring the desired outcome.
The core steps of the algorithm include:
- Initialization: Start with a superposition of all possible states.
- Oracle: Mark the desired states using an oracle, which can be any unitary operator that distinguishes the marked states from others.
- Amplification: Apply a sequence of operations to amplify the amplitude of the marked states:
- Inversion about the average amplitude (Grover diffusion operator).
- Repeating the oracle and diffusion steps.
Applications
Amplitude amplification has applications in:
- Quantum search algorithms.
- Quantum machine learning for data classification and clustering.
- Quantum algorithms for solving NP-hard problems.
- Quantum error correction.
Limitations and Extensions
While amplitude amplification provides a significant speedup, it has its limitations:
- It requires knowledge of the number of solutions to achieve optimal performance.
- It does not provide an exponential speedup for all problems.
Extensions of amplitude amplification include:
- Quantum Walks which can be used for more complex searches.
- Amplitude estimation for estimating the probability of a particular outcome.
References
See Also