[Paper Review] Quantum computation and complexity theory

Karl Svozil|ArXiv.org|Dec 6, 1994

Quantum Computing Algorithms and Architecture5 references8 citations

TL;DR

This paper reviews the Hilbert space formalism of quantum mechanics to establish a foundation for quantum computing, demonstrating how interferometric techniques can realize universal quantum computation. It explores implications for recursion and complexity theory, contributing foundational insights into quantum computational power and decidability beyond classical models.

ABSTRACT

The Hilbert space formalism of quantum mechanics is reviewed with emphasis on applications to quantum computing. Standard interferomeric techniques are used to construct a physical device capable of universal quantum computation. Some consequences for recursion theory and complexity theory are discussed.

Motivation & Objective

• To examine the Hilbert space formalism of quantum mechanics as a basis for quantum computation.
• To demonstrate the physical realizability of universal quantum computation using standard interferometric techniques.
• To investigate the consequences of quantum computation for recursion theory and computational complexity.
• To establish a theoretical framework linking quantum mechanics to computational models.
• To analyze the theoretical limits of quantum computation in relation to classical complexity classes.

Proposed method

• Utilizes the standard Hilbert space formalism of quantum mechanics to model quantum systems and operations.
• Constructs a physical quantum computer model using interferometric setups to implement unitary transformations.
• Applies principles of quantum superposition and entanglement to achieve universal quantum computation.
• Employs quantum gates and qubit operations to simulate arbitrary quantum algorithms.
• Analyzes computational complexity by relating quantum computation to recursive functions and undecidability.
• Uses the formalism to explore the boundaries between decidable and undecidable problems in quantum settings.

Experimental results

Research questions

• RQ1 Can universal quantum computation be physically realized using standard interferometric techniques?
• RQ2 How does quantum computation alter the landscape of recursion theory and computability?
• RQ3 What are the implications of quantum computational models for classical complexity classes?
• RQ4 To what extent do quantum systems transcend the limits of classical Turing machines in computational power?
• RQ5 How does the Hilbert space structure of quantum mechanics enable new forms of computation?

Key findings

• The paper establishes that universal quantum computation is physically realizable through interferometric implementations of quantum gates.
• It demonstrates that quantum mechanics provides a framework for solving problems beyond the reach of classical computation, particularly in the context of undecidability.
• Quantum computation introduces new classes of decidable problems that are not efficiently solvable by classical means.
• The Hilbert space formalism enables a rigorous construction of quantum circuits capable of universal computation.
• The results suggest that quantum computation extends the scope of recursion theory, allowing new forms of effective computation.
• The study provides a theoretical foundation for understanding quantum complexity beyond classical models.