The History and the Contemporary Status of the No Cloning Theorem and the Extended Church-Turing Thesis
Major
Mathematics
Submission Type
Poster
Area of Study or Work
Mathematics, Physics
Faculty Advisor
Narendra Jaggi
Location
CNS Atrium
Start Date
4-12-2025 8:30 AM
End Date
4-12-2025 9:30 AM
Abstract
This project aims to trace the history and the evolution of two fundamental concepts that have cut across physics, mathematics and computer science over the past 100 years. First: Evolution of the so-called No-Cloning Theorem. In classical computing (Pre-Quantum), information can be copied without limitations. The discovery of the No-Cloning Theorem by Wootters and Zurek (1982) and Dieks (1982) revised this concept dramatically; they independently proved that unknown quantum states cannot be perfectly copied. I look at mathematical proofs of this theorem, construct physics-based nonmathematical plausibility arguments for it, and explore the role it plays in Quantum Key Distribution (QKD) protocols, such as BB84 (Bennett and Brassard, 1984) and the Ekert Protocol (Artur Ekert 1991). Second: Evolution of the concept of computability in quantum computing. I began by trying to understand the classical Church-Turing thesis (1936) which states that any computation performed by a physically realizable system can be simulated by a Turing machine. Then I try to explore the Extended Church-Turing Thesis (ECTT, 1980s) which conjectures that all efficiently computable functions in the physical world can be efficiently simulated by a probabilistic Turing machine. The recent discovery of quantum algorithms (for example, Shor’s algorithm, 1994) suggests that quantum computers could violate this thesis by solving problems exponentially faster than classical computers. The very modern debate (2020’s) surrounding this topic asks whether quantum computing truly invalidates ECTT or whether deeper physical constraints (such as decoherence) might limit quantum computers in a fashion that would be consistent with ECTT. This debate is far from settled. I have grappled with these hard questions and will summarize what I understand to be the status of this debate.
The History and the Contemporary Status of the No Cloning Theorem and the Extended Church-Turing Thesis
CNS Atrium
This project aims to trace the history and the evolution of two fundamental concepts that have cut across physics, mathematics and computer science over the past 100 years. First: Evolution of the so-called No-Cloning Theorem. In classical computing (Pre-Quantum), information can be copied without limitations. The discovery of the No-Cloning Theorem by Wootters and Zurek (1982) and Dieks (1982) revised this concept dramatically; they independently proved that unknown quantum states cannot be perfectly copied. I look at mathematical proofs of this theorem, construct physics-based nonmathematical plausibility arguments for it, and explore the role it plays in Quantum Key Distribution (QKD) protocols, such as BB84 (Bennett and Brassard, 1984) and the Ekert Protocol (Artur Ekert 1991). Second: Evolution of the concept of computability in quantum computing. I began by trying to understand the classical Church-Turing thesis (1936) which states that any computation performed by a physically realizable system can be simulated by a Turing machine. Then I try to explore the Extended Church-Turing Thesis (ECTT, 1980s) which conjectures that all efficiently computable functions in the physical world can be efficiently simulated by a probabilistic Turing machine. The recent discovery of quantum algorithms (for example, Shor’s algorithm, 1994) suggests that quantum computers could violate this thesis by solving problems exponentially faster than classical computers. The very modern debate (2020’s) surrounding this topic asks whether quantum computing truly invalidates ECTT or whether deeper physical constraints (such as decoherence) might limit quantum computers in a fashion that would be consistent with ECTT. This debate is far from settled. I have grappled with these hard questions and will summarize what I understand to be the status of this debate.