[Paper Review] Computation in Finitary Quantum Processes

Karoline Wiesner, James P. Crutchfield|arXiv (Cornell University)|Mar 1, 2006

Quantum Computing Algorithms and Architecture56 references5 citations

TL;DR

This paper introduces quantum finite-state generators as a computational model for describing time-evolving quantum systems, establishing a hierarchy of finitary process languages that compare deterministic and nondeterministic quantum machines to classical stochastic counterparts. The key contribution is demonstrating that quantum systems' information processing capacity depends critically on measurement protocols, as shown in physical examples like the beam splitter and ion trap implementations of the Deutsch algorithm.

ABSTRACT

We introduce quantum finite-state generators as a first step toward completing a computational description of observing individual quantum systems over time. We develop the mathematical foundations of quantum finite-state machines and compare nondeterministic and deterministic versions to stochastic generators and recognizers, summarizing their relative computational power via a hierarchy of finitary process languages. Quantum finite-state generators are explored via several physical examples, including the iterated beam splitter, the quantum kicked top, and atoms in an ion trap--a special case of which implements the Deutsch quantum algorithm. We show that the behavior of these systems, and so their information processing capacity, depends sensitively on measurement protocol.

Motivation & Objective

• To develop a computational framework for describing individual quantum systems observed over time.
• To formalize quantum finite-state machines and compare their computational power to classical stochastic generators and recognizers.
• To establish a hierarchy of finitary process languages that classify the expressive power of quantum and classical models.
• To analyze how measurement protocols influence the information processing capacity of quantum systems.
• To demonstrate the model's applicability through physical realizations like the iterated beam splitter and ion trap quantum algorithms.

Proposed method

• Formalizing quantum finite-state generators using mathematical structures from quantum automata theory.
• Defining deterministic and nondeterministic versions of quantum finite-state machines and comparing them to stochastic finite-state generators.
• Constructing a hierarchy of finitary process languages based on the computational power of different machine types.
• Applying the model to physical systems such as the iterated beam splitter, the quantum kicked top, and trapped ions.
• Using the Deutsch quantum algorithm as a special case in ion trap systems to validate the model's predictive power.
• Analyzing how different measurement strategies alter the observed behavior and information processing capacity of quantum processes.

Experimental results

Research questions

• RQ1 How can quantum finite-state generators model the time evolution of individual quantum systems?
• RQ2 What is the relative computational power of deterministic versus nondeterministic quantum finite-state machines compared to classical stochastic models?
• RQ3 How does the choice of measurement protocol affect the information processing capacity of a quantum system?
• RQ4 In what ways do physical quantum systems like the beam splitter and ion trap realize the proposed computational model?
• RQ5 What role does the hierarchy of finitary process languages play in classifying quantum and classical computational behaviors?

Key findings

• The computational behavior of quantum systems, including their information processing capacity, is highly sensitive to the choice of measurement protocol.
• Quantum finite-state generators provide a formal framework that enables a systematic comparison between quantum and classical finite-state models.
• The hierarchy of finitary process languages reveals distinct levels of computational expressiveness, with quantum models surpassing classical stochastic counterparts in certain contexts.
• Physical implementations such as the iterated beam splitter and the quantum kicked top exhibit behavior consistent with the theoretical model.
• The ion trap realization of the Deutsch algorithm demonstrates that the model can capture known quantum computational advantages within a finite-state framework.
• Measurement-induced collapse and contextuality in quantum processes are shown to fundamentally shape the observable dynamics and computational outcomes.