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Predicative Quantum Programming

Anya Tafliovich

Published Sep 1, 2010• 1 citation
OpenAlex

Abstract

This work presents Quantum Predicative Programming --- a theory ofquantum programming that encompasses many aspects of quantum computation and quantum communication. The theory provides a methodology to specify, implement, and analyse quantum algorithms, the paradigm of quantum non-locality, quantum pseudotelepathy games, computing with mixed states, and quantum communication protocols that use both quantum and classical communication channels.

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This work presents Quantum Predicative Programming --- a theory ofquantum programming that encompasses many aspects of quantum computation and quantum communication. The theory provides a methodology to specify, implement, and analyse quantum algorithms, the paradigm of quantum non-locality, quantum pseudotelepathy games, computing with mixed states, and quantum communication protocols that use both quantum and classical communication channels.
The subject of this work is quantum predicative programming -- the study of developing of programs intended for execution on a quantum computer. We look at programming in the context of formal methods of program development, or programming methodology. Our work is based on probabilistic predicative programming, a recent generalisation of the well-established predicative programming. It supports the style of program development in which each programming step is proven correct as it is made. We inherit the advantages of the theory, such as its generality, simple treatment of recursive programs, time and space complexity, and communication. Our theory of quantum programming provides tools to write both classical and quantum specifications, develop quantum programs that implement these specifications, and reason about their comparative time and space complexity all in the same framework.
Quantum information theory allows us to use quantum mechanics resources for processing and sending information. Since the rules of quantum mechanics are in many aspects counterintuitive, the development of new quantum algorithms and protocols is complicated. At the moment only a few algorithms exist which can be executed on a quantum computer with significant improvement in speed or memory usage. \n \nThe main goal of this thesis is to describe the methods of using quantum programming languages for developing new quantum algorithms. We present the original implementation of a classical game where quantum programming was used to develop and analyse quantisation of the game. We also show how quantum data types, ie. data types allowing for storing superposition of values, can be used to develop new quantum algorithms.
Quantum pseudo-telepathy is an intriguing phenomenon which results from the application of quantum information theory to communication complexity. To demonstrate this phenomenon researchers in the field of quantum communication complexity devised a number of quantum non-locality games. The setting of these games is as follows: the players are separated so that no communication between them is possible and are given a certain computational task. When the players have access to a quantum resource called entanglement, they can accomplish the task: something that is impossible in a classical setting. To an observer who is unfamiliar with the laws of quantum mechanics it seems that the players employ some sort of telepathy; that is, they somehow exchange information without sharing a communication channel. This paper provides a formal framework for specifying, implementing, and analysing quantum non-locality games.
The two main notions of control in quantum programming languages are often referred to as "quantum control" and "classical control". With the latter, the control flow is based on classical information, potentially resulting from a quantum measurement, and this paradigm is well-suited to mixed state quantum computation. Whereas with quantum control, we are primarily focused on pure quantum computation and there the "control" is based on superposition. The two paradigms have not mixed well traditionally and they are almost always treated separately. In this work, we show that the paradigms may be combined within the same system. The key ingredients for achieving this are: (1) syntactically: a modality for incorporating pure quantum types into a mixed state quantum type system; (2) operationally: an adaptation of the notion of "quantum configuration" from quantum lambda-calculi, where the quantum data is replaced with pure quantum primitives; (3) denotationally: suitable (sub)categories of Hilbert spaces, for pure computation and von Neumann algebras, for mixed state computation in the Heisenberg picture of quantum mechanics.