Semantics for a Quantum Programming Language by Operator Algebras

Mingsheng Ying

Kenta Cho

This paper presents a novel semantics for a quantum programming language by operator algebras, which are known to give a formulation for quantum theory that is alternative to the one by Hilbert spaces. We show that the opposite category of the category of W*-algebras and normal completely positive subunital maps is an elementary quantum flow chart category in the sense of Selinger. As a consequence, it gives a denotational semantics for Selinger's first-order functional quantum programming language QPL. The use of operator algebras allows us to accommodate infinite structures and to handle classical and quantum computations in a unified way.

Yuxin Deng, Yuan Feng

Kinnari Vijay Dave

The focus of this thesis is on laying the theoretical foundation for designing an all-encompassing programming language for quantum computing. The idea inspiring all the contributions of this thesis was to mirror the development of programming languages and logic done for classical computing for its quantum counterpart. Following the groundwork of the Curry-Howard-Lambek correspondence, we try to delve into the three cornerstones of this correspondence - programming languages, logic and category theory with a focus on combining two paradigms of control flow in programs - quantum and classical into one framework. We have tried to achieve more. We have also touched upon the two complementary view of quantum mechanics- Heisenberg and Schrödinger, and adopted one view to define a physically intuitive semantics for the languages we discuss. There are two main contributions that this thesis makes to the existing literature on quantum programming languages. We first introduce a quantum lambda calculus which combines quantum and classical control, whose design is guided by the mathematical structure of its physically intuitive denotational semantics. The operational semantics for this language, is based on an adaptation of quantum configurations which we introduce, and is tied to the denotational model through an adequacy result. The language also provides a prototype of quantum inductive types through the type qnat- a type for quantum natural numbers. The denotational model for this calculus is based on von Neumann algebras- which are at the heart of the mathematical formulation of the Heisenberg view of quantum mechanics. The second contribution is a proof language for the IMALL fragment of Linear logic which builds on some key observations from the calculus introduced above. This language is one of the first such works to be able to support programs which combine constructs such as the quantum switch with measurement. In this aspect, this work is a strict improvement on the previous calculus, since it supports a more fine grained version of quantum control, thus enabling expression of algorithms such as the well-known quantum switch. The syntax is also linear algebraic in nature, supporting linear combinations of terms. A key consequence of this, along with the semantics of the B modality that it introduces is that measurement is no longer a ground construct of the language. In fact, it can now be written as just another program in the language. Finally, the denotational semantics for this work is based on C*-algebras - a generalization of von Neumann algebras. The denotational model is shown to be adequate for the operational semantics, whose linearity turns out to be a natural consequence of this. Finally, we conclude this thesis with limitations of these works and possible directions for future work.

Ellie D’Hondt

Robert I. Booth, Cole Comfort

The stabiliser fragment of quantum theory is a foundational building block for quantum error correction and the fault-tolerant compilation of quantum programs. In this article, we develop a sound, universal and complete denotational semantics for stabiliser operations which include measurement, classically-controlled Pauli operators, and affine classical operations, in which quantum error-correcting codes are first-class objects. The operations are interpreted as certain affine relations over finite fields. This offers a conceptually motivated and computationally-tractable alternative to the standard operator-algebraic semantics of quantum programs (whose time complexity grows exponentially as the state space increases in size). We demonstrate the power of the resulting semantics by describing a small, proof-of-concept assembly language for stabiliser programs with fully-abstract denotational semantics.