A general theory of quantum codes connecting quantum computation, complexity and physics

Ciarán M. Lee, Matty J. Hoban

What kind of object is a quantum state? Is it an object that encodes an exponentially growing amount of information (in the size of the system) or more akin to a probability distribution? It turns out that these questions are sensitive to what we do with the information. For example, Holevo's bound tells us that n qubits only encode n bits of classical information but for certain communication complexity tasks there is an exponential separation between quantum and classical resources. Instead of just contrasting quantum and classical physics, we can place both within a broad landscape of physical theories and ask how non-quantum (and non-classical) theories are different from, or more powerful than quantum theory. For example, in communication complexity, certain (non-quantum) theories can trivialise all communication complexity tasks. In recent work [C. M. Lee and M. J. Hoban, Proc. Royal Soc. A 472 (2190), 2016], we showed that the immense power of the information content of states in general (non-quantum) physical theories is not limited to communication complexity. We showed that, in general physical theories, states can be taken as "advice" for computers in these theories and this advice allows the computers to easily solve any decision problem. Aaronson has highlighted the close connection between quantum communication complexity and quantum computations that take quantum advice, and our work gives further indications that this is a very general connection. In this work, we review the results in our previous work and discuss the intricate relationship between communication complexity and computers taking advice for general theories.

Stephen Fenner

Megha Khandelwal, Subho Sankar Chatterjee

Quantum computing is a subject that assembles ideas from classical quantum physics, information theory, and computer science. This paper describes the connection between information theory and quantum mechanics. Explaining their relationship, the review concocts introduction to classical information theory and computer science, including Shannon's theorem, error correcting codes, Turing machines and computational complexity. The principles of quantum mechanics are then outlined, and the EPR experiment described. The EPR-Bell correlations and quantum entanglement in general, form the essential new ingredient which distinguishes quantum from classical information theory, and arguably, quantum from classical physics. Basic Quantum information ideas are described, including key distribution, teleportation, data compression, quantum error correction, the universal quantum computer and quantum algorithms. The common theme of all these ideas is the use of quantum entanglement as a computational resource. Experimental methods for small quantum processors are briefly sketched, concentrating on ion traps, high Q cavities, and NMR. The review concludes with an outline of the main features of quantum information physics, and avenues for future research.

Sankalpa Khadka

Jincheng Zhang

Traditional methods for analyzing the computational complexity of algorithms, predominantly rooted in classical computer models, struggle to accurately characterize the potential advantages offered by quantum computation. This work proposes a novel approach utilizing category theory to provide a more fundamental and abstract framework for assessing the complexity of quantum algorithms. Instead of focusing on the specifics of quantum hardware, we examine the underlying categorical structures that define the computational processes inherent in these algorithms. This allows us to identify and analyze complexity properties that are independent of the physical implementation. We define key concepts like quantum states as objects and quantum gates as morphisms, establishing a formal connection between quantum computation and category theory. This framework facilitates a more rigorous and generalized understanding of quantum complexity, potentially revealing new insights and providing a basis for comparing different quantum algorithms. The core claim is that traditional complexity analysis is inadequate, and the mechanism is the application of category theory for a more abstract and generalized framework.