Courses

Topological quantum computing and modular tensor categories

Zhenghan Wang

Topological quantum computing is a framework for fault-tolerant quantum computation based on topological phases of matter and their quasiparticle excitations, known as anyons. Quantum information is encoded and processed in global properties of multiple anyons, providing intrinsic protection against local noise.

This course will introduce the mathematical foundations of the subject, with emphasis on modular tensor categories and modular functors which are part of topological quantum field theories. We will also discuss the basic architecture of topological quantum computation, including qubit encoding, braiding gates, and measurement, and examine different anyon models and their computational capabilities.

Quantum Picturalism: A New Language for Quantum

Bob Coecke

Over the past twenty years, we have developed a diagrammatic formalism for quantum theory known as quantum picturalism. Rather than expressing quantum processes through matrices, vectors, and complex equations, the formalism represents them using pictures governed by precise mathematical rules. This approach has become an important tool in quantum computing research and industry, with applications to circuit optimization and error correction. It also makes sophisticated quantum concepts accessible to students without advanced mathematical training. Recent studies found that secondary school students taught through quantum picturalism could successfully solve problems drawn from an Oxford graduate quantum examination. The same compositional framework has also found applications in interpretable artificial intelligence, natural language processing, and music.

Topological phases, lattice models, and topological quantum computation

Shawn Cui

Topological phases are exotic gapped quantum systems that support quasiparticle excitations, known as anyons, whose exchange statistics extend beyond those of bosons and fermions. From the perspective of topological quantum computing, these phases provide physical platforms in which quantum information is encoded in global degrees of anyons. At the same time, their ground-state spaces naturally realize quantum error-correcting codes. These viewpoints illuminate the role of topological phases in fault-tolerant quantum computation.

In this course, we will study exactly solvable lattice models of topological phases, including the toric code, the Kitaev quantum double models, and the Levin–Wen string-net models. We will examine their Hamiltonians, ground states, and excitations in detail, and connect them with representation theory and category theory. We will also discuss concrete examples, such as the quantum double of S₃, and some critical problems in topological quantum computing, including universality, leakage-free braiding gates, and gate compilation.

Reconstructing quantum circuits

Steven Rayan

In this lecture series, we explore quantum computation through progressively richer mathematical perspectives, and explore different but interrelated definitions of “quantum circuit” and of what it means to “construct” one. Beginning with the conceptual foundations of the circuit model, we develop ways of thinking about quantum states, gates, measurement, and interference before turning to diagrammatic methods for representing and transforming circuits. We then address quantum error correction, where logical information is protected through algebraic, topological, and field-theoretic structures. Finally, we conclude by examining how topology and geometry can enlarge our understanding of quantum computation through topological field theory, exotic geometries, moduli spaces, and related directions. Throughout, the emphasis is on quantum circuits not only as computational devices, but also as mathematical objects that can be interpreted, redrawn, corrected, and ultimately geometrized.