Quantum Computing
Quantum computing is a computational model introduced in the 1980s by Feynman, Manin and others that has the potential to exponentially speed-up classical computation as manifested by Shor’s polynomial time algorithm for factoring integers. Two major open problems are the search for more quantum algorithms which demonstrate the computational power of quantum computers, and the physical realization of quantum computing. I will explain a new paradigm for quantum computing pursued by a Microsoft team including Freedman, Kitaev, Nayak, Walker which sheds new light on both problems: topological quantum computing. Topological quantum computing is an inherently fault tolerant model based on topological quantum field theories (TQFTs). TQFTs are models for new phases of matter of which the fractional quantum Hall liquids (whose discovery led to the 1998 Nobel Prize) are known examples.
The mathematical idea of topological quantum computing has matured since its introduction around 1997 by Freedman and Kitaev. The physical realization of a topological quantum computer has started since the establishment of Microsoft Station Q at UC Santa Barbara in 2005 (stationq.ucsb.edu). The conformation of the Jones representation of the braid groups at the 4th root of unity in fractional quantum Hall liquids has been pursued by experimental groups including the Charles Marcus group at Harvard University. This CBMC conference will cover the theoretical foundations of the field and explain the future challenges.
The subject is full of theoretical challenges. The mathematical formulation of the concept “quantum phase of matter” and its classification is unexplored territory, and topological quantum computing is just the tip of the iceberg. Classically a phase of matter is based on the Landau symmetry breaking theory, which utilizes the concept of a group. Topological phase of matter is characterized by a modular tensor category. A central question on the classification is my conjecture that if the number of particle types is fixed, there are only finitely many modular tensor categories. This can be thought as a generalization of the theorem: If the number of irreducible representations of a finite group is fixed, then there are only finitely many such groups. The classification of quantum phases of matter seems to be a combination of representation theory and several complex variables. A quantum phase of matter is determined by the wavefunctions of many particles such as electrons in solids. How the particles are organized into certain patterns and how to describe these emerged patterns are fascinating and difficult questions. In fractional quantum Hall liquids, these patterns (wavefunctions) are symmetric or anti-symmetric polynomials in many complex variables. A quantum phase is characterized by some long-range entanglement in the thermodynamic limit of the system, which is contained in the limit of the symmetric or anti-symmetric complex polynomials when the number of variables goes to infinity. I will explain this connection at the end of the lectures.
Topological quantum computing has been very active recently as evidenced by the many workshops, conferences and new works on the subject. Conferences are being held at the following locations: the Kavli Institute for Theoretical Physics at UC Santa Barbara (Jan.-June, 2006), IPAM at UCLA (Feb 27-March 2, 2007), the Kavli Institute for Theoretical Physics of China, Beijing (June, 2007), the Hamilton Mathematics Institute TCD, Dublin (Sep 10-14,2007), Quantum computing and topological orders (July16—20, Spain), the Aspen Physics Center at Colorado (July, 2007). Many new works have appeared in computer science, physics and mathematics: In computer science, for example, the work of V. Jones, D. Aharonov and Z. Landau on the approximation of Jones polynomials, and other partition functions. In condensed matter physics, topological quantum computing is one of the most active subfields; many excellent works have appeared, e.g., the work of E. Fradkin and P. Fendley on loop models in statistical mechanics. In mathematics, we have, for example, the classification of tensor categories by T. Hagge and Seu-moon Hong (my Ph.D students), the work of E. Rowell, S. Witherspoon, and P. Etingov on the quantum double of finite groups, and the q-spin networks of L. Kauffman and S. Lomonaco on the Fibonacci TQFT.
In summary, for a decade the topological idea in quantum computing has matured into a field that has its own theoretical problems: on the mathematical side, the formulation and classification of quantum phases of matter; on the physical side, the understanding of many body quantum physics which has long range entanglement, and the search of topological phases of matter. In this series of lectures, I will cover the theory.