Topological Phases

There is a fundamental problem in condensed matter physics: How to describe and classify different phases of matter? In Landau’s paradigm, phases (or orders) are described by symmetry. It turns out that there is a new kind of order beyond the paradigm, which is named topological order. The absence of symmetry doesn't imply topological orders are trivial or messy; on the contrary, they exhibit numerous extremely intriguing features.

One interesting feature is the existence of topological excitations (anyons). Anyons in some topological orders carry exotic nonabelian statistics, that is, exchanging (braiding) anyons are nontrivial unitary operations rather than phase factors.

On the other hand, some systems are topologically trivial with no symmetry, but they can be nontrivial if we impose some symmetries. They are named symmetry protected topological orders (SPT). They not only exhibit interesting boundary states, but also mysteriously relate to topological orders via the gauging operation, which promotes the global symmetry to a local gauge symmetry.

Countless interesting phenomena and structures emerge in topological phases, including bulk-boundary correspondence, anyon condensation, categorical explanation of symmetry, etc. I believe they may lead to unexpected revelations about the nature of quantum entanglement.

Quantum Information

Another fascinating feature of topological orders is the presence of long-range entanglement. In some sense, normal phases including superfluids and Bose-Einstein condenses are classical because the quantum states are short-range entangled. In contrast, topological orders are not in product states due to long-range entanglement.

Therefore, we might say that topological orders are the first examples of matters that are truly quantum.

All matters are quantum, but some matters are more quantum than others. – Not George Orwell

On the other hand, our current understanding of quantum entanglement isn't quite satisfactory. We can determine whether a certain quantum state is entangled; we may even quantify entanglement by entanglement spectrum and entanglement entropy. But this description seems to be an oversimplification. It's like describing linear operators only by determinants, ignoring other delicate structures.

I believe that there are intricate structures of quantum entanglement waiting to be discovered.

Quantum Computation

Humankind has designed a whole range of different systems to extract the power of quantum parallelism, but quantum errors haunt these attempts.

The scheme of topological quantum computation is fundamentally different from the existing systems. Specifically, we take the Hilbert space of anyon fusions to be the logical space and take braidings to be quantum operations.

There are three main requirements for a scheme of quantum computation to be practical: ability to perform arbitrary quantum operations, robustness against errors, and scalability. The topological feature ensures robustness against errors; it is also shown that universal quantum computation can be effectively performed by braiding anyons (arxiv:quant-ph/0001108) Furthermore, the scheme is easily scalable.