Topological quantum computing usually is referred to as the sexiest, complicated, far-reaching, abstract-nonsense, and sometimes even impractical model of quantum computing. Whether this model of quantum computing makes you excited or confused or gives you rash depends on how much you know or, at least, are willing to know. Because although one needs to understand a great deal of maths and physics to get into the depth of the model, to learn the big picture one only needs to shut up the inner adjective-making system and openly think about its big ideas.
Topological quantum computing is the junction of three areas of modern science: late 20th and 21st centuries maths, physics and computer science. In one line, it promises to build qubits from the topologically ordered phase of matter whose background theory is explained by category theory. Each of these words, indeed, carries a deep area of active research, but we decrypt them in a few words.
Qubits are building blocks of quantum computers. Take classical computers, as we know we can encode inputs and outputs in the strings binaries; 0 or 1. A qubit is a quantum version of 0 and 1, with a big difference; a qubit takes a probabilistic value between and including 0 and 1. What it means or represents might be baffling, but as Scott Aaronson always says, “quantum physics is easy once you take physics out of it’’. For now, assume we have such a capability, and we can leverage this power for doing computation.
Topologically ordered phases are phases which are sensitive to the topology of the manifold. Classical phases, as you remember from school, are liquid, solid, and gas. At the university, quantum phases such as ferromagnetism or paramagnetism are also added to the list. But these phases, which mostly can be modelled with some arrangements of spins, are insensitive to the geometrical properties of the manifold they are located on. For example, if you put an arrangement of spins, like the Ising model, on a torus or sphere, the ground state degeneracy will not change. But in a topologically ordered phase, the ground state degeneracy often increases with increasing the genus(holes).
Category theory(in the algebraic sense) is the language of maps instead of sets. Any graduate-level abstract algebra book ends with a chapter on category theory, and any algebraic homology or topology book starts with a chapter on category theory. At the end of 20st-century, a great deal of maths was revisited or reexpressed through the lens of category theory. Why? Mainly because of its motive, i.e. universal properties. How much is a collection of groups similar to a collection of topological spaces? While you would experience a headache answering this question with set theory, you can answer this question straightforwardly with basic knowledge of category theory. Why? Because the language eliminates a lot of unnecessary data along the way. I remember one of my tutors at the college would tell us, ``category theory is a tank when you have it, you do need to carry your pocket knife!’’
But what is the advantage of topological qubits in comparison to ion-trap or superconductors qubits? Now let's go back to the one-line description and review it in light of this new information. Imagine you have a qubit encoded in the ground state of a topologically ordered system. Since the ground space is only a function of topology, its properties stay the same as long as any errors act locally and do not change the topology. Topological qubits hence are, therefore, intrinsically robust against errors.
The background theory which explains this model is topological quantum field theory. A field theory which only understands and is sensitive to the basic topological properties of a manifold. Inspired by Segal's axiomatization of conformal field theory, Sir Michael Atiyah axiomatized topological quantum field theory with category theory. So you probably can guess the reason for the language shift.
By now, you have gained a fair understanding of the big picture of topological quantum computation. In future, we will further talk about Frank Wilczek and explain other weird properties of these systems such as hosting quasiparticles called anyons with fractional spins and charges. We will see more topological quantum field theories and get a glimpse of Atiyah’s school in Cairo. We will visit braids and knots and another Fields medalist, Michael Freedman, and his proposal for solving knot problems with topological quantum computers. Last but not least, we will talk about the star of topological quantum computing, Alexie Kitaev and his habit of playing with toy models!