Topological quantum computing encodes qubits in global, topological properties of certain quantum systems, making them intrinsically robust against local noise and decoherence. Here are the foundational ideas and proposals developed before 2020.

Anyons and Braiding

In two-dimensional systems, particles called anyons can exhibit statistics beyond bosons and fermions. Alexei Kitaev first proposed using non-Abelian anyons to store quantum information, where exchanging (braiding) anyons implements quantum gates depending only on the braid topology, not on local perturbations [1].

Kitaev’s Honeycomb Model and Toric Code

Kitaev introduced the toric code in 1997 as a solvable model realizing Abelian anyons on a spin lattice, demonstrating topologically protected degeneracy and error-correction properties [1]. His 2003 honeycomb lattice model extended these ideas, showing how to generate non-Abelian anyons in a realistic Hamiltonian framework [2].

Topological Quantum Computation Formalism

In 2002, Freedman, Kitaev, Larsen, and Wang formalized the theory of topological quantum computation. They proved that braiding non-Abelian anyons such as Fibonacci anyons can approximate any unitary operation, yielding a universal gate set [3].

Majorana Zero Modes

Majorana zero modes are emergent quasiparticles predicted to occur at the edges of 1D topological superconductors. Das Sarma, Freedman, and Nayak proposed using these Majorana modes to build topological qubits in 2005, outlining how braiding Majoranas implements protected quantum gates [4].

Experimental Proposals

Early experimental platforms focused on fractional quantum Hall systems and semiconductor–superconductor heterostructures to realize Majorana modes. The ν=5/2 fractional quantum Hall state was suggested as a candidate for non-Abelian anyons, and proposals emerged for nanowire devices combining InSb or InAs with superconductors to host Majoranas [5].

Advantages and Challenges

Topological encoding offers passive error protection: local noise cannot change global topology. However, detecting and braiding anyons in the lab proved challenging. By 2019, unambiguous demonstration of non-Abelian statistics remained elusive, and engineering complex topological phases required precise materials and low temperatures.

References

[1] Wikipedia contributors. (2020). Topological quantum computer. Wikipedia. https://en.wikipedia.org/wiki/Topological_quantum_computer

[2] Kitaev, A. Y. (2003). Fault-tolerant quantum computation by anyons. Annals of Physics, 303(1), 2-30.

[3] Freedman, M. H., Kitaev, A., Larsen, M. J., & Wang, Z. (2002). Topological quantum computation. Bulletin of the American Mathematical Society, 40(1), 31-38.

[4] Das Sarma, S., Freedman, M., & Nayak, C. (2005). Topologically Protected Qubits from a Possible Non-Abelian Fractional Quantum Hall State. Physical Review Letters, 94(16), 166802.

[5] Nayak, C., Simon, S. H., Stern, A., Freedman, M., & Das Sarma, S. (2008). Non-Abelian anyons and topological quantum computation. Reviews of Modern Physics, 80(3), 1083.