Topological quantum computation in condensed matter systems

It's been a long time since the last post. Today after a discussion with my labmates, I suddenly realized I already made detailed slides for this topic. (For my class Advanced Quantum Information and Computation taught by Prof. Liang Jiang back in 2021)

Compared to other candidates for quantum computers (trapped ions, color centers, superconducting qubits) topological quantum computing largely remains theoretical with very few advancements.

The key ingredient here is the Majorana fermion which is still elusive after several failed claims. In this presentation, I outline the basics of computing with Majorana fermions, mainly theory (as in fact, no experiments have been realized yet). Majorana zero modes obey non-Abelian statistics and thus can perform braiding to encode logic gates. Effectively, we can understand it as follows: Electrons are split into 2 halves. These are Majorana fermions, which will undergo braiding and recombination. The whole process is a logic gate. Depending on how the braiding is done, we can obtain different logic gates for quantum computing. Since this nontrivial process is protected by topology, any small perturbations that do not close the gap of the system cannot interfere with the braiding and the gate is immune to these noises. Hence we have the famous term "fault-tolerant computing".

Despite its promising fault-tolerant computing power, topological quantum computation is unfortunately incomplete, as it can't realize the universal set of gates with topological protection. Specifically, the T-gate, which changes the phase of a qubit by pi/4, is impossible to make by braiding alone.

Topological quantum computation in condensed matter.pdf

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Non-abelian statistics of Majorana fermions as one accumulates a phase as it revolves around one another. From: J. Phys. Soc. Jpn. 85, 022001 (2016)