Quantum computing despite decoherence - Lövdahl, Daniel (My University, 2025)
An introduction and overview of the most important concepts and results of quantum error correction (QEC) and fault-tolerant quantum computing (FTQC) is presented. The thesis aims to present both landmark results and contemporary research in QEC and FTQC. The thesis begins with a discussion of the theory of quantum mechanics and quantum entanglement. After this, the circuit model for ideal quantum computers is described. The detrimental effects of quantum decoherence on the qubits in an ideal quantum computer are presented. The solution to the problem of decoherence is QEC, which is introduced by describing the Shor code in detail. The Shor code belongs to a family of important quantum codes called stabilizer codes, and these are studied in the end of the second part because of their high relevance for both experimental and theoretical FTQC. In the last part of the thesis, the most important results of FTQC are discussed. Transversal gates and fault-tolerant gadgets are introduced, and a method of implementing a universal gate set with gate teleportation using magic states is discussed. The threshold theorem, which is the foundational theoretical result in FTQC, is introduced. Finally, an experimentally promising quantum code called the surface code is discussed, and a way of implementing a universal gate set with the surface code through lattice surgery is presented. The thesis ends with a brief discussion of recent experimental results from Google with the surface code. It is concluded that the theory of QEC and FTQC is well established, and that there are currently no fundamental theoretical obstacles preventing the construction of a fault-tolerant quantum computer. The most promising quantum code for building a fault-tolerant quantum computer is the surface code, which is very well studied and analyzed in the literature. The surface code has a very high qubit overhead, and therefore current research aims to find other codes with a high threshold value and low overhead. A promising family of codes for realizing FTQC is concluded to be the high-rate quantum LDPC codes.
Keywords: quantum computing, quantum error correction, stabilizer codes, faulttolerant quantum computing, surface code