Quantum Computation with Topological Codes, 1st ed. 2015 From Qubit to Topological Fault-Tolerance SpringerBriefs in Mathematical Physics Series, Vol. 8

This book presents a self-consistent review of quantum computation with topological quantum codes. The book covers everything required to understand topological fault-tolerant quantum computation, ranging from the definition of the surface code to topological quantum error correction and topological fault-tolerant operations. The underlying basic concepts and powerful tools, such as universal quantum computation, quantum algorithms, stabilizer formalism, and measurement-based quantum computation, are also introduced in a self-consistent way. The interdisciplinary fields between quantum information and other fields of physics such as condensed matter physics and statistical physics are also explored in terms of the topological quantum codes. This book thus provides the first comprehensive description of the whole picture of topological quantum codes and quantum computation with them.

1 Introduction to quantum computation 1.1 Quantum bit and elementary operations 1.2 The Solovay-Kitaev algorithm 1.3 Multi-qubit gates 1.4 Universal quantum computation 1.5 Quantum algorithms 1.5.1 Indirect measurement and the Hadamard test 1.5.2 Phase estimation, quantum Fourier transformation, and factorization 1.5.3 A quantum algorithm to approximate Jones polynomial 1.6 Quantum noise 
2 Stabilizer formalism and its applications 2.1 Stabilizer formalism 2.2 Clifford operations 2.3 Pauli basis measurements 2.4 Gottesman-Knill theorem 2.5 Graph states 2.6 Measurement-based quantum computation 2.7 Quantum error correction codes 2.8 Magic state distillation 2.8.1 Knill-Laflamme-Zurek protocol 2.8.2 Bravyi-Kitaev protocol 
3 Topological stabilizer codes 3.1 Z2 chain complex 3.2 A bit-flip code: exercise 3.3 Definition of surface codes 3.3.1 Surface code on a torus: toric code 3.3.2 Planar surface code 3.4 Topological quantum error correction 3.5 Error correction and spin glass model 3.6 Other topological codes 3.7 Connection to topological order in condensed matter physics 
4 Topological quantum computation 4.1 Defect pair qubits 4.2 Defect creation, annihilation, and movement4.3 Logical CNOT gate by braiding 4.4 Magic state injections and distillation 4.5 Topological calculus 4.6 Faulty syndrome measurements and noise thresholds 4.7 Experimental progress 
5 Topologically protected MBQC 5.1 Topological cluster state in 3D5.2 Vacuum, defect, and singular qubit regions5.3 Elementary operations in topological MBQC 5.4 Topological quantum error correction in 3D 5.5 Applications for MBQC on thermal states 
A Fault-tolerant quantum computation A.1 Fault-tolerant syndrome measurements A.2 Fault-tolerant gate operations A.3 Concatenated quantum computation 
B Decoding stabilizer codes 
Index 
References