Quantum Computation with Topological Codes (eBook)

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