Title Page
Abstract
Contents
Chapter 1. Introduction 48
Chapter 2. Preliminary 52
2.1. Stabilizer formalism 53
2.1.1. Stabilizer group and subspace 54
2.1.2. Unitary operation on a stabilizer subspace 56
2.1.3. Measurement on a stabilizer subspace 60
2.2. Measurement-based quantum computing 64
2.2.1. Cluster state 65
2.2.2. How measurement-based quantum computing works 68
Chapter 3. Color-code-based measurement-based quantum computing 72
3.1. Color-code-based cluster states 74
3.1.1. Two-dimensional color-code lattices 74
3.1.2. Construction of color-code-based cluster states 75
3.1.3. Stabilizer generators 78
3.1.4. Shrunk lattices and correlation surfaces 80
3.2. Measurement-based quantum computing via color-code-based cluster states 86
3.2.1. Measurement pattern 87
3.2.2. Defects and related correlation surfaces 88
3.2.3. Defining a logical qubit 92
3.2.4. Initialization and measurement of a logical qubit 94
3.2.5. Elementary logic gates 96
3.2.6. State injection 107
3.3. Error correction 109
3.3.1. Error correction in the vacuum 109
3.3.2. Error correction near defects 112
3.3.3. Error correction near Y-planes 115
3.4. Error simulations 135
3.4.1. Error model 136
3.4.2. Simulation methods 136
3.4.3. Decoding methods 138
3.4.4. Results 141
3.5. Resource analysis 143
3.5.1. Resource overheads for placing logical qubits 143
3.5.2. Resource overheads for nontrivial logic gates 145
3.6. Remarks 150
3.7. Appendices 152
3.7.1. Methods for analyzing resource overheads of placing logical qubits 152
3.7.2. Methods for analyzing resource overheads of logic gates in RTCS computation 157
3.7.3. Methods for analyzing resource overheads of logic gates in CCCS computation 162
Chapter 4. Linear-optical measurement-based quantum computing with parity-encoded multiphoton qubits 172
4.1. Constructing Raussendorf's 3D cluster states through fusions 175
4.1.1. Type-II fusion 175
4.1.2. Bayesian error tracking for nonideal fusions 177
4.1.3. Building a lattice 180
4.2. Parity-encoding-based topological quantum computing 182
4.2.1. Noise model 183
4.2.2. Generation of microclusters 185
4.2.3. Performance analysis 198
4.3. Modified concatenated Bell-state measurement scheme 207
4.3.1. Original CBSM scheme 208
4.3.2. Modified CBSM scheme for PTQC 210
4.3.3. Error probabilities of a CBSM under a lossy environment 211
4.4. Comparison with other approaches 215
4.4.1. Comparison with approach (i) 216
4.4.2. Comparison with approach (ii) 219
4.4.3. Comparison with approach (iii) 220
4.5. Remarks 222
4.6. Appendices 224
4.6.1. Calculation of the error probabilities of a CBSM when on-off detectors are used 224
4.6.2. Proofofvanishingoff-diagonalentriesofthePOVM elements of a lossy physical-level BSM 230
4.7. Derivationofthephysical-levelgraphsofpost-H microclus-ters 231
4.7.1. Details of error simulations 235
4.7.2. Details of resource analysis 239
Chapter 5. Conclusion 242
Bibliography 246
국문초록 256
Table 1. Qubits Q(b) corresponding to each element (vertex, edge, face, or cell) b in Lpc . The results for Ldc can be obtained by changing each p or d.[이미지참조] 81
Table 2. Allowed positional relations between a primal defect d and a compatible CS. The relations for dual defects are analogous. 91
Table 3. Resource overheads of RTCS and CCCS computation for various sets of implementable logic gates, evaluated by the numbers of physical... 144
Table 4. Numbers of physical qubits required for the logical CNOT gates with RTCSs or CCCSs. Only the leading order terms on d are presented.... 146
Table 5. Numbers of physical qubits required for the logical phase gates in defect-based (DB) RTCS, patch-based (PB) RTCS, or CCCS computation.... 148
Table 6. Information of the data points along the upper envelope lines in Fig. 54(b) when single-photon resolving detectors are used. N∗GHZ, N10-7,...[이미지참조] 203
Table 7. Information of the data points along the upper envelope lines in Fig. 54(b) when on-off detectors are used. N∗GHZ, N10-7, and d10-7 at η=1%...[이미지참조] 204
Figure 1. Examples of cluster states. Orange dots and lines indicate the vertices and edges of the graphs, respectively. (a) A cluster state on a simple... 67
Figure 2. Implementation of the Hadamard gate in two-dimensional MBQC. MX (MY) means the measurement in the X-basis (Y-basis).[이미지참조] 68
Figure 3. Two typical examples of color-code lattices: (a) 4-8-8 and (b) 6-6-6 lattices. The lattices are 3-valent and have 3-colorable faces. 74
Figure 4. (a) Red and (b) blue shrunk lattices of the 4-8-8 color-codelattice. Red or blue dots (lines) indicate their vertices (edges), which correspond to... 75
Figure 5. Structure of a single layer of a color-code-based cluster state (CCCS) based on the 4-8-8 color-code lattice L2D. Each black circle is a...[이미지참조] 76
Figure 6. Stack of multiple identical layers along the simulating time axis for a CCCS. Each pair of two CQs adjacent along the time axis is connected... 77
Figure 7. Four types of stabilizer generators in a CCCS defined in Definitions 3.1–3.3: (a) A-, (b) C-, (c) L-, and (d) J-type stabilizer generators.... 78
Figure 8. Unit cells of the primal shrunk lattices of a 4-8-8CCCS: (a) a blue cell in the primal red shrunk lattice Lpr(a green cell is identical except the...[이미지참조] 80
Figure 9. (a) Timelike joint of primal correlation surfaces (CSs) originated from a J-type stabilizer generator. The X or Z operators on the qubits in-... 84
Figure 10. Example of the construction of a spacelike joint of three primal CSs. A primal layer of a 4-8-8 CCCS is presented. We first assume a... 85
Figure 11. (a) Schematic diagram of a defect (pb-D) and a db-CS S ending at the defect. The defect is defined as Eq. (3.3) with a 2-chain h₂db ∈ H₂db...[이미지참조] 89
Figure 12. Definition of a primal logical qubit and its initialization and measurement. (a) Schematic diagram of a primal logical qubit composed... 93
Figure 13. (a) XL- and (b) ZL-initialization. A logical qubit prepared in the output layers QOUT (t0- and (t0+1)-layer). For the XL -initialization, the...[이미지참조] 95
Figure 14. Logical identity gate of a primal logical qubit between the input layer QIN (t0-layer) and the output layers QOUT (t₁- and (t₁+1)-layer). The...[이미지참조] 97
Figure 15. Construction of a CNOT gate between a primal logical qubit (target) and a dual one (control). Each colored single (double) line indicates... 100
Figure 16. (a) Construction of the primality-switching gate changing a primal logical qubit to a dual one. ZLp is transformed into ZLd' via the presented...[이미지참조] 101
Figure 17. Construction of a Hadamard gate from a primal logical qubit to a dual one. Each colored single (double) line is the primal (dual) defect of... 103
Figure 18. Construction of a logical phase gate on a primal logical qubit. The input logical-X operator (XL) is transformed into the output logical-Y...[이미지참조] 105
Figure 19. Placement of a Y-plane on thet2-layer of Fig. 18. In (a), the colored circles indicate the timelike defects penetrating the layer, and the thick... 106
Figure 20. State injection procedure. (a) An unencoded state is injected into an injection qubit qinj, which is the only input qubit, in the pr-D which...[이미지참조] 108
Figure 21. (a) Explicit structure of a parity-check operator (PC), specifically a pb-PC in a 4-8-8 CCCS. Purple triangles indicate its X-support... 110
Figure 22. PCs deformed or created due to a (a) timelike or (b) spacelike pb-D in a 4-8-8 CCCS. Each purple triangle with a solid (or dashed) border... 113
Figure 23. (a) Primal hybrid PC for error correction in a primal Y-plane, constructed by multiplying a primal PC and the dual A-type SG around its... 116
Figure 24. Error correction during the process for a Hadamard gate, particularly near the Y-planes. (a) The Y-planes should be wide enough since... 118
Figure 25. Configuration of the system SI introduced to verify error correction in a Hadamard gate, where the primal defects are just extended...[이미지참조] 119
Figure 26. Correspondences of (a), (b) hybrid PCs, (c) merged hybrid PCs, and (d) merged defect PCs in the original system SH for a Hadamard...[이미지참조] 120
Figure 27. Error correction when the vacuum and a primal Y-plane on a dual layer are separated by a pb-D. Defect (Y-plane) qubits in the layer are... 128
Figure 28. Microscopic structures near where (a) red and blue defects or (b) blue and green defects are closest for a logical phase gate. The colored... 129
Figure 29. Nontrivial undetectable primal error chains regarding a logical phase gate. The colored circles indicate the timelike parts of the defects and... 131
Figure 30. Removal of some primal PCs near where (a) red and blue defects or (b) green and blue defects are closest to verify that local nontrivial... 132
Figure 31. Structure of a layer in the simplified defect model for the simulation regarding (a) RTCSs, (b) 4-8-8 CCCSs, or (c) 6-6-6 CCCSs, partic-... 137
Figure 32. ZL error rate per two layers Plog versus nontrivial physical-level error rate pphy, for different code distances with respect to (a) 4-8-8 CCCSs, (b) 6-6-6 CCCSs, and (c) RTCSs. The small graphs show the results near the error thresholds....[이미지참조] 142
Figure 33. (a) Implementation of a logical phase gate SL with an ancilla logical state |YL〉:=(|0L〉+i|1L〉)/ √2. ZLSL or SL is applied on the input...[이미지참조] 147
Figure 34. Estimated numbers of physical qubits required for (a) an identity gate, (b) a CNOT gate, or (c) a phase gate versus the logical error rate... 149
Figure 35. Arrangement of timelike primal defects for calculating the resource overheads of MBQC via (a) RTCSs or (b) CCCSs. Their pro-... 154
Figure 36. Checkerboard architecture in patch-based RTCS computation. Blue (grey) squares are patches for logical data (ancilla) qubits. (a)... 158
Figure 37. Arrangement of defects for the logical CNOT gate in defect-based RTCS computation. (a) The control primal logical qubit is first switched to a dual one (grey squares). Then one of the dual defects proceeds to wrap around a defect of... 160
Figure 38. Arrangement of defects for the logical CNOT gate in CCCS computation. The control primal logical qubits are first switched to dual ones (colored squares with dashed boundaries) as shown in (a) and (c), then the braiding operations are... 165
Figure 39. Arrangement of defects for a logical Hadamard gate. The Y-plane (orange square) covering the logical qubit should be wide enough so... 167
Figure 40. Arrangement of defects for a logical phase gate in CCCS computation. Defects of width α0 are extended spacelikely to surround the Y-...[이미지참조] 168
Figure 41. Example of a type-II fusion. A type-II fusion is done by measuring Z0X0' and X0Z0' on the two graph states. In (a), two stabilizers (green...[이미지참조] 176
Figure 42. BSM scheme for single-photon polarization qubits. It uses three polarizing Beam splitters (PBSs), 90° and 45° wave plates, and four (A–D)... 178
Figure 43. Lattice building process with microclusters. The orange boxes indicate fusions. In step 1, side and central microclusters are fused to form... 181
Figure 44. Structure and generation of post-H microclusters for PTQC. (a) Schematic of central and side post-H microclusters used in PTQC for the... 184
Figure 45. Physical-level graphs of post-H microclusters for the HIC and HIS configurations when the (n,m) parity encoding is used for PTQC. The... 186
Figure 46. Examples of graph notations used in Fig. 45. (a) and (b) respectively show examples of a blue dashed box and a number inside a circle,... 187
Figure 47. Examples of the two types of merging operations on two GHZ states: (a) a BSM on the root photon of one state and a leaf photon of the... 189
Figure 48. Decomposition of a graph state done by separating recurrent subgraphs that are connected with multiple vertices. 190
Figure 49. Decomposition of post-H microclusters for the HIC configuration. Different types of post-H microclusters are decomposed by the method... 191
Figure 50. Decomposition of post-H microclusters for the HIS configuration. Different types of post-H microclusters are decomposed by the method... 192
Figure 51. Construction of merging graphs from a physical-level graph. v is the only vertex with a degree larger than two in the original graph. The upper and lower processes differ in the selection of the seed vertex for the decomposition of v. 195
Figure 52. Loss threshold ηth for various parameters on the encoding size (n,m), the type of detectors, the post-selection (PS) of star clusters, and the H-configuration. "SPRD" stands for single-photon resolving detector. The values of j are...[이미지참조] 200
Figure 53. Resource overhead N10-6 (calculated at η = 0.01) for various parameters on the encoding size (n,m), the type of detectors, the post-selection (PS) of star clusters, and the H-configuration. "SPRD" stands for single-photon resolving...[이미지참조] 201
Figure 54. Photon loss thresholds ηth as a function of the number N∗GHZ of GHZ-3 states required per central qubit. N∗GHZ is calculated at η=0.01...[이미지참조] 202
Figure 55. Comparison of different strategies for generating a post-H microcluster. It shows the distribution of the calculated overhead NGHZside of a...[이미지참조] 206
Figure 56. Simulation results for the approach using single-photon qubits with fusions assisted by ancillary photons. It shows the photon loss thresh-... 217
Figure 57. Simulation results for the approach using the simple repetition codes. It shows the photon loss thresholds ηth as a function of n for MTQC,...[이미지참조] 220
Figure 58. (a) Transformation of a graph state by applying a Hadamard gate followed by applying a CZ gate. (b) Physical-level graph structure of... 232
Figure 59. Encoding circuit of the state |+L〉:=|0L〉+|1L〉 in the (3,3) parity encoding. It employs multiple copies of the state |+〉:= |H〉+|V〉,...[이미지참조] 233
Figure 60. Circuit to implement the lattice-level Hadamard gate of the (3,3) parity encoding. 235
Figure 61. Structure of a logical identity gate for simulations where the code distance is d = 5 and the length along the simulated time (t) axis is T... 236