Quantum Computing

  1. Challenges in Building Quantum Computers
    1. Decoherence and Noise
      1. Sources of Noise
        1. Environmental Coupling
          1. Control Errors
            1. Crosstalk Between Qubits
              1. Charge Noise
                1. Magnetic Field Fluctuations
                2. Decoherence Mechanisms
                  1. T1 (Relaxation) Time
                    1. Energy Relaxation
                      1. Measurement and Significance
                      2. T2 (Dephasing) Time
                        1. Pure Dephasing
                          1. Echo Sequences
                          2. T2* vs T2 Echo
                          3. Noise Models
                            1. Pauli Noise
                              1. Amplitude Damping
                                1. Phase Damping
                                  1. Depolarizing Channel
                                  2. Quantum Volume as a Performance Metric
                                    1. Definition and Use
                                      1. Benchmarking Quantum Computers
                                    2. Quantum Error Correction (QEC)
                                      1. The No-Cloning Theorem
                                        1. Implications for Error Correction
                                        2. Principles of QEC
                                          1. Redundancy and Syndrome Measurement
                                            1. Error Detection vs Correction
                                              1. Stabilizer Formalism
                                              2. Simple Quantum Codes
                                                1. Repetition Codes
                                                  1. Bit-Flip Code
                                                    1. Phase-Flip Code
                                                    2. The Shor Code
                                                      1. Encoding and Decoding
                                                        1. Error Detection and Correction
                                                          1. Syndrome Extraction
                                                        2. Stabilizer Codes
                                                          1. Stabilizer Generators
                                                            1. Code Distance
                                                              1. Encoding and Decoding Circuits
                                                              2. Surface Codes
                                                                1. Lattice Structure
                                                                  1. Thresholds for Fault Tolerance
                                                                    1. Logical Qubit Implementation
                                                                      1. Syndrome Measurement
                                                                      2. Fault-Tolerant Quantum Computation
                                                                        1. Logical Qubits
                                                                          1. Fault-Tolerant Gate Implementation
                                                                            1. Threshold Theorem
                                                                              1. Magic State Distillation
                                                                              2. Quantum Error Correction Overhead
                                                                              3. Scalability and Connectivity
                                                                                1. Increasing Qubit Count
                                                                                  1. Fabrication Challenges
                                                                                    1. Control Electronics
                                                                                      1. Wiring and Packaging
                                                                                      2. Qubit-to-Qubit Connectivity
                                                                                        1. Coupling Map
                                                                                          1. Impact on Circuit Design
                                                                                            1. All-to-All vs Limited Connectivity
                                                                                            2. Crosstalk
                                                                                              1. Sources of Crosstalk
                                                                                                1. Mitigation Strategies
                                                                                                  1. Characterization Methods
                                                                                                  2. Control and Readout
                                                                                                    1. Classical Control Systems
                                                                                                      1. Real-Time Feedback
                                                                                                        1. Multiplexed Readout

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                                                                                                    7. Physical Realizations of Quantum Computers

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                                                                                                    9. Applications and Impact