Principles of Superconducting Quantum Computers. Daniel D. Stancil

of a transmon qubit...Figure 5.11 Equivalent circuits for transmission line...Figure 5.12 Magnitude and phase of the reflected signals...

      6 Chapter 6Figure 6.1 Classical harmonic oscillator consisting of a mass...Figure 6.2 Quantum mechanical harmonic oscillator...Figure 6.3 LC resonant circuit.Figure 6.4 Lumped circuit model for an open-circuited...Figure 6.5 Capacitively-coupled LC resonant circuits.

      7 Chapter 7Figure 7.1 The Fermi-Dirac distribution gives the probability...Figure 7.2 The Bose-Einstein distribution gives the expectedFigure 7.3 Depiction of a periodic potential created by the sumFigure 7.4 Cubic lattice in k-space...Figure 7.5 Spring-mass model for phonons on a 1-D lattice.Figure 7.6 Graphical depiction of electron...Figure 7.7 Visualizing conduction as the...Figure 7.8 A rectangular wire used to relate the voltage...Figure 7.9 A superconductor expels magnetic flux (top)...Figure 7.10 The potential is negative (attractive)...Figure 7.11 Diagrammatic representation of two electrons...Figure 7.12 The delayed response to the presence of an electron...Figure 7.13 The probability that a pair state is occupied...Figure 7.14 Geometry used to understand the London gauge.Figure 7.15 Behavior of the magnetic field and superconducting...Figure 7.16 Geometry of a superconducting sample...

      8 Chapter 8Figure 8.1 Geometry for a particle incident from...Figure 8.2 Geometry for a particle incident from...Figure 8.3 A Josephson junction is formed by two superconductors...Figure 8.4 Circuit symbols for a Josephson junction...Figure 8.5 Non-linear resonant circuit formed...Figure 8.6 Two Josephson junctions in parallel...Figure 8.7 Example geometry for applying a current to tune...Figure 8.8 Parametric reflection amplifiers...

      9 Chapter 9Figure 9.1 Measurement of T₁ relaxation time.Figure 9.2 Measurement of T₂ dephasing time.Figure 9.3 Measurement error calibration matrix...Figure 9.4 Mitigating measurement errors...Figure 9.5 Randomized benchmarking circuit...Figure 9.6 Randomized benchmarking experiment on IBM Q hardware.Figure 9.7 Interleaved Randomized Benchmarking (IRB) adds...Figure 9.8 Optimized spectrum with a notch at the anharmonicity...Figure 9.9 Error-mitigated results for a single-qubit...Figure 9.10 Error-mitigated results for traversal of a path...Figure 9.11 Error-mitigated results for finding the ground...Figure 9.12 Sample control pulses generated by deep...Figure 9.13 Implementation of a three-qubit Toffoli gate...

      10 Chapter 10Figure 10.1 Classical noisy channel.Figure 10.2 Encoding data for error detection/correction...Figure 10.3 Error correction using the 3-bit repetition code.Figure 10.4 Hamming [7,4,3] code. Data bits...Figure 10.5 Quantum channel with Pauli errors.Figure 10.6 Quantum error correction: encoding, error, correction.Figure 10.7 Three-qubit error correction code for bit flips only.Figure 10.8 Three-qubit error correction code for phase flips only.Figure 10.9 Shor’s [[9,1,3]] code that...Figure 10.10 Correction circuit for Shor’s [[9,1,3]] code.Figure 10.11 Partial rotation around the x-axis.Figure 10.12 Measuring Z₁Z₀...Figure 10.13 Projective measurement for...Figure 10.14 Stabilizer version of 3-qubit bit.Figure 10.15 Error correction circuit for the.Figure 10.16 Encoding circuit for the [[7, 1, 3]] Steane code.Figure 10.17 The logical X operator is transversal...Figure 10.18 Performing the T operation on encoded data.Figure 10.19 Estimated logical error rates for a...Figure 10.20 Surface code, composed by a square...Figure 10.21 Surface code error cycle. The top...Figure 10.22 When a bit flip (X) error...Figure 10.23 A phase flip error affects...Figure 10.24 When multiple errors occur in the same vicinity...Figure 10.25 Logical X and Y...Figure 10.26 Any path of X operators connecting one...Figure 10.27 Rough merge of two planar qubits.Figure 10.28 Smooth split of a planar qubits.Figure 10.29 Preparation for a CNOT operation on control qubit...Figure 10.30 State injection. (a) Prepare a single data qubit...Figure 10.31 Errors in surface code for Exercise 10.4.

      11 Chapter 11Figure 11.1 Three-qubit quantum adder.Figure 11.2 A NOT gate is reversible. By using another NOT gate...Figure 11.3 The reversible XOR gate requires...Figure 11.4 To make an AND gate reversible, we must remember...Figure 11.5 The Toffoli gate has two...Figure 11.6 The Toffoli gate is universal for Boolean logic...Figure 11.7 Examples of multi-controlled Toffoli (MCT) gates.Figure 11.8 Reversible four-input AND circuit...Figure 11.9 Reversible four-input AND circuit; ancilla bits...Figure 11.10 Reversible fanout using a CNOT. While this works...Figure 11.11 General reversible circuit...Figure 11.12 Sketch of a generalized reversible circuit...Figure 11.13 Reuse of ancilla bits (dashed lines)...Figure 11.14 A CNOT can be used to replicate...Figure 11.15 Action of Toffoli gate on superposition states.Figure 11.16 An entangled Bell state.Figure 11.17 Implementation of Toffoli gate [9].Figure 11.18 Qubit topologies for some....Figure 11.19 The direction of a CNOT gate can be reversed...Figure 11.20 The SWAP gate exchanges the state of two qubits...Figure 11.21 SWAP gates may be needed to move qubit states...Figure 11.22 Classical non-reversible one-bit full adder.Figure 11.23 Simple quantum adder. Carry output from bit...Figure 11.24 Three-bit ripple-carry adder.Figure 11.25 Three-bit ripple-carry adder with reclaimed ancillas.Figure 11.26 Three-bit ripple-carry adder, redrawn to better...Figure 11.27 MAJ (majority) and UMA...Figure 11.28 Full adder built with MAJ and UMA.Figure 11.29 Three-bit ripple adder with one ancilla bit.Figure 11.30 Optimized three-bit ripple adder with one ancilla bit.Figure 11.31 Carry-lookahead circuit for 8-bit adder...Figure 11.32 Quantum carry-lookahead adder...Figure 11.33 Controlled-���� (CZ)...Figure 11.34 CZ gate acting on an equal...Figure 11.35 Phase kickback using controlled-T gate...Figure 11.36 Selective phase rotation of basis states...Figure 11.37 Selective phase flip...Figure 11.38 Phase kickback using CZ and CNOT gates.Figure 11.39 Phase logic gates: performs phase rotation...Figure 11.40 Flipping the phase when...Figure 11.41 Interleave function for Exercise 11.4.Figure 11.42 Max function for Exercise 11.6.

      12 Chapter 12Figure 12.1 Grover search initialization. All...Figure 12.2 First iteration of Grover search...Figure 12.3 Second iteration of Grover search...Figure 12.4 Third iteration of Grover...Figure 12.5 Grover search algorithm for...Figure 12.6 QFT of the state |2⟩.Figure 12.7 QFT of the state...Figure 12.8 Result of QFT applied to...Figure 12.9 Circuit for creating a quantum-encoded square wave...Figure 12.10 Quantum-encoded square wave state with frequency...Figure 12.11 Output of QFT applied to square wave.Figure 12.12 Magnitudes of basis states from...Figure 12.13 QFT for four qubits.Figure 12.14 Relative phase rotations for each qubit...Figure 12.15 First stage of 4-qubit QFT.Figure 12.16 Inverse QFT for four qubits.Figure 12.17 Quantum Phase Estimation...Figure 12.18 Factoring integer...Figure 12.19 For a = 3 and M = 35, the period...Figure 12.20 Modular exponentiation circuit...Figure 12.21 Quantum circuit for period-finding...Figure 12.22 Components of a Variational Quantum Algorithm (VQA)...Figure 12.23 Expressibility of various ansatzes...Figure 12.24 Examples of problem-agnostic...Figure 12.25 Example of MaxCut: Partition a graph...Figure 12.26 Circuit to rotate qubits i and j...Figure 12.27 QAOA circuit for MaxCut example...

      List of Tables

      1 Chapter 4Table 4.1 Useful ABCD matrices.

      2 Chapter 6Table 6.1 Classical parameters and the...

      3 Chapter 7Table 7.1 Energy and momentum conservation in electron-phonon...Table 7.2 Comparison of...

      4 Chapter 10Table 10.1 Effects of bit-flip error channel...Table 10.2 Effects of phase-flip error...Table 10.3 Stabilizer generators for Steane [[7, 1, 3]] code.Table 10.4 Syndrome measurements for all single-qubit errors...

      5 Chapter 11Table 11.1 Boolean algebra operators.Table 11.2 Comparison of in-place...Table 11.3 Comparison of adder circuits...Table 11.4 Controlled-Z (CZ) gate...

      6 Chapter 12Table 12.1 Result of conditional phase rotations...

      Guide

      1  Cover

      2  Title page

      3  Copyright

      4