Quantum Information and Technology

Qubits, Algorithms, Codes and Devices in 19 chapters — from “Qubits, registers and the Bloch sphere” to “Quantum metrology and sensing”.

Chapters

  1. Qubits, registers and the Bloch sphere
    • Dirac notation
    • Qubit state, global and relative phase
    • Bloch sphere and projective measurement
    • Single-qubit gates as rotations
    • The tensor-product register
    • Entangled states and partial measurement
    • The no-cloning theorem
    • What the qubit state leaves out
    • Problems
  2. Mixed states, generalised measurement and telling states apart
    • Trace, positive operators and the density operator
    • Bloch ball and ensemble ambiguity
    • The partial trace
    • Schmidt decomposition and purification
    • Generalised measurements and POVMs
    • Naimark and Stinespring dilations
    • Trace distance and fidelity
    • What the density operator settles
    • Problems
  3. Nonlocality: EPR, Bell inequalities and the CHSH game
    • Two-station correlations and the EPR argument
    • Local hidden-variable models and the CHSH inequality
    • The quantum CHSH violation
    • Tsirelson bound and no-signalling
    • The CHSH game
    • GHZ paradox and contextuality
    • Bell-test loopholes
    • What the CHSH inequality rules out
    • Problems
  4. Entanglement as a resource
    • Shared entangled states
    • LOCC and the ebit
    • Superdense coding
    • Teleportation
    • Entanglement swapping
    • Entropy of entanglement
    • Majorization and pure-state conversion
    • Concentration and dilution
    • Mixed-state entanglement
    • Entanglement distillation
    • Teleportation, superdense coding and exchange rates
    • Problems
  5. The quantum circuit model
    • Asymptotic notation and the cost of logic
    • Reversible classical computation
    • Circuit notation and cost
    • Controlled unitaries and phase kickback
    • Toffoli decomposition
    • Exact universality
    • Approximate universality
    • The Solovay-Kitaev theorem
    • The counting argument
    • Deferred measurement
    • Universality and the counting argument
    • Problems
  6. Oracles, quantum parallelism and the first algorithms
    • The query model
    • Costs in the query model
    • Phase oracles
    • The Hadamard transform
    • Quantum parallelism and interference
    • The Deutsch algorithm
    • The Deutsch-Jozsa algorithm
    • The Bernstein-Vazirani algorithm
    • The Simon algorithm
    • Linear algebra mod 2
    • Separations in the query model
    • Problems
  7. The quantum Fourier transform and phase estimation
    • The quantum Fourier transform
    • The QFT product form
    • The QFT circuit and its cost
    • Why the QFT is not an FFT
    • Phase estimation
    • Phase-estimation accuracy and register size
    • Eigenstate-superposition input and energy estimation
    • Controlled-power cost and one ancilla
    • Problems
  8. Order finding and Shor's factoring algorithm
    • Modular arithmetic and continued fractions
    • From factoring to order finding
    • Modular-multiplication eigenstates
    • The order-finding algorithm
    • Continued-fraction recovery of the order
    • Shor resource count against classical factoring
    • Demonstrations and compilation
    • The post-quantum consequence
    • Problems
  9. Grover search, amplitude amplification and counting
    • Unstructured search
    • The Grover iteration
    • The Grover iteration as a rotation
    • Grover iteration count and overshooting
    • The diffusion operator in gates
    • Grover optimality and the hybrid argument
    • Amplitude amplification
    • Quantum counting and searching blind
    • The value of quadratic speedup
    • Problems
  10. What quantum computers can compute efficiently
    • Models and Church-Turing theses
    • Classical complexity classes
    • The class BQP
    • BQP containments and the path sum
    • Oracle separations of classes
    • Factoring, search, NP and BQP
    • Sampling advantage experiments
    • The status of claims about BQP
    • Problems
  11. Noise as a quantum operation
    • Quantum channels and complete positivity
    • The operator-sum representation
    • The Choi state and Choi's theorem
    • Pauli channels on the Bloch ball
    • The depolarising channel
    • Amplitude damping and T1​
    • Dephasing and T2​
    • Decoherence as information flow
    • The Markovian limit and circuit fidelity
    • Quantum channels compared
    • Problems
  12. Quantum error correction
    • Linear codes
    • Obstacles to quantum error correction
    • The three-qubit bit-flip code
    • The logical error rate of the repetition code
    • The phase-flip code
    • Shor's nine-qubit code
    • Discretisation of errors
    • The Knill-Laflamme conditions
    • Code parameters and bounds
    • CSS codes and the Steane code
    • Break-even and threshold
    • The obstacles to quantum error correction removed
    • Problems
  13. The stabilizer formalism
    • Elementary group theory
    • The Pauli group
    • Stabilizer states
    • Stabilizer codes
    • The five-qubit code
    • Syndrome and encoding circuits
    • The Clifford group
    • The Gottesman-Knill theorem
    • A surface-code preview
    • Stabilizer codes without code words
    • Problems
  14. Entropy and compression
    • Shannon entropy
    • Typical sequences
    • The noiseless coding theorem
    • Joint, conditional, mutual information
    • The von Neumann entropy
    • Relative entropy and Klein's inequality
    • Subadditivity and strong subadditivity
    • Negative conditional entropy
    • Schumacher compression
    • What the von Neumann entropy counts
    • Problems
  15. Quantum key distribution and quantum networks
    • Fock and coherent states
    • The key distribution problem
    • The BB84 protocol
    • Estimating the error rate from a sample
    • Information reconciliation
    • Privacy amplification
    • The BB84 key rate
    • Entanglement-based and device-independent QKD
    • Practical QKD with weak pulses
    • The repeaterless bound
    • Quantum repeaters and networks
    • The BB84 key rate at 100 km
    • Problems
  16. What a qubit must do: requirements, tomography and benchmarking
    • The likelihood of measured counts
    • Rabi oscillations and the Ramsey sequence
    • DiVincenzo criteria
    • Coherence-time measurements
    • Gate-fidelity measures
    • State tomography
    • Maximum-likelihood reconstruction
    • Quantum process tomography
    • SPAM errors and readout correction
    • Randomised benchmarking
    • Error budgets and system-level benchmarks
    • Summary
    • Problems
  17. Superconducting qubits and circuit QED
    • The macroscopic superconducting phase
    • The quantised LC circuit
    • The Josephson nonlinear inductor
    • The Cooper-pair box and charge noise
    • The transmon
    • Circuit QED
    • Dispersive readout
    • Microwave control and leakage
    • Superconducting two-qubit gates
    • Superconducting limits at scale
    • Summary
    • Problems
  18. Trapped ions, neutral atoms, photons, NMR and spins in solids
    • Trapped-ion qubits
    • The motional bus and MS gate
    • Neutral-atom arrays and Rydberg blockade
    • Photonic qubits
    • Linear optics and measurement-induced gates
    • NMR ensemble computing
    • NV-centre spins
    • Semiconductor spin qubits
    • Platform comparison
    • Summary
    • Problems
  19. Quantum metrology and sensing
    • Quadratures and squeezed vacuum
    • Interferometric phase estimation
    • Fisher information and Cramer-Rao
    • The standard quantum limit
    • Heisenberg limit and GHZ states
    • How noise limits quantum enhancement
    • Squeezed-light interferometry
    • Spin squeezing and clocks
    • Solid-state and atomic sensors
    • What survives decoherence
    • Problems