Programming Quantum Computers. Essential Algorithms and Code Samples

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• Preface
  • How This Book Is Structured
  • Conventions Used in This Book
  • Using Code Examples
  • OReilly Online Learning
  • How to Contact Us
  • Acknowledgments
• 1. Introduction
  • Required Background
  • What Is a QPU?
  • A Hands-on Approach
    • A QCEngine Primer
      • Running code
      • Debugging code
  • Native QPU Instructions
    • Simulator Limitations
    • Hardware Limitations
  • QPU Versus GPU: Some Common Characteristics
• I. Programming for a QPU
• 2. One Qubit
  • A Quick Look at a Physical Qubit
  • Introducing Circle Notation
    • Circle Size
    • Circle Rotation
  • The First Few QPU Operations
    • QPU Instruction: NOT
    • QPU Instruction: HAD
    • QPU Instruction: READ
    • QPU Instruction: WRITE
    • Hands-on: A Perfectly Random Bit
    • QPU Instruction: PHASE()
    • QPU Instructions: ROTX() and ROTY()
  • COPY: The Missing Operation
  • Combining QPU Operations
    • QPU Instruction: ROOT-of-NOT
  • Hands-on: Quantum Spy Hunter
  • Conclusion
• 3. Multiple Qubits
  • Circle Notation for Multi-Qubit Registers
  • Drawing a Multi-Qubit Register
  • Single-Qubit Operations in Multi-Qubit Registers
    • Reading a Qubit in a Multi-Qubit Register
  • Visualizing Larger Numbers of Qubits
  • QPU Instruction: CNOT
  • Hands-on: Using Bell Pairs for Shared Randomness
  • QPU Instructions: CPHASE and CZ
    • QPU Trick: Phase Kickback
  • QPU Instruction: CCNOT (Toffoli)
  • QPU Instructions: SWAP and CSWAP
    • The Swap Test
  • Constructing Any Conditional Operation
  • Hands-on: Remote-Controlled Randomness
  • Conclusion
• 4. Quantum Teleportation
  • Hands-on: Lets Teleport Something
  • Program Walkthrough
    • Step 1: Create an Entangled Pair
    • Step 2: Prepare the Payload
    • Step 3.1: Link the Payload to the Entangled Pair
    • Step 3.2: Put the Payload into a Superposition
    • Step 3.3: READ Both of Alices Qubits
    • Step 4: Receive and Transform
    • Step 5: Verify the Result
  • Interpreting the Results
  • How Is Teleportation Actually Used?
  • Fun with Famous Teleporter Accidents
• II. QPU Primitives
• 5. Quantum Arithmetic and Logic
  • Strangely Different
  • Arithmetic on a QPU
    • Hands-on: Building Increment and Decrement Operators
  • Adding Two Quantum Integers
  • Negative Integers
  • Hands-on: More Complicated Math
  • Getting Really Quantum
    • Quantum-Conditional Execution
    • Phase-Encoded Results
  • Reversibility and Scratch Qubits
  • Uncomputing
  • Mapping Boolean Logic to QPU Operations
    • Basic Quantum Logic
  • Conclusion
• 6. Amplitude Amplification
  • Hands-on: Converting Between Phase and Magnitude
  • The Amplitude Amplification Iteration
  • More Iterations?
  • Multiple Flipped Entries
  • Using Amplitude Amplification
    • AA and QFT as Sum Estimation
    • Speeding Up Conventional Algorithms with AA
  • Inside the QPU
    • The Intuition
  • Conclusion
• 7. QFT: Quantum Fourier Transform
  • Hidden Patterns
  • The QFT, DFT, and FFT
  • Frequencies in a QPU Register
  • The DFT
    • Real and Complex DFT Inputs
    • DFT Everything
  • Using the QFT
    • The QFT Is Fast
      • Signal processing with the QFT
      • Preparing superpositions with the inverse QFT
  • Inside the QPU
    • The Intuition
    • Operation by Operation
      • Rotating each circles phase by a multiple of its value
      • Conditionally rotating by the angle n/2N × 360°
  • Conclusion
• 8. Quantum Phase Estimation
  • Learning About QPU Operations
  • Eigenphases Teach Us Something Useful
  • What Phase Estimation Does
  • How to Use Phase Estimation
    • Inputs
    • Outputs
  • The Fine Print
    • Choosing the Size of the Output Register
    • Complexity
    • Conditional Operations
  • Phase Estimation in Practice
  • Inside the QPU
    • The Intuition
    • Operation by Operation
  • Conclusion
• III. QPU Applications
• 9. Real Data
  • Noninteger Data
  • QRAM
  • Vector Encodings
    • Limitations of Amplitude Encoding
      • Caveat 1: Beware of quantum outputs
      • Caveat 2: The requirement for normalized vectors
    • Amplitude Encoding and Circle Notation
  • Matrix Encodings
    • How Can a QPU Operation Represent a Matrix?
    • Quantum Simulation
      • The basic idea
      • How it works
      • Reconstruction
      • Deconstructing H
      • The cost of quantum simulation
• 10. Quantum Search
  • Phase Logic
    • Building Elementary Phase-Logic Operations
    • Building Complex Phase-Logic Statements
  • Solving Logic Puzzles
    • Of Kittens and Tigers
  • General Recipe for Solving Boolean Satisfiability Problems
    • Hands-on: A Satisfiable 3-SAT Problem
    • Hands-on: An Unsatisfiable 3-SAT Problem
  • Speeding Up Conventional Algorithms
• 11. Quantum Supersampling
  • What Can a QPU Do for Computer Graphics?
  • Conventional Supersampling
  • Hands-on: Computing Phase-Encoded Images
    • A QPU Pixel Shader
    • Using PHASE to Draw
    • Drawing Curves
  • Sampling Phase-Encoded Images
  • A More Interesting Image
  • Supersampling
  • QSS Versus Conventional Monte Carlo Sampling
    • How QSS Works
      • The QSS lookup table
      • Confidence maps
  • Adding Color
  • Conclusion
• 12. Shors Factoring Algorithm
  • Hands-on: Using Shor on a QPU
  • What Shors Algorithm Does
    • Do We Need a QPU at All?
    • The Quantum Approach
  • Step by Step: Factoring the Number 15
    • Step 1: Initialize QPU Registers
    • Step 2: Expand into Quantum Superposition
    • Step 3: Conditional Multiply-by-2
    • Step 4: Conditional Multipy-by-4
    • Step 5: Quantum Fourier Transform
    • Step 6: Read the Quantum Result
    • Step 7: Digital Logic
    • Step 8: Check the Result
  • The Fine Print
    • Computing the Modulus
    • Time Versus Space
    • Coprimes Other Than 2
• 13. Quantum Machine Learning
  • Solving Systems of Linear Equations
    • Describing and Solving Systems of Linear Equations
    • Solving Linear Equations with a QPU
      • What HHL does
        • Inputs
        • Outputs
        • Speed and fine print
      • Inside the box
        • 1. Quantum simulation, QRAM, and phase estimation
        • 2. Invert values
        • 3. Move inverted values into amplitudes
        • 4. Amplitude amplification
        • 5. Uncompute
  • Quantum Principle Component Analysis
    • Conventional Principal Component Analysis
    • PCA with a QPU
      • Representing a covariance matrix in a QPU register
      • Fixing problem 1
      • Fixing problem 2
      • The output
      • Performance
  • Quantum Support Vector Machines
    • Conventional Support Vector Machines
      • SVM generalizations
    • SVM with a QPU
      • Using a QPU to train a quantum SVM
        • Concern 1: Is F suitable for HHL?
        • Concern 2: How can we act F1 on [ 0 , y ] ?
        • Concern 3: How do we classify data?
  • Other Machine Learning Applications
• IV. Outlook
• 14. Staying on Top: A Guide to the Literature
  • From Circle Notation to Complex Vectors
  • Some Subtleties and Notes on Terminology
  • Measurement Basis
  • Gate Decompositions and Compilation
  • Gate Teleportation
  • QPU Hall of Fame
  • The Race: Quantum Versus Conventional Computers
  • A Note on Oracle-Based Algorithms
    • Deutsch-Jozsa
    • Bernstein-Vazirani
    • Simon
  • Quantum Programming Languages
  • The Promise of Quantum Simulation
  • Error Correction and NISQ Devices
  • Where Next?
    • Books
    • Lecture Notes
    • Online Resources
• Index

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