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Advanced Quantum Algorithms Advanced Quantum Algorithms Giulia Ferrini, Anton Frisk Kockum, Laura García-Álvarez, Pontus Vikstål November 5, 2019 Contents 1 The circuit model for quantum computation4 1.1 Components of the circuit model..................................4 1.2 Quantum bits.............................................5 1.3 Single-qubit gates...........................................6 1.4 Multi-qubit gates...........................................7 1.5 Universal quantum computation..................................8 2 Comparing quantum and classical computers 10 2.1 The Solovay–Kitaev theorem.................................... 10 2.2 Complexity classes.......................................... 11 2.2.1 Complexity classes for a deterministic Turing machine.................. 11 2.2.2 Complexity classes for a probabilistic Turing machine.................. 12 2.2.3 Complexity classes for a quantum Turing machine.................... 12 2.2.4 Summary of the complexity classes............................. 13 3 Measurement Based Quantum Computation 15 3.1 Quantum computing with discrete variables (Ref. [Horodecki et al., 2006])........... 15 3.1.1 Definition of the possible operations............................ 15 3.1.2 The general paradigm in DV................................ 16 4 Adiabatic quantum computation 19 4.1 The adiabatic theorem........................................ 20 4.1.1 Argument for noncommuting Hamiltonians in AQC................... 22 5 Fast Quantum Algorithms 23 5.1 The Quantum Fourier Transform.................................. 23 5.1.1 Another definition...................................... 23 5.1.2 An efficient implementation................................. 25 5.2 Phase estimation........................................... 25 5.3 Factoring - Shor’s algorithm..................................... 27 5.3.1 Modular arithmetics..................................... 27 5.3.2 Order finding......................................... 28 5.3.3 Factoring as order finding.................................. 29 5.3.4 A quantum algorithm for order finding........................... 29 5.3.5 Performance......................................... 30 1 6 Algorithms for solving combinatorial optimization problems 31 6.1 Combinatorial optimization problems................................ 31 6.1.1 The promises of quantum computers for solving combinatorial optimization problems 32 6.2 Quantum annealing.......................................... 33 6.2.1 Heuristic understanding of quantum annealing...................... 33 6.2.2 Ising model.......................................... 34 6.2.3 Mapping combinatorial optimization problems to spin Hamiltonians.......... 35 6.2.4 Example of the solution of a practical problem on a quantum annealer: Flight-gate assignment.......................................... 37 6.3 Quantum Approximate Optimization Algorithm (QAOA).................... 42 6.3.1 Relation between QAOA and quantum annealing..................... 59 7 The variational quantum eigensolver 60 8 Sub-Universal models of quantum computation 61 8.1 Introduction: motivation for sampling models........................... 61 8.2 Boson Sampling............................................ 62 8.2.1 Definition of the Boson Sampling model.......................... 62 8.2.2 Proof that the Boson Sampling probability distribution is proportional to permanents 64 8.2.3 Sketch of the proof of computational hardness of the Boson Sampling probability dis- tribution........................................... 67 8.3 Instantaneous Quantum Polytime.................................. 68 8.3.1 Hadamard gadget...................................... 69 8.4 Random Circuit Sampling...................................... 70 9 Continuous-Variable Approach to Quantum Information 72 9.1 Quantum computing with continuous variables.......................... 72 9.1.1 Definition of the possible operations............................ 72 9.2 Measurement-based quantum computation: the general paradigm in CV............ 74 9.3 GKP Error Correction........................................ 76 9.3.1 GKP encoding and fault-tolerance............................. 76 9.3.2 Single noise realization: intermediate measurement and threshold condition...... 78 9.3.3 Single noise realization: Output state of the GKP error-correcting gadget....... 79 9.4 Sub-Universal models........................................ 81 9.4.1 Continuous-Variable Instantaneous Quantum Polytime.................. 81 9.5 Quantum annealing.......................................... 84 9.5.1 Circuit QED......................................... 84 9.5.2 Two-photon pumped Kerr-nonlinear resonator....................... 85 9.5.3 Two- & one-photon pumped Kerr-nonlinear resonator.................. 87 9.5.4 Coupled two-photon pumped Kerr-nonlinear resonators................. 89 9.5.5 Simulation of relevant combinatorial optimization problems............... 91 9.5.6 Remarks on scalability & the model............................ 93 10 Quantum machine learning 95 2 A Quantization of the electromagnetic field in a cavity 96 A.1 Quantizing the electromagnetic field................................ 96 A.1.1 Quadrature operators.................................... 99 A.2 Coherent states............................................ 100 A.3 Phase space representation..................................... 103 A.3.1 Wigner function....................................... 103 B GKP Error-correction gadget with finite resolution 104 B.0.1 Single noise realization: intermediate measurement and threshold condition with finite resolution detectors..................................... 104 B.0.2 Single noise realization: Output state of the GKP error-correcting gadget....... 105 C Superconducting quantum circuits 108 C.1 Circuit Lagrangian.......................................... 108 C.2 Transformation to the rotating frame................................ 111 C.3 Steady state & stability....................................... 113 C.4 Effect of single-photon pump.................................... 115 C.5 Coupling between two Kerr-nonlinear resonators......................... 115 C.6 Error estimation........................................... 116 C.7 Generation of cat states using a two-photon pumped KNR.................... 117 3 Chapter 1 The circuit model for quantum computation In this course, we will give an overview of various approaches to quantum computation, reflecting many of the latest developments in the field. We will cover several different models of quantum computation, from the foundational circuit model through measurement-based and adiabatic quantum computation to boson sampling. We will discuss quantum computation with both discrete and continuous variables. When it comes to the algorithms that we study, they include both classics like Shor’s algorithm and newer, heuristic approaches like the quantum approximate optimization algorithm (QAOA). We will also see how quantum computing can be combined with machine learning. We assume that the students taking this course already have some familiarity with quantum physics (superposition, entanglement, etc.) and some basic concepts in quantum computation. We will repeat some of these basic concepts at the beginning of the course, but perhaps give a more thorough justification for why they can be used in quantum computation. In this first chapter, we will study the circuit model of quantum computation. This introduces quantum bits, quantum gates, and other components in close similarity with concepts in classical computing and gives us the tools to begin investigating whether quantum computers can ever outperform classical computers. For this chapter, we have borrowed parts from Refs. [Nielsen and Chuang, 2000, Aaronson, 2018, Kockum and Nori, 2019]. 1.1 Components of the circuit model Loosely speaking, a computation requires a system that can represent data, a way to perform manipulation of that data, and a method for reading out the result of the computation. In the circuit model of quantum computation, we use: Quantum bits (qubits) to represent the data. • State preparation to initialize the qubits in the input state we need to begin the computation. • Quantum gates on the qubits to manipulate the data. • Measurements on the qubits to read out the final result. • Below, we first say a few words about what qubits are. We then discuss various quantum gates, and what is required of such gates to allow us to perform any quantum computation we would like. We assume 4 Figure 1.1: The Bloch-sphere representation of a qubit state. The north pole is the ground state 0 and the | i south pole is the excited state 1 . To convert an arbitrary superposition of 0 and 1 to a point on the | i | i | i sphere, the parametrization ψ = cos θ 0 + eiϕ sin θ 1 is used. | i 2 | i 2 | i for now that it is possible to initialize our quantum computer in some simple state, and that we can read out the state of the qubits at the end of a computation. 1.2 Quantum bits In a classical computer, the most basic unit of information is a bit, which can take two values: 0 and 1. In a quantum computer, the laws of quantum physics allow phenomena like superposition and entanglement. When discussing information processing in a quantum world, the most basic unit is therefore a quantum bit, usually called qubit, a two-level quantum system with a ground state 0 and an excited state 1 . Unlike a | i | i classical bit, which only has two possible states, a quantum bit has infinitely many states: all superpositions of 0 and 1 , | i | i α ψ = α 0 + β 1
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