Quantum Algorithm Implementations for Beginners

As quantum computers become available to the general public, the need has arisen to train a cohort of quantum programmers, many of whom have been developing classical computer programs for most of their careers. While currently available quantum computers have less than 100 qubits, quantum computing hardware is widely expected to grow in terms of qubit count, quality, and connectivity. This review aims at explaining the principles of quantum programming, which are quite different from classical programming, with straightforward algebra that makes understanding of the underlying fascinating quantum mechanical principles optional. We give an introduction to quantum computing algorithms and their implementation on real quantum hardware. We survey 20 different quantum algorithms, attempting to describe each in a succinct and self-contained fashion. We show how these algorithms can be implemented on IBM’s quantum computer, and in each case, we discuss the results of the implementation with respect to differences between the simulator and the actual hardware runs. This article introduces computer scientists, physicists, and engineers to quantum algorithms and provides a blueprint for their implementations. Abstract Contents 1 Introduction 1.1 The quantum computing programming model 1.2 Implementations on a real quantum computer 1.3 Classes of quantum algorithms 2 Grover's Algorithm 2.1 Problem definition and background 2.2 Algorithm description 2.3 Algorithm implemented on IBM's 5-qubit computer 3 Bernstein-Vazirani Algorithm 3.1 Problem definition and background 3.2 Algorithm description 3.3 Algorithm implemented on IBM's 5-qubit and 16-qubit computers 4 Linear Systems 4.1 Problem definition and background 4.2 Algorithm description 4.3 Phase estimation 4.4 Algorithm implemented on IBM's 5 qubit computer 5 Shor's Algorithm for Integer Factorization 5.1 Problem definition and background 5.2 Algorithm description 5.3 Algorithm implemented on IBM's 5-qubit computer 6 Matrix Elements of Group Representations 6.1 Problem definition and background 6.2 Algorithm description 6.3 Algorithm implemented on IBM's 5-qubit computer 7 Quantum Verification of Matrix Products 7.1 Problem definition and background 7.2 Algorithm description 8 Group Isomorphism 8.1 Problem definition and background 8.2 Algorithm description 8.3 Algorithm implemented using Qiskit 9 Quantum Random Walks 9.1 Problem definition and background 9.2 Example of a quantum random walk 9.3 Algorithm implementation using Qiskit on IBM Q 10 Quantum Minimal Spanning Tree 10.1 Problem definition and background 10.2 Algorithm description 11 Quantum Maximum Flow Analysis 11.1 Problem definition and background 11.2 Algorithm description 12 Quantum Approximate Optimization Algorithm 12.1 Problem definition and background 12.2 Algorithm description 12.3 QAOA MaxCut on ibmqx2 12.4 A proof-of-concept experiment 13 Quantum Principal Component Analysis 13.1 Problem definition and background 13.2 Algorithm description 13.3 Algorithm implemented on IBM's 5-qubit computer 14 Quantum Support Vector Machine 15 Quantum Simulation of the Schrödinger Equation 15.1 Problem definition and background 15.2 Algorithm description 15.3 Algorithm implemented on IBM's 5-qubit computer 16 Ground State of the Transverse Ising Model 16.1 Variational quantum eigenvalue solver 16.2 Simulation and results 17 Quantum Partition Function 17.1 Background on the partition function 17.2 A simple example 17.3 Calculating the quantum partition function 17.4 Implementation of a quantum algorithm on the IBM Quantum Experience 18 Quantum State Preparation 18.1 Single qubit state preparation 18.2 Schmidt decomposition 18.3 Two-qubit state preparation 18.4 Two-qubit gate preparation 18.5 Four qubit state preparation 19 Quantum Tomography 19.1 Problem definition and background 19.2 Short survey of existing methods 19.3 Implementation of the Maximum Likelihood method on 5-qubit IBM QX 20 Tests of Quantum Error Correction in IBM Q 20.1 Problem definition and background 20.2 Test 1: errors in single qubit control 20.3 Test 2: errors in entangled 3 qubits control 20.4 Discussion Acknowledgments References