Universidade do Minho Escola de Engenharia Departamento de Informática Afonso Rodrigues Validation of quantum simulations Assessing efficiency and reliability in experimental implementations November 2018 Universidade do Minho Escola de Engenharia Departamento de Informática Afonso Rodrigues Validation of quantum simulations Assessing efficiency and reliability in experimental implementations Master dissertation Master Degree in Engineering Physics Dissertation supervised by Luís Barbosa Carlos Tavares November 2018 ACKNOWLEDGEMENTS This work was only possible thanks to the contribution of several people to which I want to express my gratitude. To my supervisor prof. Luís Barbosa, for giving me the opportunity to undertake this project, and for his constant support, availability and patience; also to my co-supervisor Carlos Tavares, for his early insight. In no particular order of importance, to Miguel Nogueira, Ricardo Alves, Rui Costa, João Fonseca, Miguel Sanches, Joaquim Santos, and José Bruno, my friends and colleagues who accompanied me in this course and provided invaluable help. To my parents and sister, and Maria Coelho, André Souto and Gil Pimentel for their unconditional encouragement on the difficult days. Finally, I wish to thank University of Minho for these five years of learning, INESC TEC and specifically the HASLab team for supporting me in my first steps as a researcher. I acknowledge use of the IBM Q for this work. i ABSTRACT Quantum simulation is one of the most relevant applications of quantum computation for the near future, due to its scientific impact and also because quantum simulation algorithms are typically less demanding than generalized quantum computations. Ultimately, the success of a quantum simulation depends on the amount and reliability of information one is able to extract from the results. In such a context, this work reviews the theory behind quantum simulation, with a focus on digital quantum simulation. The concepts of efficiency and reliability in quantum simulations are discussed, particularly for implementations of digital simulation algorithms in state-of-the-art quantum computers. A review of approaches for quantum characterization, verification and validation techniques (QCVV) is also presented. A digital quantum simulation of the Schrödinger equation for a single particle in 1 spatial dimension was experimentally implemented and analyzed, along with a quantum state tomography procedure for characterization of the final quantum state and evaluation of simulation reliability. From the literature, it is shown that digital quantum simulation is theoretically sound and experimentally feasible, with several applications in a wide range of physics-related fields. Nonetheless, a number of conditions arise that must be observed for a truly efficient im- plementation of a digital quantum simulation, from theoretical conception to experimental circuit design. The review of QCVV techniques highlights the need for characterization and validation techniques that could be efficiently implemented for current models of quantum computation, particularly in instances where classical verification is not tractable. However, there are proposals for efficient verification procedures when a set of parameters defining the final result of the simulation is known. The experimental simulation demonstrated partial success in comparison with an ideal quantum simulation. From the results it is apparent that better coherence times, better reliability and finer control are as decisive for the advancement of quantum computing power as the more-publicized number of qubits of a given device. ii RESUMO A simulação quântica é uma das aplicações mais relevantes da computação quântica num fu- turo próximo, não só devido ao seu impacto científico como também porque os algoritmos de simulação quântica são tipicamente menos exigentes do que algoritmos quânticos numéricos. Em última análise, o sucesso de uma simulação quântica depende da quantidade e fiabilidade das informações que é possível extrair dos resultados. Neste contexto, este trabalho apresenta uma revisão da teoria da simulação quântica, com ênfase na simulação quântica digital. Os conceitos de eficiência e fiabilidade em simulações quânticas são discutidos, particularmente para implementações de algoritmos de simulação digital. Uma revisão de técnicas de caracter- ização, verificação e validação de sistemas quânticos (QCVV) é também apresentada. Uma simulação quântica digital da equação de Schrödinger para uma única partícula a uma dimen- são espacial foi implementada experimentalmente e analisada, juntamente com um método de tomografia de estado quântico para a caracterização do estado quântico final e avaliação da fiabilidade da simulação. A partir da literatura, é demonstrado que a simulação quântica digital é teoricamente sólida e experimentalmente viávei, com várias aplicações em diversas áreas da física. No entanto, existem várias condições a ter em conta para uma implementação verdadeiramente eficiente de uma simulação quântica digital, da sua concepção teórica até à implementação experimental de circuitos. A revisão de técnicas QCVV destaca a necessidade de técnicas de caracterização e validação que possam ser eficientemente implementadas para modelos atuais de computação quântica, particularmente em instâncias em que a verificação clássica não é possível ou desejável. No entanto, existem propostas para técnicas de verificação que são eficientes quando se conhece, a priori, um conjunto de parâmetros característicos do resultado final da simulação. A simulação experimental demonstrou sucesso parcial relativamente a uma simulação quân- tica ideal. A partir dos resultados, evidencia-se que melhores tempos de coerência, maior fiabilidade e controlo mais refinado são tão decisivos para o avanço da computação quântica quanto o número de qubits de um dispositivo. iii CONTENTS 1 introduction 1 1.1 The context: quantum simulation1 1.2 Schrödinger equation3 1.3 Objectives5 1.4 Outline6 2 quantum simulation 8 2.1 Digital quantum simulation 11 2.2 Towards an efficient implementation of DQS 13 2.3 Analog quantum simulation 16 2.4 Physical realizations 17 2.5 Applications 20 2.6 Summary 23 3 quantum characterization, verification and validation 24 3.1 Quantum state tomography 25 3.2 Quantum process tomography 30 3.3 Randomized benchmarking 32 3.4 Other verification and validation methods 34 3.5 Summary 37 4 experimental procedure 38 4.1 Quantum devices 38 4.2 Simulation algorithm 41 4.2.1 2-qubit implementation 45 4.2.2 3-qubit implementation 47 4.3 Quantum state tomography 49 4.4 Procedure 51 5 results and discussion 54 5.1 2-qubit simulation 55 5.2 3-qubit simulation 60 5.3 Discussion 61 6 conclusions 65 6.1 Future work 67 iv Contents v a quantum computing 80 a.1 Hilbert space and the bra-ket notation 80 a.2 Quantum computing programming model 81 a.3 Quantum gates 83 b qiskit implementation 85 b.1 2-qubit algorithms 85 b.2 3-qubit algorithms 101 c qiskit results 122 c.1 Device parameters 122 c.2 Output quantum circuits 136 c.2.1 2-qubit implementation 136 c.2.2 3-qubit implementation 148 1 INTRODUCTION 1.1 the context: quantum simulation The possibility of performing computational tasks deemed inefficient, or even impossible, on available classical computing power has increased the momentum on quantum computation and simulation research over the past decade. However, there are still major milestones to be reached before the first fault-tolerant, universal quantum computer is built. As of 2018, available quantum devices work by approximating quantum computations on physical qubits; quantum information and computation research is entering the NISQ (Noisy intermediate- scale quantum) era. Before a noise-resilient logical qubit - one that performs as theoretically predicted, holding its state in arbitrarily long quantum algorithms - is reached, error rates and coherence times need to be further improved, and error correcting codes allowing for implementation of a universal set of gates while keeping low overhead, need to be devised (Campbell et al., 2017). Quantum simulation is currently one of the most relevant applications of quantum computa- tion. This is true not only due to its scientific and industrial impact, but also because quantum simulation algorithms are typically less demanding than general quantum computations. For example, a quantum simulator with tens of qubits could already perform useful simulations under current technology, whereas thousands of qubits would be needed to factorize modest numbers using Shor’s algorithm (Buluta and Nori, 2009). In fact, quantum simulators could even explore the presence of environmental errors and decoherence to simulate the presence of same phenomena on the simulated system (Lloyd, 1996). It is also believed that no known classical algorithm can, without compromises, efficiently simulate the dynamics of a quantum system (Preskill, 2018). The biggest demonstrated classical numerical simulation of a quantum system was per- formed by a team of researchers from IBM on a conventional supercomputer, in October 2017. The team managed to effectively simulate a 56-qubit quantum system, which implicates that a scenario of quantum supremacy Boixo et al.(2018) would be achieved on a quantum computer with a greater amount of qubits and reasonable fidelity. Of course, quantum supremacy is dependent on many factors other than qubit number (e.g. universality, fidelity, entanglement 1 1.1. The context: quantum simulation 2 capabilities, decoherence), and, as classical computational