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Quantum Algorithms for Portfolio Management FACULDADE DE ENGENHARIA DA UNIVERSIDADE DO PORTO Quantum Algorithms for Portfolio Management Duarte Frazão Mestrado Integrado em Engenharia Informática e Computação Supervisor: Koen Bertels Second Supervisor: José Matos July 19, 2021 Quantum Algorithms for Portfolio Management Duarte Frazão Mestrado Integrado em Engenharia Informática e Computação Approved in oral examination by the committee: Chair: Prof. João Paulo Fernandes External Examiner: Prof. Imran Ashraf Supervisor: Prof. Koen Bertels July 19, 2021 Abstract Quantum Computing is the process of leveraging quantum phenomena such as superposition and entanglement to solve problems that are intractable for classical computers. These properties al- low quantum computers to process an exponential amount of states in parallel in relation to the number of quantum bits, which is the main limitation of classical computing. With recent suc- cessful innovation on building more powerful quantum computers, the research community has started to explore different areas that will be disrupted when operational error-corrected quantum computers become a reality. One recent area of research is Quantum Finance, intended to be the bridge between Quantum Computing and Finance. The focus lies on understanding which problems in this area will be impacted by quantum computers. These can either be problems that can be solved more efficiently or problems that have never been solved due to classical computer limitations. Finance is one of the largest industries in the world, impacting every citizen in many ways, from our savings and investments to the companies that we rely upon to live our lives. Port- folio Management is the sub-field of Finance that tries to find the optimal investment portfolio for each individual investor. Several contributions have been made to understand the future of Portfolio Management with quantum computing, mainly using quantum annealing, a restricted family of quantum computers with the objective of finding global minima in combinatorial opti- mization problems. However, gate-based quantum computers will allow a much wider range of possibilities, which are still under-explored. More research on Quantum Finance using gate-based quantum computers is warranted to understand what will be possible in the future. In this work, we solve the Portfolio Management problem of finding the optimal portfolio that minimizes risk and maximizes returns for a given risk profile on a gate-based quantum computer simulator. The quantum algorithms chosen were QAOA and its new version QAOA+. QAOA is the basic ver- sion, it is already available and contains a large body of literature covering it. QAOA+ is a recent proposed version, it still has few pieces of literature associated. To understand how and when should we use QAOA and QAOA+ for Portfolio Management, an in-depth comparison between the two algorithms for this topic was realized. We implemented our solution using IBM’s quantum platform, Qiskit, using a perfect quantum simulator to test the algorithms. We analyzed a possible scenario of choosing a portfolio from 15 assets to understand the quality of the results, where we found that the implemented QAOA+ yielded better solutions than QAOA. Finally, several metrics were gathered to compare both algorithms, such as execution time, circuit depth, number of gates, number of qubits, and number of classical loop iterations. Even though Qiskit did not provide all the circuit resources needed to reach the most efficient version, we concluded that our implemen- tation still achieved a better performance than QAOA in most metrics evaluated. Possibilities of further work were also identified throughout the dissertation for future research. Keywords: Quantum finance, Portfolio Management, Quantum Computing i ii Resumo Computação Quântica é o processo de aproveitar fenómenos quânticos como sobreposição de estados e entrelaçamento quântico para resolver problemas que são inviáveis para computadores clássicos. Estas propriedades permitirão processar uma quantidade exponencial de estados em relação ao número de bits quânticos, a maior limitação de computação clássica. Com a recente inovação, bem sucedida, na construção de computadores quânticos mais potentes, a comunidade científica começou a explorar diferentes áreas que vão sofrer uma mudança de paradigma quando computadores quânticos operacionais e com correção de erros se tornarem realidade. Uma área recente de investigação é a de Finanças Quânticas, com o objetivo de fazer a ligação entre Com- putação Quântica e Finanças. O foco baseia-se em compreender que problemas nesta área serão impactados por computadores quânticos. Estes estão divididos em problemas que podem ser resol- vidos mais eficientemente ou que nunca foram resolvidos devido às limitações de computadores clássicos. A indústria financeira é uma das maiores do mundo, afetando os cidadãos de várias for- mas, desde as suas poupanças aos seus investimentos. A Gestão de Carteiras é uma sub-categoria de Finanças que tenta encontrar o portefólio ótimo para cada investidor. Várias contribuições já foram feitas para compreender o futuro de Gestão de Carteiras com computação quântica, prin- cipalmente usando Recozimento Quântico (Quantum Annealing). Por outro lado, computadores quânticos gate-based vão permitir mais possibilidades, que ainda estão inexploradas. Mais in- vestigação em Finanças Quânticas usando computadores quânticos gate-based é necessária para entender o que será possível no futuro. Neste trabalho, resolvemos o problema de Gestão de Carteiras de encontrar o portefólio ótimo que minimize o risco e maximize os retornos para um determinado perfil de risco usando um simulador de computadores quânticos gate-based. Os al- goritmos quânticos escolhidos foram o QAOA e a sua nova versão QAOA+. QAOA é a versão base, já disponível e com muita literatura associada. QAOA+ é uma versão proposta recentemente ainda com pouca investigação associada. Para perceber melhor como e quando devemos usar o QAOA ou o QAOA+ para Gestão de Carteiras, uma comparação detalhada foi feita entre os dois algoritmos para o tópico em questão. Implementámos a nossa solução usando a plataforma para computação quântica da IBM, Qiskit, usando um simulador de computador quântico perfeito para testar os algoritmos. Analisamos o cenário de escolher um portefólio de um grupo de 15 ati- vos para perceber a qualidade dos resultados, onde descobrimos que a nossa implementação do QAOA+ mostrou melhores soluções que o QAOA. Finalmente, várias métricas foram recolhidas para comparar ambos os algoritmos, como o tempo de execução, profundidade do circuito, número de gates e número de ciclos clássicos. Apesar do Qiskit não conter todos os recursos necessários para construir a versão mais eficiente possível, concluímos que a nossa implementação ainda con- seguiu atingir uma melhor performance que o QAOA em grande parte das métricas avaliadas. Possibilidades de investigação futura também foram identificadas ao longo da dissertação. Keywords: Quantum Finance, Portfolio Management, Quantum Computing iii iv Acknowledgements I want to thank everyone that in one way or another helped me finish my Master’s Degree and this dissertation. To Professor Bertels, who guided me into the field of Quantum Computing, giving me the tools and means to complete this work, all while pushing the Quantum community forward. To Professor Matos, who helped me bring together the two fields of Quantum and Finance, while helping me improve my knowledge and work. To Aritra Sarkar, whose expertise and advice helped me work through any roadblocks. To the rest of the QBeeX team, who provided me with a collaboratively work environment to learn and explore this new field. I also want to thank all my friends that were part of this journey and contributed to make it such a great experience. To Basia, for getting me out of the house and being the best support I could ever ask for. Finally, none of this would have been possible without my family, who guided me throughout all my life. Duarte Frazão v vi “The first principle is that you must not fool yourself, and you are the easiest person to fool.” Richard P. Feynman vii viii Contents 1 Introduction1 1.1 Context . .1 1.2 Motivation . .2 1.3 Objectives . .2 1.4 Document Structure . .3 2 Background5 2.1 Quantum computing . .5 2.1.1 Quantum bits . .5 2.1.2 Gates . .7 2.1.3 State of multiple qubits . .8 2.1.4 Entanglement . .9 2.1.5 Scaling on quantum computing . .9 2.1.6 How to make a quantum algorithm? . 10 2.1.7 State of the hardware . 10 2.1.8 Approaches to quantum research . 11 2.1.9 How to evaluate a quantum algorithm . 11 2.2 Finance . 12 2.2.1 Financial market . 12 2.2.2 Modern Portfolio Theory . 12 2.2.3 QUBO . 13 3 Literature Review 15 3.1 Different approaches to Finance using quantum computing . 15 3.2 Surveys on Quantum Finance and Combinatorial Optimization . 16 3.3 Quantum Finance, using quantum annealing . 19 3.4 Quantum Approximate Optimization Algorithm (QAOA) and Quantum Alternat- ing Operator Ansatz (QAOA+) . 22 3.5 Quantum Finance, using universal quantum computers . 25 3.6 Literature Analysis . 27 4 Problem Statement 29 4.1 Problem . 29 4.2 Main hypotheses . 30 4.3 Contribution . 30 4.4 Scope . 31 4.4.1 Perfect qubits . 31 4.4.2 Finance assumptions . 31 ix x CONTENTS 4.4.3 Qiskit . 31 5 Implementation 33 5.1 Data . 33 5.2 Portfolio Management base problem . 35 5.3 Quantum algorithms . 36 5.3.1 QAOA . 36 5.3.2 QAOA+ . 37 5.4 Constraints . 41 5.4.1 Constraints as penalty terms . 41 5.4.2 Constraints as mixer operator . 43 5.4.3 Penalty terms versus mixer encoding . 44 5.5 Overview of the solution . 45 5.6 Initial analysis . 45 5.6.1 Conservative profile . 46 5.6.2 Risky profile . 47 5.6.3 Results . 48 6 Results 51 6.1 Experimental results . 51 6.1.1 Test environment . 51 6.1.2 Tests settings . 51 6.1.3 Gates .
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