UNIVERSITY OF CALGARY Algorithmic Quantum Channel Simulation by Dongsheng Wang A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF PHYSICS AND ASTRONOMY CALGARY, ALBERTA SEPTEMBER, 2015 c Dongsheng Wang 2015 Abstract Quantum simulation, which is generically the task to employ quantum computers to simulate quantum physical models, has been one of the most significant motivations and applications of quantum computing. Quantum dynamics, unitary or nonunitary Markovian dynamics driven by local interactions, has been proved to be efficiently simulatable on quantum com- puters. Extending the underlying models in quantum computation and quantum simulation from unitary to general nonunitary evolution, and from continuous-time to discrete-time evo- lution is essential not only for quantum simulation of more general processes, e.g., dissipative processes with evident non-Markovian effects, but also for developing alternative quantum computing models and algorithms. In this thesis, we explore quantum simulation problems mainly from the following three themes. First, we extend quantum simulation framework of Hamiltonian-driven evolution to quan- tum simulation of quantum channels, combined with the scheme of algorithmic simulation that accepts a promised simulation accuracy, hence algorithmic quantum channel simulation. Our simulation scheme contains a classical preprocessing part, i.e. a classical algorithm for quantum-circuit design, and a quantum part that is the execution of a quantum circuit for the quantum channel simulator. The classical algorithm accepts the description of an arbi- trary quantum channel and a simulation error tolerance as input, and delivers the description of the implementation of a quantum circuit and an actual error as output. The quantum simulator circuit then generates final state within the actual error, which should be smaller than the tolerance in the worst case, which guarantees the accurate simulation of further observable effects on the simulator. Our quantum channel simulation framework is distinct from non-algorithmic simulation scheme, and also some dedicated simulation for particular instead of arbitrary quantum processes. Second, we employ nonstandard yet beneficial quantum simulation algorithms for arbi- i trary quantum channels beyond the dilation method. We explore the method of quantum channel decomposition in terms of convex combination of smaller channels, known as gen- eralized extreme channels, for which the channel decomposition is known as a nontrivial open problem. To attack the channel decomposition problem, we develop an optimization algorithm for approximate decomposition into a convex sum of generalized extreme channels from our construction. We provide an ansatz for generalized extreme channels that proves to be able to yield arbitrary generalized extreme channels and allows a precise quantum circuit description. Furthermore, our numerical simulation has demonstrated the validity of our optimization algorithm for low-dimensional quantum channels. Third, we also attempt to provide general definitions of quantum simulation problems by exploring the freedom of the notion of simulation. By considering quantum simulation problems beyond the rough distinction between digital and analog simulations, and beyond the quantum-state generation problem, we can define various quantum simulation problems, namely, uniform, strong, and weak quantum simulations from the point view of operator topology. Our quantum channel simulation developed above is a strong simulation, which simulates the effects of a channel on arbitrary input states. For the other two possible simulation methods, we define a general weak quantum simulation problem, which actually simulates observable effects instead of the effects on state generation. Also we study the channel simulation problem in the quantum query model, and provide the query complexity by employing uniform quantum simulation method. Acknowledgements I want to express my great acknowledgement to my PhD supervisor Dr. Barry C. Sanders, who initially provided me the opportunity to Canada, and has gradually changed my view towards mathematics, computer science, and research on quantum foundation. Particularly, I want to thank him for the training of scientific work writing, and his patience of endless writings and discussions. Also I want to thank him for providing financial support and many chances for conference, collaboration and summer school. Last but not least, I also want to thank him for choosing such a significant research topics. I want to thank Dr. Peter Høyer and Dr. David Feder for serving as my PhD committee member, and for their occasional discussions and help, which have certainly triggered many ideas, and some of them have contributed to this thesis. I also would like to thank Dr. Gilad Gour for accepting me to join his group meeting. I want to thank Dr. Dominic Berry from Macquarie university and Dr. Marcos C. de Oliveira from Universidade Estadual de Campinas for their collaborations. Particularly, I want to thank Dominic for his mathematical insights, and Marcos for his inspirations and kindness especially during my first year in Canada. I want to thank the group of Prof. Jian-Wei Pan in Shanghai, China for the collabo- ration on the experimental realization of our qubit channel simulator. I want to thank my collaborators there, particularly, Dr. Yu-Ao Chen, He Lu, and Chang Liu. I want to thank Ms. Lucia Wang and Ms. Nancy Lu for their selfless help, especially when I have trouble in my personal life in Calgary or the department. I want to thank the group members, former or current, and other candidates sitting in the same office with me, and also other classmates and friends, particularly, I want to mention Yunjiang Wang, Jianming Wen, Ran Hee Choi, Mohammad Khazali, Ehsan Zahedinejad, Khulud Almutairi, and Farokh Mivehvar for bringing joy in my research career and personal life. Particularly, I iii want to thank Varun Narasimhachar and Pantita Palittapongarnpim for reading my thesis and providing comments. Finally, I want to thank my family for their understandings of my decision to go abroad, to study physics, and to continue my career. Table of Contents Abstract ........................................i Acknowledgements .................................. iii Table of Contents . .v List of Tables . viii List of Figures . ix List of Symbols . .x 1 Introduction . .1 1.1 Quantum simulation of quantum physics . .1 1.1.1 Solving quantum physical problems . .1 1.1.2 Quantum computation vs. quantum simulation . .5 1.1.3 Digital vs. analog quantum simulations . .7 1.1.4 Algorithmic vs. non-algorithmic quantum simulations . .8 1.1.5 Classification of quantum simulations . 10 1.2 Algorithmic quantum simulation of quantum channels . 13 1.2.1 Quantum processes characterized by quantum channels . 13 1.2.2 From unitary to nonunitary quantum simulation . 17 1.2.3 Challenges and our basic methods . 18 1.2.4 Applications of quantum channel simulators . 21 1.3 Summary of results and significance . 22 1.4 Structure . 24 2 Background . 26 2.1 Quantum states and quantum channels . 26 2.1.1 Quantum states and operator basis . 26 2.1.2 Representations of quantum channels . 30 2.2 Distance measure . 35 2.2.1 Norms on operator and superoperator . 35 2.2.2 Inequalities . 37 2.3 Convex set . 40 2.3.1 Affine space and convex set . 41 2.3.2 Examples of convex sets . 42 2.3.3 The set of quantum channels . 45 2.4 Conclusion . 48 3 Single-qubit unitary quantum gate compiling . 49 3.1 Universality . 49 3.2 Solovay-Kitaev theorem . 52 3.2.1 Solovay-Kitaev algorithm for qubit gates . 53 3.3 Lookup table . 55 3.4 Discussion and conclusion . 57 4 Quantum channel decomposition and simulation . 59 4.1 Extreme quantum channels . 59 4.1.1 Mathematical characterization . 60 4.1.2 Kraus operator-sum representation of extreme channels . 64 v 4.1.3 Choi state representation of extreme channels . 69 4.1.4 Quantum circuit of extreme qudit channels . 73 4.2 Quantum channel decomposition . 77 4.2.1 Ruskai's conjecture . 78 4.2.2 Optimization for quantum channel simulation . 79 4.2.3 Space and time cost of quantum simulation circuit . 82 4.3 Alternative strategies . 83 4.3.1 Dilation approach for simulation . 84 4.3.2 Factorization decomposition . 85 4.3.3 Simulate each Kraus operator probabilistically . 86 4.4 Conclusion . 87 5 Qubit quantum channel simulation . 88 5.1 Qubit channel representations . 88 5.2 Qubit channel decomposition . 90 5.2.1 Theory of Ruskai-Szarek-Werner . 90 5.2.2 Geometry for qubit channel decomposition . 92 5.2.3 Our approach for qubit channel simulation . 94 5.3 Qubit channel simulation algorithm . 95 5.3.1 Representations of generalized extreme qubit channels . 95 5.3.2 Optimization algorithm for qubit channel simulation . 98 5.3.3 Quantum circuit cost . 99 5.4 Discussion and conclusion . 101 6 Photonic qubit-channel simulator . 103 6.1 Setup . 103 6.2 Simulation of arbitrary qubit channel . 105 6.3 Protecting superposition via weak measurement . 107 6.4 Simulation of transpose and positive mappings . 110 6.5 Conclusion . 112 7 Qutrit and two-qubit quantum channel simulations . 113 7.1 Qutrit quantum channel simulation . 113 7.1.1 Extreme qutrit channels . 113 7.1.2 Classification of extreme and quasi-extreme qutrit channels . 116 7.1.3 Simulation results . 118 7.2 Two-qubit quantum channel simulation . 121 7.2.1 Extreme two-qubit channels . 121 7.2.2 Classification of extreme and quasi-extreme two-qubit channels . 122 7.2.3 Simulation results . 123 7.3 Conclusion . 124 8 Concepts of quantum simulation . 125 8.1 Quantum simulation frameworks . 125 8.2 Weak quantum simulation . 130 8.3 Query model for quantum channel simulation . 134 8.4 Conclusion . 139 9 Conclusions . 140 9.1 Summary . 140 vi 9.1.1 Qubit-channel simulation . 140 9.1.2 Photonic qubit-channel simulator . 141 9.1.3 Qudit-channel simulation . 141 9.1.4 Alternative quantum simulation problems .