Computational Problems Related to Open Quantum Systems by Chunhao Wang A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Computer Science Waterloo, Ontario, Canada, 2018 © Chunhao Wang 2018 Examining Committee Membership The following served on the Examining Committee for this thesis. The decision of the Examining Committee is by majority vote. External Examiner: Hoi Fung Chau Professor, Department of Physics, The University of Hong Kong Supervisor(s): Richard Cleve Professor, Cheriton School of Computer Science, University of Waterloo Internal Member: Debbie Leung Professor, Department of Combinatorics and Optimization, University of Waterloo Internal Member: John Watrous Professor, Cheriton School of Computer Science, University of Waterloo Internal-External Member: Ashwin Nayak Professor, Department of Combinatorics and Optimization, University of Waterloo ii I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. iii Abstract A quantum system that has interaction with external resources, such as probability dis- tribution, dissipation, and noise, is referred to as an open quantum system. Not only do open quantum systems play a vital role in the field of quantum physics, but they are also fundamental objects in quantum information and quantum computing. In this thesis, we focus on computational problems related to open quantum systems. In particular, we study efficient constructions of open quantum systems and their algorithmic applications. A unitary 2-design is a quantum analogue of universal 2-hash functions. It is an exam- ple of open quantum systems in the sense that it is a probability distribution of unitaries. As unitary 2-designs inherit many properties of the Haar randomness on the unitary group, they have many applications in quantum information, such as benchmarking and decou- pling. We study the structures of unitary 2-designs and present efficient methods for their constructions. The continuous-time evolution of a closed quantum system can be described by the Schr¨odingerequation. A natural generalization of the Schr¨odingerequation to Markovian open quantum systems, in the sense of generating dynamical semigroups, is called the Lindblad equation. We show that it is impossible for a simple reductionist approach to simulate Lindblad evolution with gate complexity that has linear dependence in evolution time. Moreover, we use a novel variation of the \linear combination of unitaries" con- struction that pertains to quantum channels to achieve the desired linear dependence in evolution time and poly-logarithmic dependence in precision. Open quantum systems can also be used as building blocks of quantum algorithms. We present a dissipative query model, which is based on the amplitude damping process. With this dissipative query model, we provide a quantum algorithm that performs a fixed-point quantum search while preserving the quadratic speedup against classical algorithms. iv Acknowledgements First of all, I would like to thank my supervisor Richard Cleve for his advice, guidance, and support during my Ph.D. program. From every single meeting and discussion with him, I gained knowledge and insight, which are influential in my research and future career. I want to thank my committee members Hoi Fung Chau, Richard Cleve, Debbie Le- ung, Ashwin Nayak, and John Watrous for reading my thesis and giving me the valuable feedback. I am extremely grateful to Richard Cleve, Peter Høyer, Debbie Leung, Ashwin Nayak, and John Watrous for the discussion and guidance of the research presented in this thesis. I also would like to thank Li Liu and Leonard Wossnig for the productive collaborations and fruitful discussions. I am deeply indebted to the faculty members at IQC and CS who helped me, and to all my friends in Waterloo who enriched my life. I would like to give special thanks to my wife Chang Liu for her endless support. She also contributed to this thesis with love and understanding. v Dedication This thesis is dedicated to my grandparents and my wife. vi Table of Contents List of Figuresx 1 Introduction1 1.1 Unitary 2-designs................................2 1.2 Lindblad evolution...............................3 1.3 Dissipative quantum search..........................5 1.4 Organization of this thesis...........................6 2 Notation and Preliminaries8 2.1 Basics for quantum computing.........................8 2.2 Properties of finite fields............................ 16 3 An Example of Open Quantum Systems: Unitary 2-Designs 20 3.1 Previous work and main results........................ 20 3.2 Definitions of unitary 2-designs........................ 22 3.3 Pauli mixing and unitary 2-designs...................... 31 3.3.1 Pauli mixing implies a unitary 2-design................ 31 n 3.3.2 Pauli mixing and SL2(GF(2 ))..................... 35 n 3.3.3 A framework for implementing elements of SL2(GF(2 ))...... 39 3.4 Efficient constructions of unitary 2-designs.................. 42 3.4.1 Near-linear implementation based on self-dual basis for GF(2n)... 42 vii 3.4.2 Near-linear implementations based on polynomial basis for GF(2n). 49 3.4.3 Lower bounds for the size and depth of unitary 2-designs...... 58 4 Continuous-Time Evolution of Markovian Open Quantum Systems 61 4.1 Macroscopic derivation of the Lindblad equation............... 62 4.2 Examples of Lindblad evolution........................ 66 4.3 Lower-bound of simulation as Hamiltonian evolution in a larger Hilbert space 67 5 Quantum Algorithms for Simulating Markovian Open Quantum Systems 75 5.1 Previous work and main results........................ 75 5.1.1 Previous work.............................. 75 5.1.2 Main results............................... 76 5.2 Novel techniques................................ 79 5.2.1 The performance of the standard LCU method on Stinespring dilations 80 5.2.2 Brief summary of novel techniques.................. 81 5.3 New LCU method for channels and completely positive maps........ 82 5.4 Overview of the algorithm........................... 85 5.4.1 A linear map that approximates infinitesimal Lindblad evolution.. 87 5.4.2 Implementing the approximation map by the new LCU method.. 90 5.4.3 Simulation with constant success probability............. 92 5.4.4 Oblivious amplitude amplification for isometries........... 93 5.4.5 Concentration bound and encoding scheme.............. 96 5.4.6 Total number of gates and proof of the main theorem........ 100 5.5 Lindbladians with sparse Hamiltonian and Lindblad operators....... 102 6 Harnessing Open Quantum Systems: Dissipative Quantum Search 104 6.1 Previous work and main results........................ 105 6.1.1 Previous work.............................. 105 viii 6.1.2 Main results............................... 106 6.2 Review of Grover's algorithm......................... 107 6.3 The dissipative query model.......................... 108 6.4 Dissipative quantum search algorithm..................... 112 7 Conclusion 120 References 121 ix List of Figures 3.1 Illustration of a 2-query distinguishing circuit. The first query Q1 can be y U or U , likewise for the second query Q2. The initial state ρ is arbitrary, V is an arbitrary unitary, and the final measurement is also arbitrary and outputs one bit.................................. 23 3.2 Illustration of the bilateral twirl: querying U twice in parallel. The initial sate ρ is arbitrary................................ 25 3.3 An illustration of the Pascal's triangle structure of the matrix L8 . Taking the left half of an 8-level Pascal's triangle and rotating counter-clockwise by 90 degrees, Γ we obtain L8. Note that the block L4 is the horizontal reflection of the lower- diagonal block L4 with a downward shift, as described by property 2 of Lk. .. 47 3.4 An example of representation conversion circuit which demonstrates the recursive structure. ..................................... 47 5 3.5 The implementation of Πr for multiplication of a by r where a; r 2 GF(2 ). Π~ r is an implementation of Sch¨onhage'smultiplication algorithm. The input and output bits are with respect to a self-dual basis. ................. 49 3.6 Illustration of the lower-triangular Pauli mixing within the zero column (N = 2n)..................................... 56 3.7 Illustration of the lower-triangular Pauli mixing within the nonzero columns (N = 2n)..................................... 56 3.8 Illustration of the column Pauli mixing for the zero column (N = 2n).... 57 3.9 Illustration of the column Pauli mixing for the nonzero columns (N = 2n). 57 3.10 Illustration of the mixing procedure starting in the zero row (N = 2n)... 58 3.11 Illustration of the mixing procedure starting in a nonzero row (N = 2n)... 59 x 4.1 Lindblad evolution for time t approximated by unitary operations. There are N iterations and δ = t=N. This converges to Lindblad evolution as N ! 1..................................... 68 4.2 N-stage -precision discretization of the trajectory resulting from L. For each k 2 f1;:::;Ng, after k stages, the channel should be within of kT exp N L ..................................... 69 4.3 The Local Hamiltonian Approximation lemma. The first register is M- dimensional, the second register contains n qubits, and the approximation is within O(δ2) (independent of M and n)................... 70 4.4 A demonstration