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Efficient Framework for Quantum Walks and Beyond

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Efficient Framework for Quantum Walks and Beyond University of Calgary PRISM: University of Calgary's Digital Repository Graduate Studies The Vault: Electronic Theses and Dissertations 2015-12-22 Efficient Framework for Quantum Walks and Beyond Dohotaru, Catalin Dohotaru, C. (2015). Efficient Framework for Quantum Walks and Beyond (Unpublished doctoral thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/25844 http://hdl.handle.net/11023/2700 doctoral thesis University of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission. Downloaded from PRISM: https://prism.ucalgary.ca UNIVERSITY OF CALGARY Efficient Framework for Quantum Walks and Beyond by Cat˘ alin˘ Dohotaru A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY GRADUATE PROGRAM IN COMPUTER SCIENCE CALGARY, ALBERTA December, 2015 c Cat˘ alin˘ Dohotaru 2015 Abstract In the first part of the thesis we construct a new, simple framework which amplifies to a constant the success probability of any abstract search algorithm. The total query complexity is given by the quantum hitting time of the resulting operator, which we show that it is of the same order as the quantum hitting time of the original algorithm. As a major application of our framework, we show that for any reversible walk P and a single marked state, the quantum walk corresponding to P can find the solution using a number of queries that is quadratically smaller than the classical hitting time of P. Our algorithm is more general and simpler to implement than the solution known previously in the literature (Krovi, Magniez, Ozols, and Roland, 2015), which was developed specifically for quantum walks; we also prove that, for the particular case of quantum walks, we can embed their algorithm into our framework, thus simulating it exactly. Finally, we show that we can implement amplitude amplification using our tool. In the second part of the thesis we give a new lower bound in the query model which proves that Grover’s algorithm for unordered searching is exactly optimal. Similar to existing methods for proving lower bounds, we bound the amount of information we can gain from a single oracle query, but we bound this information in terms of angles. This allows our proof to be simple, self-contained, based on only elementary mathematics, capturing our intuition, while obtaining at the same an exact bound. We then turn our attention to non-adaptive algorithms for the same problem of searching an unordered set. In this model, we obtain a lower bound and we give an algorithm which matches the lower bound, thus showing that the lower bound is exactly optimal. ii Acknowledgements I would like to express my gratitude for my supervisor, Peter Høyer, for his continu- ous help and encouragement, even in the moments when I lost any hope. Next I would like to thank my committee members Hari Krovi, Philipp Woelfel, Michael J. Jacobson Jr., Gilad Gour, and Peter Høyer for reading my thesis and providing insightful com- ments. I have benefited and learned a lot from the numerous discussions I had with Donny Cheung, Jibran Rashid, Nathan Wiebe, and Philipp Woelfel, and I would like to thank them all for their help. I also thank my friends and my family for their support. Contents Abstract ii Acknowledgements iii 1 Introduction 1 1.1 Quantum computation ............................... 1 1.2 Quantum query complexity ............................ 2 1.3 Reflections and rotations .............................. 4 1.4 Mathematical preliminaries ............................ 4 1.5 Notations ....................................... 8 2 Controlled quantum amplification 9 2.1 Problem description ................................ 9 2.2 Introduction ..................................... 10 2.2.1 Related work ................................ 10 2.2.2 The circuit of Tulsi ............................. 16 2.2.3 Our contribution and organization .................... 16 2.3 The circuit ...................................... 18 2.4 The flip-flop theorem ................................ 20 2.5 Spectral analysis of our circuit ........................... 24 2.5.1 Eigenphases ................................. 24 2.5.2 Principal eigenvector ............................ 26 2.6 Cost of our circuit .................................. 29 2.6.1 Definition of the quantum hitting time .................. 29 iv Contents 2.6.2 Quantum hitting time and phase estimation .............. 32 2.6.3 Relations between quantum hitting times ................ 34 2.7 Simulation of amplitude amplification ...................... 42 2.7.1 Standard amplitude amplification .................... 42 2.7.2 Simulation of amplitude amplification .................. 43 2.8 Application to quantum walks .......................... 45 2.8.1 Classical hitting time ............................ 46 2.8.2 Definition of a quantum walk ....................... 50 2.8.3 Relation between classical and quantum hitting times ......... 51 2.9 Simulation of the interpolated quantum walk .................. 56 3 Exact lower bounds for quantum unordered search 62 3.1 Introduction ..................................... 62 3.2 Exact lower bound for adaptive quantum algorithms ............. 63 3.2.1 The query model for adaptive quantum algorithms .......... 64 3.2.2 Exact lower bound for quantum searching ............... 65 3.3 Exact lower bound for non-adaptive quantum algorithms ........... 70 3.3.1 The query model for non-adaptive quantum algorithms ....... 70 3.3.2 Lower bound for non-adaptive quantum algorithms ......... 72 3.3.3 Exact non-adaptive algorithm for quantum searching ......... 73 4 Conclusions 74 4.1 Summary of original contributions ........................ 74 4.2 Future work ..................................... 76 Bibliography 83 A Examples of source code 84 A.1 Efficient simulation of the quantum walk on the grid ............. 84 A.2 Comparison with interpolated quantum walks ................. 87 v List of Figures 2.1 Subconstant success probability of A ....................... 13 2.2 Iterations of A on a grid graph ........................... 13 2.3 Hitting times and the ed upper bound ...................... 14 2.4 The circuit of Tulsi ................................. 16 2.5 U as A and reflection. ................................ 18 2.6 U as W and reflection. ................................ 19 2.7 U as W and reflection, simplified. ......................... 19 2.8 U as A and rotation. ................................. 19 2.9 U as W and rotation. ................................ 20 2.10 Phase estimation with t bits of precision ..................... 33 2.11 U(2) with reflection about spanfj0, initi, j1,˜ gig ................. 38 2.12 U(2) with unconditional G in the second register. ................ 38 2.13 Equivalent form for U(2) .............................. 39 2.14 Simulation of amplitude amplification ...................... 44 0 2.15 Original walk P (left) and the modified walk P (right). ............ 48 vi CHAPTER 1 Introduction 1.1 Quantum computation We first outline in this section the basic concepts of quantum computation. For more details we refer the reader to the textbook [NC00]. States: The state of a quantum computer is described by a unit vector in an finite dimen- m sional Hilbert space C2 . Consider a basis B = fjii; i 2 f0, 1gmg of this space. The basis states are labeled by classical bit-strings of m bits. Any state in the space is a superposition (linear combination) of the basis states jYi a j i = ∑ i i , (1.1) jii2f0,1gm kYk2 ∑ ja j2 a 2 C where = jii2f0,1gm i = 1. The coefficients i are called amplitudes. The state jYi can be represented as a column vector with 2m dimensions. Combining two independent systems: If jui and jvi are quantum states, then the com- bined state of the system is a tensor product jui ⊗ jvi. If the state jui was on n qubits, and the state jvi was on m qubits, then jui ⊗ jvi is a state on n + m qubits. We often denote a tensor product state without the ⊗ symbol, so instead of jui ⊗ jvi we write juijvi or ju, vi. 1 Chapter 1. Introduction Measurement: Measuring the state jYi from Eq. 1.1 with respect to the basis fjii; i 2 f gmg j i ja j2 0, 1 produces outcome i with probability i . After the measurement, the state collapses to the observed basis state jii. We only described here projective measurements; more general measurements are equivalent to projective measurements on a larger space. In order to distinguish between two quantum states, they must be almost orthogonal (see Lemma 64 for a more precise statement). Unitary evolution: Quantum mechanics requires that the evolution of a quantum system is reversible. That is, the quantum state jYi is mapped to the state UjYi, where U is a linear and norm-preserving transformation (such matrices are called unitary). Unitary matrices admit an orthogonal basis of eigenvectors, and the corresponding eigenvalues have absolute value 1. Simulation of classical computation: Any quantum computer can perform any clas- sical computation. That follows from the fact that any classical computation can be made reversible by replacing any irreversible gate x ! g(x) with the reversible gate (x, y) ! (x, y ⊕ g(x)), and running it on input (x, 0). In other words, we make a classical computation reversible by storing all intermediate
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