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At the Intersection of Quantum Computing and Quantum Chemistry

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At the Intersection of Quantum Computing and Quantum Chemistry At the intersection of quantum computing and quantum chemistry A dissertation presented by James Daniel Whitfield to The Department of Chemistry and Chemical Biology in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Chemical Physics Harvard University Cambridge, Massachusetts November 2011 c 2011 - James Daniel Whitfield All rights reserved. Thesis advisor Author Al´anAspuru-Guzik James Daniel Whitfield At the intersection of quantum computing and quantum chemistry Abstract Quantum chemistry is concerned with solving the Schr¨odingerequation for chemically rele- vant systems. This is typically done by making useful and systematic approximations which form the basis for model chemistries. A primary contribution of this dissertation, is taking concrete steps toward establishing a new model chemistry based on quantum computation. Electronic energies of the system can be efficiently obtained to fixed accuracy using quan- tum algorithms exploiting the ability of quantum computers to efficiently simulate the time evolution of quantum systems. The quantum circuits for simulation of arbitrary electronic Hamiltonians are given using quantum bits associated with single particle basis functions. This model chemistry is applied to hydrogen molecule as a case study where all necessary quantum circuits are clearly laid out. Furthermore, our collaboration to experimentally realize a simplified version of the hydrogen molecule quantum circuit is also included in this thesis. Finally, alternatives to the gate model of quantum computation are pursued by exploring models based on the quantum adiabatic theorem and on the generalization of random walks. iii Citations to Previously Published Work Chapter2 is a largely unmodified contribution to an undergraduate textbook: Mathematical modeling II: Quantum mechanics and spectroscopy T. L. Story with chapter on electronic structure contributed by J. D. Whitfield Educational Publisher. To appear Fall 2011. Parts of the review article, Simulating chemistry with quantum computers I. Kassal∗, J. D. Whitfield∗ A. Perdomo-Ortiz, M.-H. Yung, and A. Aspuru- Guzik Annual Reviews of Physical Chemistry. Volume 62, Pages 185-207 (2011) appear in many places throughout the thesis. Many of the major themes of the thesis are found in, Simulation of electronic structure Hamiltonians using quantum computers J. D. Whitfield, J. D. Biamonte, and A. Aspuru-Guzik Molecular Physics. Volume 109, 735 (2011) and large portions of this publication have been incorporated into Chapters5,6, and8. The description of the experimental simulation of the hydrogen molecule in Chapter8 is largely based on Towards quantum chemistry on a quantum computer B. P. Lanyon, J. D. Whitfield, G. G. Gillet, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru- Guzik, and A. G. White Nature Chemistry. Volume 2, 106 (2010) Chapters9 and 10 apart from minor modifications appear respectively in, Adiabatic quantum simulators J. D. Biamonte, V. Bergholm, J. D. Whitfield, J. Fitzsimons, and A. Aspuru- Guzik AIP Advances. Volume 1, 022126 (2011) Quantum stochastic walks: A generalization of classical random walks and quan- tum walks J. D. Whitfield, C. A. Rodr´ıguez-Rosario,and A. Aspuru-Guzik. Physical Review A. Volume 81, 022323 (2010) Other work I was involved in but not explicitly included in the thesis are the articles iv Abstract v Simulation of classical thermal states on a quantum computer: A transfer-matrix approach M.-H. Yung, D. Nagaj, J. D. Whitfield, and A. Aspuru-Guzik Physical Review A. Volume 82, 060302(R) (2010) Solving quantum ground-state problems with nuclear magnetic resonance Z. Li, M.-H. Yung, H. Chen, D. Lu, J. D. Whitfield, X. Peng, A. Aspuru-Guzik, and J. Du Accepted to Scientific Reports. Preprint available at arXiv:1106.0440 Contents Title Page........................................i Abstract......................................... iii Citations to Previously Published Work....................... iv Table of Contents.................................... vi Acknowledgments....................................x Dedication........................................ xi 1 Introduction1 1.1 Overview.....................................1 2 Overview of electronic structure methods5 2.1 Introduction....................................5 2.2 Electronic structure: Hamiltonians and wave functions............6 2.2.1 Atomic units...............................6 2.2.2 The Born-Oppenheimer approximation.................7 2.2.3 Wave functions in electronic structure.................9 2.2.4 Matrix elements of the electronic Hamiltonian............. 17 2.3 Full configuration interaction.......................... 20 2.4 Hartree-Fock theory............................... 23 2.4.1 The Hartree-Fock equations....................... 25 2.4.2 Derivation of the Hartree-Fock equations............... 27 2.4.3 Roothann matrix equations....................... 28 2.4.4 Semi-empirical methods......................... 30 2.4.5 Correlation energy............................ 33 2.4.6 Multireference approaches........................ 35 2.5 Møller-Plesset many-body perturbation theory................ 35 2.6 Coupled-cluster methods............................. 38 2.7 Density functional theory............................ 41 2.7.1 Densities in quantum mechanics.................... 41 2.7.2 Theoretical basis for DFT: three theorems............... 43 2.7.3 The Kohn-Sham system......................... 47 2.7.4 Approximations to the exchange-correlation functional........ 49 2.8 Electronic structure calculations........................ 51 vi Contents vii 3 Quantum computation and quantum information 53 3.1 Introduction.................................... 53 3.2 Information is physical.............................. 54 3.3 Quantum information.............................. 55 3.3.1 Differences between classical and quantum information........ 56 3.3.2 Entanglement............................... 59 3.4 Quantum computation.............................. 60 3.4.1 Differences between quantum and classical computation....... 61 3.4.2 Models of quantum computation.................... 62 3.5 Experimental progress.............................. 65 4 Computational complexity in quantum chemistry 69 4.1 Introduction.................................... 69 4.2 Preliminary notions............................... 70 4.2.1 Computational problems and problem instances............ 73 4.2.2 Reductions................................ 74 4.2.3 Promise problems............................ 75 4.2.4 Relevant polynomial time complexity classes............. 76 4.2.5 Relevant non-deterministic complexity classes............. 77 4.2.6 Completeness and hardness....................... 79 4.3 Computational chemistry problems....................... 80 4.3.1 Quantum chemistry........................... 80 4.3.2 Computational problems......................... 81 4.4 Hartree-Fock is NP-complete.......................... 83 4.5 Reduced wave function approaches....................... 85 4.5.1 N-REPRESENTABILITY....................... 87 4.5.2 UNIVERSAL FUNCTIONAL..................... 88 4.5.3 Extensions to pure states........................ 90 4.6 Other topics.................................... 91 5 Measurements and the phase estimation algorithm 92 5.1 The phase estimation algorithm......................... 94 5.1.1 Iterative phase estimation........................ 95 5.1.2 Estimating overlaps........................... 97 5.2 Error analysis................................... 98 5.3 Other measurement schemes........................... 102 6 Simulation of chemical Hamiltonians 105 6.1 Introduction.................................... 105 6.2 Suzuki-Trotter formulas and the unitary propagator............. 106 6.2.1 Trotter decomposition.......................... 106 6.2.2 Suzuki-Trotter formulas......................... 108 6.3 First quantization................................ 110 6.4 Second quantization............................... 114 6.4.1 The electronic Hamiltonian in second quantization.......... 114 Contents viii 6.4.2 Representing the molecular Hamiltonian with qubits......... 115 6.4.3 Quantum circuits for quantum simulation............... 116 7 Playing Merlin: ideas and difficulties 121 7.1 General methods................................. 121 7.2 Ground states................................... 122 7.2.1 Adiabatic state preparation....................... 123 7.3 Thermal state preparation............................ 125 7.3.1 QMA-hardness and future prospects.................. 126 8 Quantum simulation of hydrogen molecule: theory and experiment 128 8.1 Quantum simulation of the hydrogen molecule................ 129 8.2 Resource count for hydrogen molecule..................... 133 8.3 Experimental implementation of molecular hydrogen simulation....... 134 8.4 Discussion..................................... 140 8.4.1 Model Hamiltonian used for experimental implementation...... 140 8.4.2 Details of classical computational methods.............. 143 8.4.3 Classical error correction techniques.................. 144 8.4.4 Count rates................................ 144 8.5 Additional experimental results......................... 145 9 Adiabatic quantum simulators 150 9.1 Introduction.................................... 150 9.2 Simulator overview................................ 152 9.2.1 Initialization............................... 153 9.2.2 Adiabatic evolution........................... 153 9.2.3 Measurement..............................
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