Exponentially More Precise Algorithms for Quantum Simulation of Quantum Chemistry The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:38811452 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA Abstract In quantum chemistry, the \electronic structure problem" refers to the process of numer- ically solving Schrodinger's equation for a given molecular system. These calculations rank among the most CPU-intensive computations that are performed nowadays, yet they are also crucial to the design of new drugs and materials. The idea behind quantum simulation of quantum chemistry is that a quantum computer could perform such calcu- lations in a manner that is exponentially more efficient than what a classical computer could accomplish. In fact, quantum simulation is quite possibly the application most naturally suited to a quantum computer. In this work we introduce novel algorithms for the simulation of quantum chemistry, algorithms that are exponentially more pre- cise than previous algorithms in the literature. Previous algorithms are based on the Trotter-Suzuki decomposition and scale like the inverse of the desired precision, while our algorithms represent the first practical application of the recently developed trun- cated Taylor series approach, and they achieve scaling that is logarithmic in the inverse of the precision. We explore algorithms for both second-quantized and first-quantized encodings of the chemistry Hamiltonian, where the second-quantized encoding requires O(N) qubits for an N spin-orbital system, and the highly compressed first-quantized encoding requires O(η) qubits, where η << N is the number of electrons in the system. We first introduce a second-quantized algorithm that uses pre-computed molecular in- tegrals, resulting in a gate count that scales like Oe(N 8t). Next we introduce algorithms that compute molecular integrals on the fly by carefully discretizing these integrals. In- deed, figuring out how to evaluate the integrals in an efficient manner is one of the main challenges of applying the truncated Taylor series approach to the chemistry problem: the complexity of a general quantum algorithm is traditionally formulated in terms of the number of queries made to an oracle whose assumption we just assume, but in this case, we actually construct the oracle for our specific problem. The evaluation of the oracle itself requires evaluating molecular integrals, which adds additional complexity to an algorithm. We find that the gate count of the second-quantized on-the-fly algorithm scales like Oe(N 5t). We also find that the first-quantized on-the-fly algorithm, which uses the configuration interaction (CI) matrix representation of the Hamiltonian, requires at most Oe(η2N 3t) gates. In general, the discovery of exponentially more precise algorithms will allow for the simulation of larger molecular systems than was previously possible under Trotter-based methods. In addition, the results here can be generalized to the simulation of other fermionic systems beyond the electronic structure problem. Acknowledgements First off I would like to thank Professor Aspuru-Guzik for his generosity and support over the time I've been working in his lab, and because I've learned so much about what it means to do interdisciplinary research|in chemistry, computer science, physics|in the time I've been there. I also owe a lot to Ryan Babbush, who has very patiently dealt with my many questions and explained things to me, and from whom I've learned a lot about chemistry and quantum computing. I'd like to thank Ian Kivlichan for many helpful conversations, and I'd like to thank all of our other collaborators on this project, Dominic Berry and Peter Love, as well. I'd also like to thank Jhonathan Romero Fontalvo for all his help and patience with running Psi4 and quantum chemistry packages; Thomas Markovich and Jarrod McClean for help with Psi4 as well; and Rafa Bombarelli, Jorge Aguilera, and Tim Hirzel for selecting and providing molecular input files to run these quantum chemistry packages on. iii Contents Abstract ii Acknowledgements iii 1 Introduction1 1.1 Motivation...................................1 1.2 Organization..................................3 1.3 Citations to Previous Work..........................4 2 Quantum Computation5 2.1 Asymptotic Notation..............................5 2.2 Quantum Mechanics..............................6 2.2.1 States and Wave Functions......................6 2.2.2 Time Evolution.............................7 2.3 Quantum Computing..............................8 2.3.1 Quantum Circuit Model........................9 2.3.1.1 A Grab Bag of Useful Gates................9 2.3.2 Adiabatic Model............................ 11 2.4 Summary.................................... 11 3 Quantum Simulation of Quantum Chemistry 12 3.1 The Chemistry Hamiltonian.......................... 12 3.1.1 First-Quantized vs. Second-Quantized................ 13 3.2 The Electronic Structure Problem...................... 15 3.3 The Canonical Algorithm........................... 16 3.3.1 State Preparation........................... 16 3.3.2 Hamiltonian Evolution......................... 17 3.3.3 Trotterization.............................. 17 3.3.3.1 Simulation of Sparse Hamiltonians............. 18 3.3.4 Phase Estimation Algorithm..................... 19 3.4 Summary.................................... 19 4 Truncated Taylor Series Algorithm 21 4.1 Overview of Algorithm............................. 21 4.2 Integral Hamiltonians............................. 28 iv Contents v 5 The Integrand Oracle 32 5.1 Basis Function Circuit Construction..................... 33 5.2 Integrand Circuit Complexity......................... 34 5.3 Integrand Circuit Construction........................ 37 5.4 Summary.................................... 38 6 Second-quantized Algorithms 39 6.1 Hamiltonian Oracle............................... 40 6.2 Simulating Hamiltonian Evolution with Pre-Computed Integrals: the Database Algorithm.................................... 42 6.3 Simulating Evolution Under Integral Hamiltonians: the On-the-Fly Al- gorithm..................................... 44 7 First-Quantized Algorithm 50 7.1 CI Matrix Encoding.............................. 50 7.2 CI Matrix Decomposition........................... 52 7.2.1 Decomposition into 1-sparse matrices................ 53 7.2.2 Decomposition into hij and hijk` ................... 54 7.2.3 Discretizing the Integrals....................... 55 7.2.4 Decomposition into Unitary Matrices................. 55 7.3 CI Matrix Hamiltonian Oracle........................ 56 7.4 Simulating Hamiltonian Evolution: the On-the-Fly Algorithm....... 58 8 Algorithms for Real Molecules 62 8.1 Second-Quantized Database Algorithm.................... 62 8.2 Future Directions................................ 64 Bibliography 67 Chapter 1 Introduction 1.1 Motivation This thesis explores the interplay between quantum chemistry and quantum computa- tion, and how advances in the latter can inform the way we approach the former. In quantum chemistry, an important problem is being able to solve Schrodinger's equa- tion for molecular systems, yielding the energy eigenvalues of the system, as this allows us to understand, from first principles, the way these systems function and evolve. Solv- ing these equations is quite difficult to do for large, multi-electron systems, and many existing methods rely on approximations. Being able to carry out these computations allows us study properties of chemical reactions like the ground-state, excited-state, and transition-state energies of reacting molecules. In particular, being able to understand electronic structures can help with the design of new drugs and materials. Meanwhile, the term \quantum computation" refers to using properties of quantum mechanical systems to develop a computing paradigm that is altogether conceptually different from classical computation. The hope is that quantum effects such as the su- perposition of quantum bits (qubits) and entanglement between qubits can allow for algorithms that are more efficient than classically possible. The idea of a quantum com- puter first originated with Richard Feynman in 1982 [1]. More specifically, Feynman's first, hypothetical quantum computer was a quantum simulator, as this would have been the most natural application for a quantum computer. Quantum simulation is defined as the use of a controllable quantum system|in this case, the quantum computer|to investigate the behavior of another, less accessible quantum system|in this case, the model chemistry. In our specific case, this would allow us to observe how a given wave function evolves under the chemistry Hamiltonian. Feynman's 1982 paper showed that a 1 Chapter 1. Introduction 2 classical Turing machine could not simulate quantum phenomena without experiencing an exponential slowdown, while a hypothetical quantum machine would not face such a problem. This leads us, next, to the question of quantum simulation, specifically, for quantum chemistry. Feynman left open the question of being able to simulate fermionic systems, where fermions are the antisymmetric half-integer-spin particles, like electrons, that make up the matter of everyday life. This