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Towards Viable Quantum Computation for
Chemistry
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Babbush, Ryan Joseph. 2015. Towards Viable Quantum Computation for Chemistry. Doctoral
dissertation, Harvard University, Graduate School of Arts & Sciences.
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Towards Viable Quantum Computation for Chemistry
A dissertation presented
by
Ryan Joseph Babbush
to
The Committee in Chemical Physics
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
in the subject of
Chemical Physics
Harvard University
Cambridge, Massachusetts
May 2015
c 2015 - Ryan Joseph Babbush
(cid:13)
All rights reserved.
Dissertation Advisor Author
Al´an Aspuru-Guzik Ryan Joseph Babbush
Towards Viable Quantum Computation for Chemistry
Abstract
Sinceitsintroductiononedecadeago,thequantumalgorithmforchemistryhasbeenamong
the most anticipated applications of quantum computers. However, as the age of industrial
quantum technology dawns, so has the realization that even “polynomial” resource over-
heads are often prohibitive. There remains a large gap between the capabilities of existing
hardware and the resources required to quantum compute classically intractable problems
in chemistry. The primary contribution of this dissertation is to take meaningful steps to-
wards reducing the costs of three approaches to quantum computing chemistry. First, we
discusshowchemistryproblemscanbeembeddedinHamiltonianssuitableforcommercially
manufacturedquantumannealingmachines. Weintroduceschemesformoreefficientlycom-
piling problems to annealing Hamiltonians and apply the techniques to problems in protein
folding, gene expression, and cheminformatics. Second, we introduce the first adiabatic
quantum algorithm for fermionic simulation. Towards this end, we develop tools which
embed arbitrary universal Hamiltonians in constrained hardware at a reduced cost. Finally,
we turn our attention to the digital quantum algorithm for chemistry. By exploiting the
locality of physical interactions, we quadratically reduce the number of terms which must
be simulated. By analyzing the scaling of time discretization errors in terms of chemical
properties, we obtain significantly tighter bounds on the minimum number of time steps
which must be simulated. Also included in this dissertation is a protocol for preparing
configuration interaction states that is asymptotically superior to all prior results and the
details of the most accurate experimental quantum simulation of chemistry ever performed.
iii
Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
1 Introduction 1
1.1 Quantum Annealing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Adiabatic Quantum Computing . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Digital Quantum Computation . . . . . . . . . . . . . . . . . . . . . . . . . 10
I Quantum Annealing 14
2 Compiling Classical Optimization Problems for Quantum Annealing 15
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Optimal reduction gadgets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Efficient encoding techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3 Lattice Protein Folding Using Quantum Annealing 43
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2 The “Turn” Encoding of Self-Avoiding Walks . . . . . . . . . . . . . . . . . 48
3.3 The “Diamond” Encoding of SAWs . . . . . . . . . . . . . . . . . . . . . . . 66
3.4 Pseudo-boolean Function to W-SAT . . . . . . . . . . . . . . . . . . . . . . 72
3.5 W-SAT to Integer-Linear Programming . . . . . . . . . . . . . . . . . . . . 76
3.6 Locality Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.7 Example Encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4 Bayesian Network Structure Learning Using Quantum Annealing 93
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.3 Mapping BNSL to QUBO . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.4 Penalty Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
iv
5 Training Robust Binary Classifiers Using Quantum Annealing 128
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.2 Cubic loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.3 Sixth-order loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.4 Explicit tensor construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
II Adiabatic Quantum Computation 154
6 Realizable Perturbative Gadgets for Encoding Quantum Problems 155
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.2 Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.3 Improved Oliveira and Terhal subdivision gadget . . . . . . . . . . . . . . . 165
6.4 Parallel subdivision and k- to 3-body reduction . . . . . . . . . . . . . . . . 170
6.5 Improved Oliveira and Terhal 3- to 2-body gadget . . . . . . . . . . . . . . 179
6.6 Creating 3-body gadget from local X . . . . . . . . . . . . . . . . . . . . . . 188
6.7 YY gadget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
6.8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
7 Adiabatic Quantum Simulation of Quantum Chemistry 214
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
7.2 Second Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
7.3 Qubit Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
7.4 Hamiltonian Gadgets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
7.5 Example Problem: Molecular Hydrogen . . . . . . . . . . . . . . . . . . . . 236
7.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
III Digital Quantum Computation 245
8 Scaling of Trotter-Suzuki Errors in Quantum Chemistry Simulation 246
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
8.2 Analysis of Trotter error operator . . . . . . . . . . . . . . . . . . . . . . . . 255
8.3 Improved simulation methods inspired by classical approaches . . . . . . . . 271
8.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
8.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
9 Exploiting Locality in Quantum Chemistry Simulation 291
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
9.2 Quantum energy estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
9.3 Using imperfect oracles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
9.4 Adiabatic computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
9.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
v
10 Quantum Chemistry Simulation in a Solid-State Spin Register 329
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
10.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
10.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
Bibliography 343
vi
Citations to previously published work
Apart from minor modifications, Chapters 2-10 appeared as the following publications:
“Resource Efficient Gadgets for Compiling Adiabatic Quantum Optimization
Problems”. Ryan Babbush, Bryan O’Gorman, and Al´an Aspuru-Guzik. An-
nalen der Physik. Volume 525, Number 10-11: 877-888. 2013.
“Construction of Energy Functions for Lattice Heteropolymer Models: Efficient
Encodings for Constraint Satisfaction Programming and Quantum Annealing”.
Ryan Babbush, Alejandro Perdomo-Ortiz, Bryan O’Gorman and Al´an Aspuru-
Guzik. Advances in Chemical Physics. Volume 155, Chapter 5: 201-243. 2014.
“Bayesian Network Structure Learning Using Quantum Annealing”. Bryan
O’Gorman, Alejandro Perdomo-Ortiz, Ryan Babbush, Al´an Aspuru-Guzik and
Vadim Smelyanskiy. European Physical Journal Special Topics. Volume 225,
Number 1: 163-188. 2015.
“Construction of Non-Convex Polynomial Loss Functions for Training a Binary
Classifier with Quantum Annealing”. Ryan Babbush, Vasil Denchev, Nan Ding,
Sergei Isakov and Hartmut Neven. arXiv preprint 1406.4203. 1-9. 2014.
“Hamiltonian Gadgets with Reduced Resource Requirements”. Yudong Cao,
Ryan Babbush, Jacob Biamonte, and Sabre Kais. Physical Review A. Volume
91, Number 1: 012315. 2015.
“Adiabatic Quantum Simulation of Quantum Chemistry”. Ryan Babbush, Pe-
ter Love and Al´an Aspuru-Guzik. Scientific Reports. Volume 4, Number 6603:
1-11. 2014.
“Chemical Basis of Trotter-Suzuki Errors in Quantum Chemistry Simulation”.
Ryan Babbush, Jarrod McClean, Dave Wecker, Al´an Aspuru-Guzik and Nathan
Wiebe. Physical Review A. Volume 91, Number 2: 022311. 2015.
“Exploiting Locality in Quantum Computation for Quantum Chemistry”. Jar-
rod McClean, Ryan Babbush, Peter Love and Al´an Aspuru-Guzik. Journal of
Physical Chemistry Letters. Volume 5, Number 24: 4368-4380. 2014.
“Quantum Simulation of Helium Hydride in a Solid-State Spin Register”. Ya
Wang, Florian Dolde, Jacob Biamonte, Ryan Babbush, Ville Bergholm, Sen
Yang, Ingmar Jakobi, Philipp Neumann, Al´an Aspuru-Guzik, James D. Whit-
field and Jørg Wrachtrup. arXiv preprint 1405.2696. 1-9. 2014.
vii
I contributed to two additional publications during the course of my doctorate which are
not related to quantum computation and thus did not make it into this dissertation [17, 91]:
“Force-Field Functor Theory: Classical Force-Fields which Reproduce Equilib-
riumQuantumDistributions”. RyanBabbush,JohnParkhill,andAl´anAspuru-
Guzik. Frontiers in Chemistry. Volume 1, Number 26: 1-10. 2013.
“BayesianSamplingUsingStochasticGradientThermostats”. NanDing,Youhan
Fang, Ryan Babbush, Changyou Chen, Robert Skeel and Hartmut Neven. Ad-
vancesinNeuralInformationProcessingSystems. Number27: 3203-3211. 2014.
viii
Acknowledgments
First and foremost, I would like to thank my advisor, teacher, and friend, Professor
Al´an Aspuru-Guzik. Al´an’s infectious enthusiasm for science, his endless excitement and
his unpretentious demeanor foster an environment where students learn because there are
no stupid questions and where creativity abounds because discussing science is as much fun
as it is serious. Working in Al´an’s lab provided me the opportunity to pursue the projects
I found most interesting, work with the people from whom I could learn the most, and
seek out opportunities (such as internships at Microsoft and Google) which advanced a
non-academic scientific career. Though I’ve learned a lot of science from Al´an, by watching
him I think I have learned even more about how to inspire people and be a good boss.
Next, I thank three of my senior coauthors whose sustained mentorship has had very
significant impact on my development as a scientist. During my second year, I worked
closely with Professor John Parkhill (then a postdoc) who taught me much about program-
ming, statistical mechanics, and research methods in general. While interning at Microsoft
Research, I spent hours each day pestering Dr. Nathan Wiebe who responded by teaching
me almost everything I know about universal quantum simulation. Last but not least, it
has been a true privilege to work closely with Professor Peter Love for the last two years.
Peter’s constant advice on everything from the politics of industry to the classification of
Lie algebras has contributed substantially to my success in graduate school.
I am also grateful for my affiliation with the Quantum A.I. Lab, which is joint between
Google and NASA. Throughout my internship and collaborations with the group at Google
I learned much about quantum annealing from my many conversations with Dr. Sergio
Boixo, Dr. Masoud Mohseni, Dr. Alireza Shabani, Dr. Sergei Isakov, Dr. Hartmut Neven
ix