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Analysis of Quantum Error-Correcting Codes: Symplectic Lattice Codes and Toric Codes

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Analysis of quantum error-correcting codes: symplectic lattice codes and toric codes Thesis by James William Harrington Advisor John Preskill In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2004 (Defended May 17, 2004) ii c 2004 James William Harrington All rights Reserved iii Acknowledgements I can do all things through Christ, who strengthens me. Phillipians 4:13 (NKJV) I wish to acknowledge first of all my parents, brothers, and grandmother for all of their love, prayers, and support. Thanks to my advisor, John Preskill, for his generous support of my graduate studies, for introducing me to the studies of quantum error correction, and for encouraging me to pursue challenging questions in this fascinating field. Over the years I have benefited greatly from stimulating discussions on the subject of quantum information with Anura Abeyesinge, Charlene Ahn, Dave Ba- con, Dave Beckman, Charlie Bennett, Sergey Bravyi, Carl Caves, Isaac Chenchiah, Keng-Hwee Chiam, Richard Cleve, John Cortese, Sumit Daftuar, Ivan Deutsch, Andrew Doherty, Jon Dowling, Bryan Eastin, Steven van Enk, Chris Fuchs, Sho- hini Ghose, Daniel Gottesman, Ted Harder, Patrick Hayden, Richard Hughes, Deborah Jackson, Alexei Kitaev, Greg Kuperberg, Andrew Landahl, Chris Lee, Debbie Leung, Carlos Mochon, Michael Nielsen, Smith Nielsen, Harold Ollivier, Tobias Osborne, Michael Postol, Philippe Pouliot, Marco Pravia, John Preskill, Eric Rains, Robert Raussendorf, Joe Renes, Deborah Santamore, Yaoyun Shi, Pe- ter Shor, Marcus Silva, Graeme Smith, Jennifer Sokol, Federico Spedalieri, Rene Stock, Francis Su, Jacob Taylor, Ben Toner, Guifre Vidal, and Mas Yamada. Thanks to Chip Kent for running some of my longer numerical simulations on a computer system in the High Performance Computing Environments Group (CCN-8) at Los Alamos National Laboratory. Some simulations were also run on the Pentium Pro based Beowulf Cluster (naegling) computer system operated by the Caltech CACR. I am very grateful for helpful comments from Charlene Ahn, Chris Lee, and Graeme Smith on draft versions of my thesis chapters. A special thanks to Francis Su, Winnie Wang, Dawn Yang, and Esther Yong for working alongside me while I was writing parts of this thesis. Also, thanks to Robert Dirks for being such a patient and fun bridge partner to play with this iv past year, which provided a welcome break from work. I would like to acknowledge some of the teachers that have encouraged me and helped lead me to this point: David Crane, Fred DiCesare, Maureen Dinero, Autumn Finlan, Dan Gauthier, Linda Hagreen, Calvin Howell, David Kraines, Jane LaVoie, Harold Layton, Claude Meyers, Mary Champagne-Myers, Kate Ross, Anthony Ruggieri, David Schaeffer, Roxanne Springer, and Stephanos Venakides. Additionally, I want to acknowledge the sharpening role of the accountabil- ity partners in my life over the past few years: James Bowen, Isaac Chenchiah, Brenton Chinn, Gene Chu, Tony Chu, Stephan Ichiriu, Frank Reyes, and Kenji Shimibukuro. I also appreciate my brothers and sisters in the Sedaqah groups I have belonged to at Evergreen, as well as my fellow grad students in the Caltech Christian Fellowship and UCLA Graduate Christian Fellowship. Soli Deo Gloria. v Analysis of quantum error-correcting codes: symplectic lattice codes and toric codes by James William Harrington In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Abstract Quantum information theory is concerned with identifying how quantum mechan- ical resources (such as entangled quantum states) can be utilized for a number of information processing tasks, including data storage, computation, communi- cation, and cryptography. Efficient quantum algorithms and protocols have been developed for performing some tasks (e.g., factoring large numbers, securely com- municating over a public channel, and simulating quantum mechanical systems) that appear to be very difficult with just classical resources. In addition to iden- tifying the separation between classical and quantum computational power, much of the theoretical focus in this field over the last decade has been concerned with finding novel ways of encoding quantum information that are robust against er- rors, which is an important step toward building practical quantum information processing devices. In this thesis I present some results on the quantum error-correcting properties of oscillator codes (also described as symplectic lattice codes) and toric codes. Any harmonic oscillator system (such as a mode of light) can be encoded with quantum information via symplectic lattice codes that are robust against shifts in the system’s continuous quantum variables. I show the existence of lattice codes whose achievable rates match the one-shot coherent information over the Gaussian quantum channel. Also, I construct a family of symplectic self-dual lattices and vi search for optimal encodings of quantum information distributed between several oscillators. Toric codes provide encodings of quantum information into two-dimensional spin lattices that are robust against local clusters of errors and which require only local quantum operations for error correction. Numerical simulations of this system under various error models provide a calculation of the accuracy threshold for quantum memory using toric codes, which can be related to phase transitions in certain condensed matter models. I also present a local classical processing scheme for correcting errors on toric codes, which demonstrates that quantum information can be maintained in two dimensions by purely local (quantum and classical) resources. vii Contents 1 Introduction 1 1.1 Overview . 1 1.2 Introduction to quantum error correction . 2 1.3 Key features of lattices . 6 2 Achievable rates for the Gaussian quantum channel 9 2.1 Abstract . 9 2.2 Introduction . 9 2.3 The Gaussian quantum channel . 11 2.4 Lattice codes for continuous quantum variables . 16 2.5 Achievable rates from efficient sphere packings . 20 2.6 Improving the rate . 22 2.7 Achievable rates from concatenated codes . 26 2.8 The Gaussian classical channel . 33 2.9 Conclusions . 37 3 Family of symplectic self-dual lattice codes 39 3.1 Abstract . 39 3.2 Introduction . 39 3.3 Constructing oscillator codes . 40 3.4 Program . 42 3.4.1 Parameterization . 42 3.4.2 Algorithm . 44 viii 3.5 Results . 46 3.6 Error models . 50 3.6.1 The square lattice Z2 ..................... 50 3.6.2 The hexagonal lattice A2 ................... 51 3.6.3 The checkerboard lattice D4 . 53 3.6.4 Bavard’s symplectic lattice F6 . 56 3.6.5 The exceptional Lie algebra lattice E8 . 56 3.7 Achievable rates . 57 3.8 Conclusion . 60 4 Accuracy threshold for toric codes 61 4.1 Abstract . 61 4.2 Introduction . 62 4.3 Models . 65 4.3.1 Random-bond Ising model . 65 4.3.2 Random-plaquette gauge model . 73 4.3.3 Further generalizations . 77 4.4 Accuracy threshold for quantum memory . 78 4.4.1 Toric codes . 78 4.4.2 Perfect measurements and the random-bond Ising model . 84 4.4.3 Faulty measurements and the random-plaquette gauge model 86 4.5 Numerics . 90 4.5.1 Method . 90 4.5.2 Random-bond Ising model . 92 4.5.3 Random-plaquette gauge model . 98 4.5.4 Anisotropic random-plaquette gauge model . 102 4.5.5 The failure probability at finite temperature . 104 4.6 Conclusions . 106 ix 5 Protecting topological quantum information by local rules 109 5.1 Abstract . 109 5.2 Introduction . 110 5.3 Robust cellular automata . 110 5.4 Toric codes and implementation . 112 5.4.1 Toric code stabilizers . 112 5.4.2 Hardware layout . 113 5.4.3 Error model . 113 5.5 Processor memory . 114 5.5.1 Memory fields . 114 5.5.2 Memory processing . 117 5.6 Local rules . 118 5.7 Error decomposition proof . 124 5.8 Lower bound on accuracy threshold . 128 5.8.1 Correction of level-0 errors . 128 5.8.2 Correction of higher level errors . 128 5.8.3 Classical errors . 134 5.9 Numerical results . 135 5.10 Conclusion . 137 A Translation of “Hyperbolic families of symplectic lattices” 140 A.1 Introduction . 140 A.2 General study of hyperbolic families . 143 A.2.1 Symplectic lattices and hyperbolic families . 143 A.2.2 Symplectic actions on the families . 146 A.2.3 Geometric study of lengths . 148 A.2.4 Relative eutaxy . 150 A.2.5 Principal length functions. Principal points . 153 A.2.6 Dirichlet-Vorono¨ıand Delaunay decompositions (associated with the principal points) . 156 x A.2.7 Study in the neighborhood of points. Vorono¨ı’s algorithm and finitude . 161 A.2.8 Morse’s theory . 166 A.3 Examples . 168 A.3.1 Families An. Forms F2n . 168 A.3.2 Families An (continued). Forms G2n . 170 A.3.3 Families An (continued). Forms H2n(ϕ)(ϕ ∈ SL2(Z)) . 173 A.3.4 Families An (continued). Forms J4m−2 . 177 A.3.5 Extremal points of families An for 1 ≤ n ≤ 16 . 178 A.3.6 An interesting hyperbolic family . 183 A.3.7 Other examples . 184 xi List of Figures 2.1 Two ways to estimate the rate achieved by a lattice code. 24 2.2 Rates achieved by concatenated codes, compared to the one-shot coherent information optimized over Gaussian input states. 30 2 2 2 2.3 The slowly varying function C , defined by R = log2(C /σ ), where R is the rate achievable with concatenated codes. 31 2.4 Rates for the Gaussian classical channel achievable with concate- nated codes, compared to the Shannon capacity. 36 3.1 The normalizer lattice of the Z2 encoding. 52 3.2 The normalizer lattice of the A2 encoding. 54 3.3 Failure probability of several lattice codes over the Gaussian channel 57 3.4 Achievable rates of several lattice codes over the Gaussian channel 59 4.1 The chain E of antiferromagnetic bonds (darkly shaded) and the chain E0 of excited bonds (lightly shaded), in the two-dimensional random-bond Ising model.
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