Quantum Operations and Codes Beyond the Stabilizer-Clifford Framework by MASSACHUSETTS INSTIfUTE by OF TECHNOLOGY Bei Zeng I AUG 0 2009 B.Sc., Tsinghua University (2002) M.Sc., Tsinghua University (2004) LIBRARIES Submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2009 ARCHIVES © Bei Zeng, MMIX. All rights reserved. The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document in whole or in part. Author .... Department of Physics June 2009 Certified by ....... Isaac L. Chuang Associate Professor of Physics Associate Professor of Electric Engineering and Computer Science Thesis Supervisor Accepted by ................... ......... / / Thomas J. Greytak Professor of Physics Quantum Operations and Codes Beyond the Stabilizer-Clifford Framework by Bei Zeng Submitted to the Department of Physics on June 2009, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics Abstract The discovery of quantum error-correcting codes (QECCs) and the theory of fault-tolerant quantum computation (FTQC) have greatly improved the long-term prospects for quantum communication and computation technology. Stabilizer codes, a quantum analogue of clas- sical additive codes, are the most important class of QECCs. These codes have dominated the study of quantum error-correction and fault-tolerance ever since the discovery of QECC- FTQC in the mid-1990s. The dominance of stabilizer codes is not only due to their simple construction based on Abelian groups, but also because of the simplicity with which oper- ations in the Clifford group can be performed on them. However, practical QECC-FTQC schemes neither achieve the maximum theoretical communication rate despite the noise nor do they allow sufficient gate failure probability to meet the needs of our current technol- ogy. Therefore, it is necessary to go beyond this framework to seek for new QECC-FTQC techniques. This thesis develops a systematic study of QECC-FTQC beyond the Stabilizer-Clifford framework, building on both well-known results and recent theoretical advances. Our new systematic theory includes two main parts. The first part provides further understanding of quantum operations on stabilizer codes, that is, the "minimal subcodes+semi-Clifford operations" theory for studying non-Clifford operations on stabilizer codes. The other part deals with the construction of QECCs, that is, the codeword stabilized (CWS) quantum codes framework combined with the generalized concatenation method for constructing good nonadditive (and stabilizer) quantum codes. Our theory successfully tackles some important open problems in the field (such as a proof of the "transversality versus universality problem for stabilizer codes", and a systematic method for constructing good nonadditive QECCs), and makes progress on many other related open problems (such as the LU-LC equivalence problem for stabilizer codes, and the Ck structure problem). We believe that our theory sheds light on deriving novel QECC-FTQC schemes that will bring us closer to implementing reliable quantum communication systems and building large-scale quantum computers. Thesis Supervisor: Isaac L. Chuang Title: Associate Professor of Physics Associate Professor of Electric Engineering and Computer Science Acknowledgments *This thesis is dedicated to my Little Elephant.* My debts are many. I thank my academic advisor David Pritchard and research advisor Isaac Chuang, who patiently guide me, support me and help me throughout my years' academic life at MIT. I am also very grateful to MIT professors Edward Farhi, David Forney, Seth Lloyd, Peter Shor, and Xiaogang Wen for their guidance and help. I give my many thanks to my many colleagues, coauthors, collaborators, and friends - at MIT, at IBM; in US, in China, and all around the world: Panos Aliferis, David Ba- con, Salman Beigi, Charles Bennett, Sergey Bravyi, Xie Chen, Hyeyoun Chuang, Andrew Cross, David DiVincenzo, Luming Duan, Runyao Duan, Andrew Fletcher, Daniel Gottes- man, Markus Grassl, Martin Roetteler, Zhengcheng Gu, Aram Harrow, Zhengfeng Ji, Su- peng Kou, Debbie Leung, Mary Beth Ruskai, Yaoyun Shi, Graeme Smith, John Smolin, Changpu Sun, Barbara Terhal, Zhan Xu, Beni Yoshida, Li You, Duanlu Zhou... I thank my husband and my parents for their love, understanding and support. Above all, I thank God for His inestimable love, and all the blessings of this life. Contents I Background 8 1 The Stabilizer-Clifford Framework for Quantum Codes 9 1.1 Introduction .... ... ... .... .. .... .. ... 9 1.2 Classical error correction .................... ....11 1.2.1 Principle of error correction .. .. .. .. .. .. .. .. 12 1.2.2 Linear codes and dual codes . ... .... .... .. ... 12 1.2.3 Weight and distances . .. .. ... .. ... .. .. .. 13 1.2.4 Equivalence of codes . .. .. .. .. .. .. .. .. .. 14 1.3 Quantum error correction ................... 14 1.3.1 Quantum bits and quantum codes .. .. .. .. 15 1.3.2 Quantum error model ................. ... 16 1.3.3 Error detection and correction conditions . .. .. 16 1.3.4 Weight and distances .................. ... 17 1.3.5 Code equivalence/automorphism and fault-tolerance .. 18 1.4 The Stabilizer-Clifford framework . .. ... .... ... .. .. 20 1.4.1 The stabilizer codes .................. ... 20 1.4.2 Clifford group and operations on stabilizer codes .. .. 22 1.5 Quantum codes of minimum distance two .. .. .. .. .. .. 24 1.5.1 Distance two codes of even length . .. .. .. .. .. 24 1.5.2 Distance two codes of odd length .. .. .. .. .. .. 26 1.6 The perspective and organization of this thesis .. .. .. 28 1.6.1 Non-Clifford operations on stabilizer codes .. .. .. 30 1.6.2 Nonadditive quantum codes .. .. .. .. .. .. .. .. 31 II Quantum Operations 2 Local Equivalence of Stabilizer Codes 2.1 Introduction . .................. 2.2 Preliminaries . ................. 2.2.1 Stabilizers states and graph states 2.2.2 Minimal supports ............ 2.3 The main theorem ............... 2.3.1 d > 2 and d = 2 graphs and Case V4 . 2.3.2 Case V1 U V2: The standard procedure 2.3.3 Case V3 ... ... ... .... .. .. 2.3.4 Some remarks . ............. 2.3.5 Algorithm for constructing /Cn .............. .. .. 50 2.3.6 d = 2 graphs beyond the main theorem . ............. .. 52 2.4 d > 2 graph states beyond the MSC . ................ .. 52 2.4.1 d > 2 graphs beyond the MSC for minimal n . ............ 52 2.4.2 Graphs derived from codes with non-Clifford transversal gates . 52 2.4.3 LU , LC property for d > 2 graphs . ................. 54 2.5 Conclusion and discussion ...... .. ...................... 56 3 Transversality versus Universality for Stabilizer Codes 57 3.1 Introduction ....... ........................... 57 3.2 Preliminaries ........ ..... .. ....................... 59 3.2.1 Stabilizers and stabilizer codes ................... .. 59 3.2.2 Universality . ..................... .. ... ... 60 3.2.3 Automorphisms of stabilizer codes . .................. 61 3.2.4 Fault-tolerance and multiple encoded blocks . ............. 62 3.2.5 Codes stabilized by minimal elements and the minimal support con- dition ... ... .. .. ... .. .. ... .. ... .. .. 63 3.2.6 Coordinates not covered by minimal supports . ............ 67 3.3 Transversality versus universality ................ ...... 68 3.3.1 The single block case (r = 1) ................... ... 68 3.3.2 The multiblock case (r > 1) ................ ... 71 3.4 The effect of coordinate permutations . ............ .......... 74 3.5 Applications and examples . .................. ... 75 3.6 Conclusion and discussion . .................. .. .. .. 78 4 The Ck Hierarchy for Quantum Teleportation 79 4.1 Introduction .......... ........... ............. 79 4.2 Semi-Clifford operations and its generalization . .............. 82 4.3 The structure of Ck ......... ........ .. ... .. .. .. .. 83 4.3.1 Basic properties ................. ..... ..... 83 4.3.2 Structure theorems .............. .... ............ 84 4.3.3 Open problems ............ ....... .. .. ..... 86 4.4 Single qubit Ck gates . ............... ... .. ... 87 4.4.1 Single qubit gates with eigenvalues +1 . .............. 87 4.4.2 Gate series associated with 1I(Tp) and F2(0,) . ... ............ 87 4.4.3 Gates in Ck \ Ck-1 for single qubit ............. .... .. 89 4.5 Detailed analysis about C3 . .. .. ... .. .. .. ... ... .. .. ... 90 4.5.1 Notations . .................. .............. 90 4.5.2 Some facts for calculating C3 structure . ............... 91 4.5.3 n = 1 case ................... .... .. 91 4.5.4 n = 2 case .. ................................ ..... 92 4.5.5 n = 3 case ....... .. ............. ... ....... 92 4.5.6 Diagonal gates in C3 . .. ... .. .. .. ... ... .. .. .. .. 92 4.6 The teleportation depth . .. .................. ....... 92 4.6.1 Definition of the teleportation depth . ......... ........ 93 4.6.2 Teleportation depth of semi-Clifford Ck gates . ............ 95 4.6.3 Teleportation depth beyond semi-Clifford Ck gates . ......... 98 4.6.4 Teleportation beyond Ck ............................ 99 4.7 Conclusion and discussion . ... ....................... 100 III Quantum Codes 103 5 Codeword Stabilized Quantum Codes 105 5.1 Introduction ..................... ... ... 105 5.2 General construction and properties . ....... 107 5.2.1 Relation to stabilizer codes .. .... .. 110 5.3 Examples ....................... 111 5.3.1 The [5,1,3] code . .............