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UC Riverside UC Riverside Electronic Theses and Dissertations UC Riverside UC Riverside Electronic Theses and Dissertations Title Codeword Stabilized Quantum Codes and Their Error Correction Permalink https://escholarship.org/uc/item/67h4j8p8 Author Li, Yunfan Publication Date 2010 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA RIVERSIDE Codeword Stabilized Quantum Codes and Their Error Correction A Dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Electrical Engineering by Yunfan Li August 2010 Dissertation Committee: Dr. Ilya Dumer, Chairperson Dr. Leonid Pryadko Dr. Alexander N. Korotkov Copyright by Yunfan Li 2010 The Dissertation of Yunfan Li is approved: Committee Chairperson University of California, Riverside Acknowledgments I would like to thank all people who have helped and inspired me during my doctoral study. I especially want to thank my advisor, Professor Ilya Dumer, for his important and valuable guidance during my research and study at UC Riverside. His perpetual energy and enthusiasm in research had motivated me. In addition, he was always willing to help his students with their research. As a result, research life became smooth and rewarding for me. I was delighted to interact with Professor Leonid Pryadko by attending his classes and having him as my co-advisor. I want to thank him for his guidance and joint effort in my research. I would like to thank Professor Alexander N. Korotkov. I got deep understanding about quantum computation and quantum error correcting codes from his lectures. I also want to express my thanks to Bei Zeng for the detailed explanation of the CWS code construction, to Markus Grassl for the important joint effort. My graduate career was supported in part by the NSF grant No. 0622242. iv ABSTRACT OF THE DISSERTATION Codeword Stabilized Quantum Codes and Their Error Correction by Yunfan Li Doctor of Philosophy, Graduate Program in Electrical Engineering University of California, Riverside, August 2010 Dr. Ilya Dumer, Chairperson Quantum decoherence and errors represent some of the major challenges arising in quan- tum computations. Quantum error correcting codes protect quantum states against these errors and make quantum computing more reliable. One of the main problems of quantum error correction is the design of feasible decoding algorithms that can sim- plify error-correction for general quantum codes. The dissertation addresses decoding of general Codeword Stabilized (CWS) codes. This class of quantum codes also includes some other important classes such as additive Stabilizer codes and non-additive Union Stabilizer (USt) codes. We first design a generic error-correcting algorithm for CWS codes and analyze the number of decoding measurements and quantum gates. This algorithm performs exhaustive screening of different error patterns, similar to decoding of classical non- linear codes. For an n-qubit quantum code correcting up to t erroneous qubits, this brute-force approach consecutively tests all correctable error patterns and employs a separate n-qubit measurement in each test. The main result is a new error grouping technique that enables simultaneous testing of large groups of errors in a single measurement. To achieve this reduction, v we first proceed with a new error-correction algorithm for the USt codes. Secondly, we design an algorithm that converts generic non-linear CWS codes into the simpler quasi-linear USt codes. Each decoding measurement can then either locate the actual error in a given group of errors or entirely eliminate this group. This technique yields a much simpler algorithm that exponentially reduces the number of measurements about 3t times in any t-error-correcting CWS code. vi Contents List of Figures ix 1 Introduction 1 1.1 Quantum Computation and Error Correcting Codes . .... 1 1.2 ClassicalErrorCorrectingCodes . ... 5 1.2.1 CommunicationSystem . .. .. .. .. .. .. 5 1.2.2 BasicsofErrorCorrectingCodes . 10 1.2.3 BinaryLinearBlockCodes . 11 1.2.4 Bounds ................................ 13 1.3 BasicsofQuantumComputation . 14 1.3.1 Wave Function and Schr¨odinger Equations . 14 1.3.2 Observable .............................. 17 1.3.3 QubitsandQuantumGates . 17 1.3.4 BipartiteSystem ........................... 21 2 Stabilizer Quantum Codes 25 2.1 Notations ................................... 25 2.2 General QuantumErrorCorrectingCodes . 27 2.3 StabilizerCodes ............................... 29 2.4 DualSpaceofStabilizer . 31 3 Codeword Stabilized Codes 35 3.1 UnionStabilizercodes . .. .. .. .. .. .. .. 35 3.2 Graphstates ................................. 36 3.3 CWScodesandStandardForm. 38 3.4 GenericrecoveryforCWScodes . 41 4 Union Stabilizer Quantum Codes 46 4.1 Algebraofmeasurements .......................... 46 4.2 Error detection for union stabilizer codes . ...... 50 5 Clustered Error Correct 52 5.1 StructuredmeasurementforCWScodes . 52 5.1.1 Groupingcorrectableerrors . 52 5.1.2 Complexity of a combined measurement . 55 5.1.3 AdditiveCWScodes......................... 56 5.1.4 GenericCWScodes ......................... 58 vii 5.1.5 GeneralUStcodes .......................... 60 5.1.6 Error correction beyond t ...................... 60 6 Conclusions 62 Bibliography 68 viii List of Figures 1.1 Analogmodulations ............................. 6 1.2 Digitalmodulations. .. .. .. .. .. .. .. .. 8 1.3 Block diagram of a communication system. ... 9 1.4 PauliGates .................................. 18 1.5 Hadamardgate................................. 19 1.6 Hadamardgateonvectors. ......................... 20 1.7 Hadamardgate................................. 24 1.8 CNOTgate................................... 24 2.1 Measurement of an observable M with all eigenvalues 1......... 26 ± 3.1 Union of 3 shifts of a 2-dimensional G.................... 36 3.2 Ring graphs with (a) 3 vertices and (b) 5 vertices. ...... 37 3.3 Circuit for generating 3 vertices graph states. ........ 38 3.4 Operators Mci , i = 0,..., 5 for [[5, 6, 2]] code in example 3. 43 3.5 Operator M for [[5, 6, 2]]codeinexample3. 43 C 3.6 Decomposition of n-qubit controlled-Z gate CZ in terms of (n 1)- n − controlled CNOT gate (n-qubit Toffoli gate) for n =3........... 43 3.7 Generic measurement of the classical part M of the CWS code stabilizer C M [Eq. (3.14)] uses K (n + 1)-qubit controlled-Z gates. 44 Q 4.1 Measurement M M by performing logical AND operation on two an- 1 ∧ 0 cillas. ..................................... 47 4.2 Simplified measurement for M M . ................... 47 1 ∧ 0 4.3 Measurement M1⊞M0 by performing logical XOR gate on the two ancillas and subsequent recovery of the first ancilla. 47 4.4 Simplified measurement for M1 ⊞ M0. ................... 48 4.5 Implementation of the controlled-H gate based on the identity H = exp( iY π/8)Z exp(iY π/8). ......................... 48 − ix Chapter 1 Introduction 1.1 Quantum Computation and Error Correcting Codes Quantum computation makes it possible to achieve polynomial complexity for many classical problems that are believed to be hard [1, 2]. One important feature of quantum computation is the superposition of quantum states. The quantum bit (qubit)is a superposition of state 0 and state 1 , which is different from the classical | i | i case. This distinct feature enables quantum computer to carry on massive parallel computation. One example showing this advantage is to determine whether the binary mapping f works as f(0) = f(1) or f(0) = f(1) where f(0),f(1) 0, 1 . With a 6 ∈ { } classical computer, we need to run the function twice to know both f(0) and f(1) in order to learn whether f(0) = f(1) or f(0) = f(1). With a quantum computer, we use a 6 superposition of 0 and 1 as the input to know f(0) = f(1) or f(0) = f(1) by running | i | i 6 the algorithm only one time. One celebrated outcome of these parallel computations is a quantum algorithm to factorize numbers, which has been discovered by Shor[1] and performs in a polynomial time in the number of digits. Another distinct feature is the measurement of observable, whose outcome is 1 generally a random variable. The possible values are eigenvalues of the operator that is related to the observable. After the measurement, the state collapses to one of the eigenstate of the operator, giving the eigenvalue corresponding to the eigenstate as the outcome. A famous example illustrating the feature of measurement is Shr¨odiner’s cat. Before the measurement, the cat was in the state of superposition of alive and dead 1 ( Alive + Dead ). Once we make the measurement, the outcome is either alive or √2 | i | i dead. At the same time, its state collapses to either alive or dead correspondingly. | i | i We may say that the measurement saves the cat or kills the cat, but we should never say that the measurement enables us to know whether it was in Alive or Dead . | i | i Quantum errors and decoherence is one of the major challenges in quantum computation. To preserve coherence, quantum states need to be protected by quantum error correcting codes (QECCs) [3, 4, 5]. With error probabilities in elementary gates below a certain threshold, one can use multiple layers of encoding (concatenation) to reduce errors at each level and ultimately make arbitrarily-long quantum computation possible [6, 7, 8, 9, 10, 11, 12, 13, 14]. The actual value of the threshold error probability strongly depends on the assumptions of the error model and on the chosen architecture, and presently varies 3 from 10− % for a chain of qubits with nearest-neighbor
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