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Performance of Various Low-Level Decoder for Surface Codes in the Presence of Measurement Error Air Force Institute of Technology AFIT Scholar Theses and Dissertations Student Graduate Works 3-2021 Performance of Various Low-level Decoder for Surface Codes in the Presence of Measurement Error Claire E. Badger Follow this and additional works at: https://scholar.afit.edu/etd Part of the Computer Sciences Commons Recommended Citation Badger, Claire E., "Performance of Various Low-level Decoder for Surface Codes in the Presence of Measurement Error" (2021). Theses and Dissertations. 4885. https://scholar.afit.edu/etd/4885 This Thesis is brought to you for free and open access by the Student Graduate Works at AFIT Scholar. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of AFIT Scholar. For more information, please contact [email protected]. Performance of Various Low-Level Decoders for Surface Codes in the Presence of Measurement Error THESIS Claire E Badger, B.S.C.S., 2d Lieutenant, USAF AFIT-ENG-MS-21-M-008 DEPARTMENT OF THE AIR FORCE AIR UNIVERSITY AIR FORCE INSTITUTE OF TECHNOLOGY Wright-Patterson Air Force Base, Ohio DISTRIBUTION STATEMENT A APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED. The views expressed in this document are those of the author and do not reflect the official policy or position of the United States Air Force, the United States Department of Defense or the United States Government. This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. AFIT-ENG-MS-21-M-008 Performance of Various Low-Level Decoders for Surface Codes in the Presence of Measurement Error THESIS Presented to the Faculty Department of Electrical and Computer Engineering Graduate School of Engineering and Management Air Force Institute of Technology Air University Air Education and Training Command in Partial Fulfillment of the Requirements for the Degree of Master of Science in Computer Science Claire E Badger, B.S.C.S., B.S.E.E. 2d Lieutenant, USAF March 26, 2021 DISTRIBUTION STATEMENT A APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED. AFIT-ENG-MS-21-M-008 Performance of Various Low-Level Decoders for Surface Codes in the Presence of Measurement Error THESIS Claire E Badger, B.S.C.S., B.S.E.E. 2d Lieutenant, USAF Committee Membership: Laurence D. Merkle, Ph.D Chair Brett J. Borghetti, Ph.D Member David E. Weeks, Ph.D Member AFIT-ENG-MS-21-M-008 Abstract Quantum error correction is a research speciality within the area of quantum computing that constructs quantum circuits that correct for errors. Decoding is the process of using measurements from an error correcting code, known as error syn- drome, to decide corrective operations to perform on the circuit. High-level decoding is the process of using the error syndrome to perform corrective logical operations, while low-level decoding uses the error syndrome to correct individual data qubits. Research on machine learning-based decoders is increasingly popular, but has not been thoroughly researched for low-level decoders. The type of error correcting code used is called surface code. A neural network-based decoder is developed and com- pared to a partial lookup table decoder and a graph algorithm-based decoder. The effects of increasing error correcting code size and increasing measurement errors on the error syndrome are analyzed for the decoders. The results demonstrate that there are advantages in terms of average execution time and resistance to increasing mea- surement error with the neural network-based decoder when compared to the two other decoders. iv Table of Contents Page Abstract . iv List of Figures . viii List of Tables . .x I. Introduction . .1 1.1 Motivation . .1 1.2 Research Objectives . .3 1.3 Assumptions and Limitations . .5 1.4 Summary and Document Overview . .6 II. Background and Literature Review . .8 2.1 Overview . .8 2.2 Quantum Computation . .8 2.2.1 Qubits . .9 2.2.2 Basis States . 10 2.2.3 Single-Qubit Operations . 11 2.2.4 Multi Qubit Operations . 14 2.3 Quantum Error Correction . 14 2.3.1 Noise Models . 16 2.3.2 Stabilizer Formalism . 18 2.3.3 Surface Code . 19 2.4 Decoding . 24 2.4.1 Decoder Characteristics . 24 2.4.2 Examples of Decoders . 26 2.5 Machine Learning . 29 2.5.1 ROC Curves . 31 2.5.2 Cross Validation. 32 2.5.3 Random Forest. 34 2.5.4 kNN . 35 2.5.5 Multi-Output Classifier. 36 2.5.6 Artificial Neural Networks . 36 2.6 Hypothesis Testing . 39 2.6.1 Paired T-Test . 39 2.6.2 5x2 Cross Validation . 39 2.6.3 Matthew's Correlation Coefficient . 40 2.6.4 McNemar's Test . 40 2.7 Neural Network Decoders (Previous Work). 41 2.8 Summary . 45 v Page III. Methodology . 47 3.1 Overview . 47 3.2 Approach . 47 3.3 Evaluation Metrics . 48 3.3.1 Accuracy and F1 . 48 3.3.2 Speed . 50 3.3.3 Scalability . 50 3.3.4 Fault Tolerance . 51 3.4 Neural Network Task . 51 3.5 Data Simulation . 56 3.6 Machine Learning Pipeline . 60 3.6.1 Feature Selection and Data Prepocessing . 60 3.6.2 Model Selection and Design . 65 3.7 Experimental Design . 70 3.7.1 Significance Test . 70 3.7.2 Decoder Evaluation . 72 3.8 Summary . 75 IV. Results and Analysis . 76 4.1 Overview . 76 4.2 Performance in Relation to Code Depth . 76 4.2.1 Accuracy Results and Analysis . 76 4.2.2 F1 Results and Analysis . 79 4.2.3 Summary of Performance for Increasing Code Depths . 81 4.3 Noise Results and Analysis . 82 4.3.1 Increasing Measurement Error on Depth 3 . 83 4.3.2 Increasing Measurement Error on Depth 5 . 83 4.3.3 Increasing Measurement Error on Depth 7 . 84 4.3.4 Summary of Performance for Increasing Probability of Measurement Error . 85 4.4 Execution Time Results and Analysis . 87 4.5 Analysis of P-Values . 89 4.5.1 McNemar's Test with Depth 5 and 7 . 89 4.5.2 McNemar's Test with.
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