This thesis titled, “Performance Analysis of Extension of 3- and 5-bit Quantum Repetition Codes”, submitted by the group as mentioned below has been accepted as satisfactory in partial fulfillment of the requirements for the degree B.Sc. in EEE in March 2025. This is to certify that the work presented in this thesis is the outcome of the investigation and research carried out by us under the supervision of Dr Md Saifur Rahman. Group Members:Masuk Ridwan Saumo, ID: S201906106Nafis Faisal, ID: S201906033 Supervisor:Dr Md Saifur RahmanProfessorDepartment of Electrical and Electronic EngineeringBangladesh University of Engineering and Technology Abstract: Quantum computers are becoming an engineer's reality from a physicist's dream. Microsoft, IBM, Google are some of the key players who are pushing the quantum boundaries and bringing us closer to actual quantum computers. Although research in Quantum Computing is focused mostly on creating viable Qubits, a large portion of research is also dedicated to reducing errors due to decoherence, noise and errors from various sources so that the calculations done using Qubits are not flawed. Transmitting Quantum Information from one place to another is what Quantum Communication is about. Here too, the channel induces errors in the transmitted Qubits leading to loss in quantum information. Quantum Error Correction came as an analogue to classical error correction to solve the same problems of bit errors in quantum computation and communication. This work investigates a type of error correction code, the repetition code, common to both classical and quantum domain. In this work, extended 3-bit and 5-bit quantum repetition codes (QRC) in tripartite and pentapartite GHZ states were investigated to probe their efficacy in error correction. 3- and 5-bit QRC can completely protect the Ideal GHZ state, i.e. resulting fidelity remains at one at all conditions. The research indicates that, in case of generalized GHZ states, all of them performed better than their conventional counterparts, especially extension of 5-bit QRC showing the highest fidelity. Modified 5 bit quantum repetition code in pentapartite GHZ state displayed a fidelity margin of 0.075-0.1 over traditional 3-bit QRC on a single state. However small this margin may seem, this fidelity increase is quite remarkable in large-scale quantum computations and practical implementations.
This thesis titled, “Performance Analysis of Extension of 3- and 5-bit Quantum Repetition Codes”, submitted by the group as mentioned below has been accepted as satisfactory in partial fulfillment of the requirements for the degree B.Sc. in EEE in March 2025. This is to certify that the work presented in this thesis is the outcome of the investigation and research carried out by us under the supervision of Dr Md Saifur Rahman. Group Members:Masuk Ridwan Saumo, ID: S201906106Nafis Faisal, ID: S201906033 Supervisor:Dr Md Saifur RahmanProfessorDepartment of Electrical and Electronic EngineeringBangladesh University of Engineering and Technology Abstract: Quantum computers are becoming an engineer's reality from a physicist's dream. Microsoft, IBM, Google are some of the key players who are pushing the quantum boundaries and bringing us closer to actual quantum computers. Although research in Quantum Computing is focused mostly on creating viable Qubits, a large portion of research is also dedicated to reducing errors due to decoherence, noise and errors from various sources so that the calculations done using Qubits are not flawed. Transmitting Quantum Information from one place to another is what Quantum Communication is about. Here too, the channel induces errors in the transmitted Qubits leading to loss in quantum information. Quantum Error Correction came as an analogue to classical error correction to solve the same problems of bit errors in quantum computation and communication. This work investigates a type of error correction code, the repetition code, common to both classical and quantum domain. In this work, extended 3-bit and 5-bit quantum repetition codes (QRC) in tripartite and pentapartite GHZ states were investigated to probe their efficacy in error correction. 3- and 5-bit QRC can completely protect the Ideal GHZ state, i.e. resulting fidelity remains at one at all conditions. The research indicates that, in case of generalized GHZ states, all of them performed better than their conventional counterparts, especially extension of 5-bit QRC showing the highest fidelity. Modified 5 bit quantum repetition code in pentapartite GHZ state displayed a fidelity margin of 0.075-0.1 over traditional 3-bit QRC on a single state. However small this margin may seem, this fidelity increase is quite remarkable in large-scale quantum computations and practical implementations.
Quantum Entanglement is a key to establish secure communication between two nodes in a communication channel. Ensuring high fidelity entanglement has always been a challenging task owing to interaction with the hostile channel environment. Several methods have been proposed in this paper to mitigate the problem. It has been shown that fidelity can be improved by encoding bell pairs using Quantum Repetition Codes (QRCs). In this regard, various QRC-based techniques are also analyzed mathematically, and their performances have been compared.
Quantum computing exists at the intersection of mathematics, physics, chemistry, and engineering; the main goal of quantum computing is the creation of devices and algorithms which use the properties of quantum mechanics to store, manipulate and measure information. There exist many families of algorithms, which, using non-classical logical operations, can outperform traditional, classical algorithms in terms of memory and processing requirements. In addition, quantum computing devices are fundamentally smaller than classical processors and memory elements; since the physical models governing their performance are applicable on all scales, as opposed to classical logic elements, whose underlying principles rely on the macroscopic nature of the device in question. \n \nQuantum algorithms, for the most part, are predicated on a theory of resources. It is often assumed that quantum computers can be placed in a precise fiducial state prior to computation, and that logical operations are perfect, inducing no error on the system which they affect. These assumptions greatly simplify algorithmic design, but are fundamentally unrealistic. In order to justify their use, it is necessary to develop a framework for using a large number of imperfect devices to simulate the action of a perfect device, with some acceptable probability of failure. This is the study of fault-tolerant quantum computing. In order to pursue this study effectively, it is necessary to understand the fundamental nature of generic quantum states and operations, as well as the means by which one can correct quantum errors. Additionally, it is important to attempt to minimize the use of computational resources in achieving error reduction and fault-tolerant computing. \n \nThis thesis is concerned with three projects related to the use of error-prone quantum systems to transmit and manipulate information. The first of these is concerned with the use of imperfectly-prepared states in error-correction routines. Using optimal quantum error correction, we are able to deduce a method of partially protecting encoded quantum information against preparation errors prior to encoding, using no additional qubits. The second of these projects details the search for entangled states which can be used to transmit classical information over quantum channels at a rate superior to classical states. The third of these projects concerns the transcoding of data from one quantum code into another using few ancillary resources. The descriptions of these projects are preceded by a brief introduction to representations of quantum states and channels, for completeness. \n \nThree techniques of general interest are presented in appendices. The first is an introduction to, and a minor advance in the development of optimal error correction codes. The second is a more efficient means of calculating the action of a quantum channel on a given state, given that the channel acts non-trivially only on a subsystem, rather than the entire system. Finally, we include documentation on a software package developed to aid the search for quantum transcoding operations.
Quantum systems operate using quantum bits (qubits), which are highly sensitive to external noises and disturbances and they are easily prone to errors. In the Noisy Intermediate-Scale Quantum (NISQ) era, these errors make difficult to build reliable, fault tolerant quantum circuits for real world deployment. This work provides an overview of need for Quantum Error Correction (QEC) in NISQ devices, as qubits commonly get affected by bit-flip, phase-flip, and combined errors that degrades system performance and examines Quantum Error Correction Code (QECC) techniques usage in communication, computation, and quantum multimedia applications such as image and video transmission. In addition, a simple three-qubit repetition circuit is introduced to demonstrate an error-mitigation approach. These findings, also show how QEC improves fidelity, reduces logical errors, and improves noise tolerance. The analysis also identifies challenges and outlines the future research direction aimed at achieving efficient and scalable fault tolerant quantum circuits.
Quantum systems operate using quantum bits (qubits), which are highly sensitive to external noises and disturbances and they are easily prone to errors. In the Noisy Intermediate-Scale Quantum (NISQ) era, these errors make difficult to build reliable, fault tolerant quantum circuits for real world deployment. This work provides an overview of need for Quantum Error Correction (QEC) in NISQ devices, as qubits commonly get affected by bit-flip, phase-flip, and combined errors that degrades system performance and examines Quantum Error Correction Code (QECC) techniques usage in communication, computation, and quantum multimedia applications such as image and video transmission. In addition, a simple three-qubit repetition circuit is introduced to demonstrate an error-mitigation approach. These findings, also show how QEC improves fidelity, reduces logical errors, and improves noise tolerance. The analysis also identifies challenges and outlines the future research direction aimed at achieving efficient and scalable fault tolerant quantum circuits.
In this thesis, we investigated a theory of quantum error correction (QEC) in nearterm quantum devices.Due to the recent development of quantum technologies, QEC with several tens of qubits is expected to be demonstrated in the next few years.On the other hand, since such a large quantum system cannot be tracked with current classical computers, several problems about practical QEC experiments emerge as quantum devices are scaled up.We provided solutions to the following two essential problems among them.First, we extended a simulatability of quantum error correcting code under coherent noise using matchgate circuits.This extension enables an efficient and accurate evaluation of an effective physical error probability suppressed with QEC under practical errors.Since this probability should be known before we design quantum devices for QEC, this result provides vital information for experiments.Second, we proposed a general framework to use machine learning as a part of decoding algorithm in QEC.QEC requires a fast and high-performance classical decoder for its implementation.Using machine learning for decoding is one of the most promising solutions to this.However, what part of a decoding algorithm should be delegated to machine learning in order to achieve the near-optimal performance has not been discussed.We constructed a general framework to discuss it, and showed fundamental relations between the decoding problem and the task of machine learning.Furthermore, we proposed a criterion for achieving high performance, and showed specific formulation of tasks of machine learning.Then, we confirmed that the performance of the proposed machine-learning-based decoder is superior to the known decoders in various situations.This thesis is written based on various supports and advice from many people.I would like to thank Prof. Masato Koashi, who has totally supervised my research during undergraduate and graduate school.His thoughtful advice and questions are helpful for me to obtain clear viewpoint of my research.I am frequently impressed by his outstanding perspective of quantum information science, and the discussions with him strongly encourage my interest in physics.I also would like to thank Assoc.Prof. Keisuke Fujii, who supervised both of the two main results of the thesis.My knowledge about quantum computation is supported by his deep understanding of it.I am very motivated by his strong enthusiasm for realizing quantum computer.Mr. Amarsanaa Davaasuren is a collaborator of the result about quantum error correction using machine learning discussed in Chapter 4 of this thesis.His many creative ideas about both machine learning and quantum information have amazed me.The result of Chapter 4 is largely supported by his collaboration.Assist.Prof. Takanori Sugiyama gave me lectures about the latest and essential problems in near-term quantum devices.In particular, his lectures were vital for us to find the research topic of an efficient simulation of quantum circuits of quantum error correction under practical noise models, which is discussed in Chapter 3 of this thesis.I have been financially supported by Advanced Leading Graduate Course for Photon Science (ALPS).During this program, I have had an opportunity to discuss with Prof. Akira Furusawa.His advice from the standpoint of an experimental researcher helps my practical understanding of experimental systems.His philosophy as a professional researcher shapes my attitude to research.I also would like to thank the members and alumni of Koashi's group.In particular, I thank Assoc.Prof. Haruka Tanji-Suzuki and Assist.Prof. Toshihiko Sasaki for advice about from fundamental physics to presentation skills.My skills as a researcher are trained through discussions with them.I also thank Dr. Yoshi
Quantum entanglement is a unique criterion of the quantum realm and an essential tool to secure quantum communication. Ensuring high-fidelity entanglement has always been a challenging task owing to interaction with the hostile channel environment created due to quantum noise and decoherence. Though several methods have been proposed, correcting almost all arbitrary errors is still a gigantic task. As one of the main contributions of this work, a new model for 'large distance communication' has been proposed, which may correct all bit flip errors or other errors quite extensively if proper encoding and subspace measurements are used. To achieve this purpose, at the very first step, the idea of differentiating the 'long-distance communication' and 'short-distance applications' has been introduced. Short-distance is determined by the maximum range of applying unitary control gates by the qubit technology. How the error correcting ability of Quantum codes change for short and long-distance application is investigated in this work, which was not explored in previous literatures as far as we know. At the beginning, we have applied stabilizer formalism and Repetition Code for decoding to distinguish the error correcting ability in long and short distance communication. Particularly for short-distance communication, it has been demonstrated that a 'properly encoded' bell state can identify all the bit flip, or phase flip errors with 100% accuracy theoretically. In contrast, if the bell states are used in long-distance communication, the error-detecting and correcting ability reduces at huge amounts. To increase the fidelity significantly and correct the errors quite extensively for long-distance communication, a new model based on classical communication protocol has been suggested. All the required circuits in these processes have been generalized for arbitrary (even) numbers of ancilla qubits during encoding. Proposed analytical results have also been verified with the Simulation results of IBM QISKIT QASM.