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International Conference on Electrical and Computer Engineering (ICECE)

Effect of Quantum Repetition Code on Fidelity of Bell States in Bit Flip Channels

Syed Emad Uddin Shubha, Md. Saifur Rahman

Published Dec 21, 2022• 1 citation
Publisher

Abstract

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.

DOI
10.1109/icece57408.2022.10088665

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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 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.
This paper investigates entanglement distillation protocols assisted by quantum codes under bit-flip noise. Three protocols are compared: repetition code in correction and detection modes, and the [[4, 2, 2]] code in detection mode. We derive output fidelity and success probability for each, showing that the detection-based repetition code achieves the highest fidelity, while the [[4, 2, 2]] code provides better success rates. These results illustrate trade-offs between fidelity and yield, and highlight the potential of quantum codes in realistic noisy channels.
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.
We study the performance of simple quantum error correcting codes with respect to correlated noise errors characterized by a finite correlation strength. Specifically, we consider bit flip (phase flip) noisy quantum memory channels and use repetition and noiseless quantum codes. We characterize the performance of the codes by means of the entanglement fidelity as function of the error probability and degree of memory. Finally, comparing the entanglement fidelities of repetition and noiseless quantum codes, we find a threshold for the correlation strength that allows to select the code with better performance.
The future of quantum repeater networking will require interoperability between various error-correcting codes. A few specific code conversions and even a generalized method are known, however, no detailed analysis of these techniques in the context of quantum networking has been performed. In this paper we analyze a generalized procedure to create Bell pairs encoded heterogeneously between two separate codes used often in error-corrected quantum repeater network designs. We begin with a physical Bell pair and then encode each qubit in a different error-correcting code, using entanglement purification to increase the fidelity. We investigate three separate protocols for preparing the purified encoded Bell pair. We calculate the error probability of those schemes between the Steane [[7,1,3]] code, a distance-3 surface code, and single physical qubits by Monte Carlo simulation under a standard Pauli error model and estimate the resource efficiency of the procedures. A local gate error rate of ${10}^{\ensuremath{-}3}$ allows us to create high-fidelity logical Bell pairs between any of our chosen codes. We find that a postselected model, where any detected parity flips in code stabilizers result in a restart of the protocol, performs the best.