Quantum Error Correction (QEC) is essential for achieving fault-tolerant quantum computation, enabling the protection of fragile quantum information from decoherence and noise. While classical Forward Error Correction (FEC) codes have been highly effective in digital communication systems, their direct application to quantum systems is inadequate mainly due to the no cloning theorem and the continuous nature of quantum errors. Unlike classical FECs, QEC is designed to detect and correct both bit-flip and phase-flip errors without disturbing the encoded quantum state. This paper presents the code structure design and simulation of two quantum error correction codes—the Planar surface code and the 5-qubit Perfect code—using Qiskit and Stim on Google Colab. It is worth mentioning here that the performance of the Steane [ [7], [1], [3]] code is included in the current work as a benchmark. The codes are evaluated under statistical noise models, including amplitude damping and dephasing, which closely mirror the physical behavior of quantum hardware. The primary performance metric, for performance assessment, is the logical error rate, evaluated across a range of physical error rates and code distances under each noise model. The simulation results show that increasing the distance of the Planar code significantly reduces the logical error rates, demonstrating superior performance compared to other similar codes under this study.
Quantum Error Correction (QEC) is essential for achieving fault-tolerant quantum computation, enabling the protection of fragile quantum information from decoherence and noise. While classical Forward Error Correction (FEC) codes have been highly effective in digital communication systems, their direct application to quantum systems is inadequate mainly due to the no cloning theorem and the continuous nature of quantum errors. Unlike classical FECs, QEC is designed to detect and correct both bit-flip and phase-flip errors without disturbing the encoded quantum state. This paper presents the code structure design and simulation of two quantum error correction codes—the Planar surface code and the 5-qubit Perfect code—using Qiskit and Stim on Google Colab. It is worth mentioning here that the performance of the Steane [ [7], [1], [3]] code is included in the current work as a benchmark. The codes are evaluated under statistical noise models, including amplitude damping and dephasing, which closely mirror the physical behavior of quantum hardware. The primary performance metric, for performance assessment, is the logical error rate, evaluated across a range of physical error rates and code distances under each noise model. The simulation results show that increasing the distance of the Planar code significantly reduces the logical error rates, demonstrating superior performance compared to other similar codes under this study.
Quantum error correction (QEC) represents a critical enabler for fault-tolerant quantum computing, protecting fragile quantum states from decoherence and operational errors. This paper provides a comprehensive analysis of leading quantum error correction schemes, focusing on surface codes and topological codes within the context of Noisy Intermediate-Scale Quantum (NISQ) era processors. We examine the theoretical foundations of stabilizer codes, compare surface code implementations with code distances ranging from d=3 to d=17, and analyze topological codes including toric codes and color codes. Our investigation encompasses syndrome extraction circuits, logical qubit encoding strategies, and threshold analysis for contemporary quantum hardware platforms. Performance metrics reveal that surface codes achieve error thresholds of approximately 1% with realistic noise models, while topological codes offer architectural advantages for specific qubit connectivity constraints. We discuss implementation challenges on current NISQ processors including IBM Quantum, Google Sycamore, and IonQ systems, addressing qubit overhead, gate fidelity requirements, and decoding algorithms. This analysis provides insights for selecting appropriate error correction schemes based on application requirements and hardware capabilities.
Quantum Error Correction (QEC) is essential for overcoming the fragility of qubits, enabling reliable and scalable quantum computing by mitigating the effects of noise, decoherence, and operational errors. This chapter explores QEC's foundational principles, including error types, noise impact, and syndrome measurement techniques for identifying and correcting errors. It covers key codes such as Shor and Steane, which provide basic error protection, and advanced codes like concatenated codes, surface codes, and quantum Low-Density Parity-Check (LDPC) codes. Concatenated codes employ hierarchical encoding to exponentially reduce logical error rates, while surface codes, including Toric codes, utilize topological properties of qubits arranged on a 2D lattice to detect and correct errors with high fault tolerance and scalability. Quantum LDPC codes leverage sparse parity-check matrices to offer efficient and scalable error correction, closely approaching theoretical performance limits. The chapter concludes by exploring current research efforts to optimize QEC codes.
Fault-tolerant quantum computation requires quantum error correction (QEC), and while the performance of various QEC codes has been well characterized under idealized noise conditions, it has not been sufficiently characterized by simulation under realistic noise conditions.In this paper, the performance of three basic QEC codes (3-qubit Bit-Flip, 3-qubit Phase-Flip and Shor's 9-qubit [[9,1,3]]) under simulated depolarizing noise, by using Qiskit Aer, is analyzed systematically in three stages.Stage 1 is a validation of all three implementations under ideal circumstances.Stage 2 tests for 6 different error rates of 0% to 5%, and runs 2048 shots per error rate.Stage 3 brings the analysis from stage 2 to 0% to 10%, but with an uncorrected baseline for comparison.Results verify that Shor's > Bit-Flip > Phase-Flip is consistent over practically relevant error rates.At 1% noise, within current IBM Quantum hardware range, all three codes achieve a logical fidelity of over 98.7%.An error rate of 7-8% is used to identify a cross-over threshold above which the simpler Bit-Flip code can match Shor's code in fidelity with only one-third the number of qubits.
Abstract Quantum error correction (QEC) is essential for mitigating decoherence in quantum computations, but many QEC schemes rely heavily on mid-circuit measurements, which introduce significant noise due to their high error rates in current hardware. To overcome this limitation, we propose a measurement-free QEC scheme based on the surface code, leveraging its topological robustness while eliminating the need for physical measurements during error correction. Our key innovation is a three-dimensional patch-based architecture that interlinks three logical qubit patches, enabling coherent error extraction and feedback without measurements. By transforming syndrome information into geometric correlations across patches, our design inherently suppresses measurement-induced errors while preserving the surface code’s fault-tolerant thresholds. We analyze the scheme’s performance under realistic noise models and demonstrate its advantages over traditional flag-based or Steane-type QEC in regimes where measurements dominate error rate. This work bridges the gap between measurement-free QEC and scalable surface code architectures, offering a path toward fault-tolerant quantum computing in systems where fast, high-fidelity measurements remain impractical.
Fault-tolerant Quantum Computing (FTQC) relies on Quantum Error Correction (QEC) codes that encode logical qubits across many physical qubits to detect and correct errors. The surface code is among the most widely studied codes due to its high error threshold, the existence of efficient decoders, and hardware-friendly properties: a planar, two-dimensional layout with nearest-neighbor connectivity. In practice, however, the fabrication of solid-state quantum processors introduces hardware defects, resulting in defective qubits and couplers that must be discarded. Adapting the surface code to these defects often requires measuring the X- and Z-type checks in separate rounds rather than simultaneously. In this work, we investigate the optimal X-to-Z checks round-scheduling ratio under biased noise systems. Our results characterize how key architectural parameters, such as noise bias, code distance, and defect rate, impact the logical error rate. We provide insights into how to determine the optimal scheduling ratio directly from device calibration data, enabling manufacturers to maximize performance without extensive simulations. Our approach reduces the logical error rate by up to 4.25× at a 1% defect rate and up to 8.46× at a 2% defect rate for a distance-13 surface code under moderately biased noise. Furthermore, we demonstrate that the benefits of round-scheduling extend beyond the defective-hardware setting. In biased-noise architectures subject to CNOT crosstalk, separating X and Z measurement rounds yields up to 4.5× reduction in logical error rate.
Current quantum processors are fragile, noisy and fairly limited in both quantity and quality with tens of qubits and physical error rates of around 10^-3. To realize practical quantum applications, however, error rates need to be below 10^-15 across millions of qubits. To bridge this gap and fully harness the potential of quantum computers, quantum error correction (QEC) is essential. QEC codes are designed to protect quantum information by redundantly encoding it onto multiple physical qubits. This encoding allows for the detection and correction of local errors affecting individual qubits, e.g., through stabilizer measurements. Importantly, if the physical error rates are below a specific threshold, QEC codes can exponentially suppress logical error rates by increasing the number of physical qubits involved. This is essential for achieving fault-tolerant computations, which are key to unlocking the full potential of quantum computers. The work presented in this thesis focuses on the implementation and optimization of small-scale QEC experiments using the surface code and flux-tunable superconducting qubits (Transmons). It addresses several key challenges: enhancing two-qubit gate fidelity in Surface-4 (Chapter 2), implementing an error-detection code with Surface-7 (Chapter 3), automating the calibration and benchmarking of the building blocks in Surface-17 (Chapter 4), reducing leakage into higher excited states with leakage reduction units (Chapter 5), assessing and enhancing the performance of logical qubits (Chapters 7 and 8).