Quantum has many unique characteristics for future applications in the sixth telecommunication generation (6G) 2030 based on entanglement and quantum teleportation. However, quantum communications are still vulnerable to errors and require quantum error corrections (QECC). This paper proposes quantum accumulate codes, where idea is coming from the classical accumulate codes that can provide good performances to remove or minimize the error-floor in classical communications. The proposed quantum codes can exactly satisfy both Hamming and Singleton bounds resulting in one of the smallest perfect quantum codes. This paper explains in details the steps to construct the codewords, syndrome extraction, error correction, and the quantum circuits. We perform a series of computer simulations to evaluate the quantum word error rate (QWR) and quantum fidelity performances under the depolarizing channels. Our results confirm that the stabilizers of the proposed codes can extract the syndrome and distinguish all error patterns of any single qubit errors of Pauli X, Y, and Z providing performances confirmed by the theoretical QWER curves. The results are expected to provide contributions to the development of quantum communications and future security issues.
Quantum has many unique characteristics for future applications in the sixth telecommunication generation (6G) 2030 based on entanglement and quantum teleportation. However, quantum communications are still vulnerable to errors and require quantum error corrections (QECC). This paper proposes quantum accumulate codes, where idea is coming from the classical accumulate codes that can provide good performances to remove or minimize the error-floor in classical communications. The proposed quantum codes can exactly satisfy both Hamming and Singleton bounds resulting in one of the smallest perfect quantum codes. This paper explains in details the steps to construct the codewords, syndrome extraction, error correction, and the quantum circuits. We perform a series of computer simulations to evaluate the quantum word error rate (QWR) and quantum fidelity performances under the depolarizing channels. Our results confirm that the stabilizers of the proposed codes can extract the syndrome and distinguish all error patterns of any single qubit errors of Pauli X, Y, and Z providing performances confirmed by the theoretical QWER curves. The results are expected to provide contributions to the development of quantum communications and future security issues.
This paper proposes short but high rate quantum codes, derived from the classical accumulate codes, having capability of correcting multiple errors simultaneously. The high coding rate is obtained from k = 2 with minimum blocklength n, but still satisfying both Singletone and Hamming bounds to have error correction capability of t = 2 qubits, We perform an intensive computer search to find the best matrix of parity check that match with the accumulator matrix to provide zero simplectic inner product (SIP) mod 2. We found that it is difficult to obtain a perfect parity check matrix that provide zero diagonal element in SIP to guarantee completely unique syndromes. However, since it may be difficult, we also present in this paper a theoretical quantum word error rate (QWER), when some syndromes do not have unique values to identify the quantum errors of X, Z, and Y. The results in this paper are expected to provide groundbreaking foundation on the development of high rate quantum codes withk≥2, especially for short blocklength.
Quantum technology offers transformative potential for future applications, including secure communications and high-speed computing. However, quantum systems are highly susceptible to decoherence caused by environment interactions, making error correction essential for reliable operation. This paper proposes a new 7-qubit quantum error correction code constructed from classical Hamming codes through a distinct derivation of the parity matrix. Unlike the well-known Steane code, our scheme employs a different parity matrix within the stabilizer framework, resulting in a novel code structure. We detail the step-by-step construction of codewords, syndrome extraction using stabilizer measurements, and the corresponding quantum circuits. Performance evaluation is conducted through simulations under depolarizing channels across error probabilities from to 0, using Quantum Word Error Rate (QWER) as the metric. Simulation results confirm that the proposed stabilizers can uniquely identify all single-qubit error patterns of Pauli X, Y, and Z. Moreover, the scheme achieves a significant error reduction of approximately 28 times (about 96%) compared to uncoded transmission. These findings demonstrate that deriving quantum codes from modified classical matrices offers an effective strategy for enhancing the reliability of future quantum communication systems.
The employment of quantum error correction codes (QECCs) within quantum computers potentially offers a reliability improvement for both quantum computation and communications tasks. However, the laws of quantum mechanics prevent us from directly invoking the mature family of classical error correction codes in the quantum domain. In order to circumvent the associated problems, the notion of quantum stabilizer codes (QSCs) was proposed in conjunction with syndrome-based decoding in the quantum regime. However, most of the powerful QSC schemes require long codewords for achieving a high performance, which potentially imposes additional challenges for implementation concerning their implementation, since their decoding may require longer than the quantum circuit's coherence time. Hence at the time of writing, QSCs exhibiting short to moderate codeword lengths are preferable. We commence by describing the pivotal problem encountered by classical error-correction codes, which also emerges when designing the QSCs, namely the intrinsic trade-off between the minimum distance versus coding rate. The complete formulation of this particular trade-off does not exist, but several lower and upper bounds can be found in the literature. It has been shown that a substantial gap can be observed between the upper and lower bound of the minimum distance, given the codeword length and the quantum coding rate. Hence, we propose an appealingly simple and invertible analytical approximation, for characterizing the trade-off between the quantum coding rate and the minimum distance of QSCs as well as their corresponding quantum bit error rate (QBER) performance upper-bound. For example, for a half-rate QSC having a codeword length of n = 128, the minimum distance is bounded by 11 < d < 22, while our approximation yields a minimum distance of d = 17 for the above-mentioned code. Next, we link this parametric study of the minimum distance versus quantum coding rate to the popular QSCs, namely to the family of quantum topological error correction codes (QTECCs). In order to construct the classical-to-quantum isomorphism, we conceive and investigate the family of classical topological error correction codes (TECCs), assuming that the bits of a codeword can be arranged in a lattice structure. We then present the classical-to-quantum isomorphism to pave the way for constructing their dual pairs in the quantum domain, which are the QTECCs. Finally, we characterize the performance of QTECCs in the face of the quantum depolarizing channel in terms of both their QBER and fidelity. Specifically, we demonstrate that for quantum coding rate rQ ~ 0, the threshold probability of the QBER below which the colour, rotated-surface, surface, and toric codes become capable of improving the uncoded QBER are given by 1.8 ×10-2, 1.3 ×10-2, 6.3 ×10-2 and 6.8 ×10-2, respectively. Furthermore, we also demonstrate that we can achieve beneficial fidelity improvements above the minimum fidelity of 0.94, 0.97 and 0.99 by employing the rQ = 1/7 colour code, the rQ = 1/9 rotated-surface code, and the rQ = 1/13 surface code, respectively. However, QSCs require additional quantum gates for their employment. Incorporating more quantum gates for performing error correction potentially introduces further sources of quantum decoherence into quantum computers. In this scenario, the primary challenge is to find the sufficient condition required by each of the quantum gates for beneficially employing QECCs in order to yield reliability improvements, given that the quantum gates utilized by the QECCs also introduce quantum decoherence. In this treatise, we approach this problem by firstly presenting the general framework of protecting quantum gates by the amalgamation of the transversal configuration of quantum gates and QSCs, which can be viewed as syndrome-based QECCs. Secondly, we provide examples of the advocated framework by invoking QTECCs for protecting both transversal Hadamard gates and controlled (CNOT) gates. Both our simulation and analytical results explicitly show that by utilizing QTECCs, the fidelity of the quantum gates can be beneficially improved, provided that quantum gates satisfying a certain minimum depolarization fidelity threshold (Fth) are available. For instance, for protecting transversal Hadamard gates, the minimum fidelity values required for each of the gates in order to attain fidelity improvements are 99.74%, 99.73%, 99.87%, and 99.86%, when they are protected by colour, rotated-surface, surface, and toric codes, respectively. Unfortunately, these specific Fth values can only be obtained for a very large number of physical qubits (n → ∞), when the quantum coding rate of the QTECCs approaches zero (rQ → 0). Finally, in order to conceive QSCs exhibiting a high quantum coding rate, we modify the construction of QTECCs for conceiving a low-complexity concatenated quantum turbocode (QTC). The above-mentioned high quantum coding rate is obtained by combining the quantum-domain version of short-block codes (SBCs) also known as single parity-check (SPC) codes as the outer codes and quantum unity-rate codes (QURCs) as the inner codes. Despite its design simplicity, the proposed QTC yields a near-hashing-bound error correction performance. For instance, compared to the best half-rate QTC known in the literature, namely the quantum irregular convolutional codes (QIrCCs) combined with the QURC scheme, which operates at the distance of D = 0.037 from the quantum hashing bound, our novel QSBC-QURC scheme can operate at the lower distance of D = 0:029. It is also worth mentioning that this is the first instantiation of QTCs capable of adjusting the quantum encoders according to the quantum coding rate required for mitigating the Pauli errors imposed by the time-variant depolarizing probabilities of the quantum channel.
In the era of sixth-generation (6G) communications, quantum technologies are expected to serve as foundational components for future network infrastructure. However, the inherently fragile nature of quantum states, susceptible to environmental noise and hardware operational errors, creates an urgent demand for effective mechanisms to protect information. This article provides a comprehensive survey of quantum error correction codes (QECCs), spanning from the inception of the field through 2026, with a specific focus on their role as an essential reliability layer within the 6G ecosystem. The survey presents an overview of the transition from classical ECC to QECC, while systematically organizing the foundations, evolution, and classification of key QECC families. We conduct an in-depth analysis of their structural properties in relation to scalability, locality, and distributed operation. Furthermore, the article delves into a detailed discussion of encoding techniques, syndrome extraction, and practical implementation challenges. In particular, we emphasize the approach of hardware-software co-design and analyze the bandwidth-latency trade-offs inherent in distributed quantum communication environments. By identifying open challenges and emerging trends leading up to 2026, this article outlines a strategic roadmap for integrating reliability mechanisms into the infrastructure of the future quantum internet.
Quantum parallel processing techniques are capable of solving certain complex problems at a substantially lower complexity than their classical counterparts. From the perspective of telecommunications, this quantum-domain parallel processing provides a plausible solution for achieving full-search based multi-stream detection, which is vital for future gigabit-wireless systems. The peculiar laws of quantum mechanics have also spurred interest in the absolutely secure quantum-based communication systems. Unfortunately, quantum decoherence imposes a hitherto insurmountable impairment on the practical implementation of quantum computation as well as on quantum communication systems, which may be overcome with the aid of efficient error correction codes. In this thesis, we design error correction codes for the quantum domain, which is an intricate journey from the realm of classical channel coding theory to that of the Quantum Error Correction Codes (QECCs). Since quantum-based communication systems are capable of supporting the transmission of both classical and quantum information, we initially focus our attention on the code design for entanglementassisted classical communication over the quantum depolarizing channel. We conceive an Extrinsic Information Transfer (EXIT) chart aided near-capacity classical-quantum code design, which invokes a classical Irregular Convolutional Code (IRCC) and a Unity Rate Code (URC) in conjunction with our proposed soft-decision aided SuperDense Code (SD). Hence, it is referred to as an ‘IRCC-URCSD’ arrangement. The proposed scheme is intrinsically amalgamated both with 2-qubit as well as 3-qubit SD coding protocols and it is benchmarked against the corresponding entanglement-assisted classical capacity. Since the IRCC-URC-SD scheme is a bit-based design, it incurs a capacity loss. As a further advance, we design a symbol based concatenated code design, referred to as a symbol-based ‘CC-URC-SD’, which relies on a single-component classical Convolutional Code (CC). Additionally, for the sake of reducing the associated decoding complexity, we also investigate the impact of the constraint length of the convolutional code on the achievable performance. Our initial designs, namely IRCC-URC-SD and CC-URC-SD, exploit redundancy in the classical domain. By contrast, QECCs relying on the quantum-domain redundancy are indispensable for conceiving a quantum communication system supporting the transmission of quantum information and also for quantum computing. Therefore, we next provide insights into the transformation from the family of classical codes to the class of quantum codes known as ‘Quantum Stabilizer Codes’ (QSC), which invoke the classical syndrome decoding. Particularly, we detail the underlying quantum-to classical isomorphism, which facilitates the design of meritorious families of QECCs from the known classical codes. We further study the syndrome decoding techniques operating over classical channels, which may be exploited for decoding QSCs. In this context, we conceive a syndrome-based block decoding approach for the classical Turbo Trellis Coded Modulation (TTCM), whose performance is investigated for transmission over an Additive White Gaussian Noise (AWGN) channel as well as over an uncorrelated Rayleigh fading channel. Pursuing our objective of designing efficient QECCs, we next consider the construction of Hashingbound-approaching concatenated quantum codes. In this quest, we appropriately adapt the conventional non-binary EXIT charts for Quantum Turbo Codes (QTCs) by exploiting the intrinsic quantumto- classical isomorphism. We further demonstrate the explicit benefit of our EXIT-chart technique for achieving a Hashing-bound-approaching code design. We also propose a generically applicable structure for Quantum Irregular Convolutional Codes (QIRCCs), which can be dynamically adapted to a specific application scenario with the aid of the EXIT charts. More explicitly, we provide a detailed design example by constructing a 10-subcode QIRCC and use it as an outer code in a concatenated quantum code structure for evaluating its performance. Working further in the direction of iterative code structures, we survey Quantum Low Density Parity Check (QLPDC) codes from the perspective of code design as well as in terms of their decoding algorithms. Furthermore, we propose a radically new class of high-rate row-circulant Quasi-Cyclic QLDPC (QC-QLDPC) codes, which can be constructed from arbitrary row-circulant classical QC LDPC matrices. We also conceive a modified non-binary decoding algorithm for homogeneous Calderbank-Shor-Steane (CSS)-type QLDPC codes, which is capable of alleviating the problems imposed by the unavoidable length-4 cycles. Our modified decoder outperforms the state-of-the-art decoders in terms of their Word Error Rate (WER) performance, despite imposing a reduced decoding complexity. Finally, we intricately amalgamate our modified decoder with the classic Uniformly-ReWeighted Belief Propagation (URW-BP) for the sake of achieving further performance improvement.
With rapid developments in quantum hardware comes a push towards the first practical applications. While fully fault-tolerant quantum computers are not yet realized, there may exist intermediate forms of error correction that enable practical applications. In this work, we consider the idea of post-processing error decoders using existing quantum codes, which mitigate errors on logical qubits using post-processing without explicit syndrome measurements or additional qubits beyond the encoding overhead. This greatly simplifies the experimental exploration of quantum codes on real, near-term devices, removing the need for locality of syndromes or fast feed-forward. We develop the theory of the method and demonstrate it on an example with the perfect [[5, 1, 3]] code, which exhibits a pseudo-threshold of p ≈ 0.50 under a single qubit depolarizing channel applied to all qubits. We also provide a demonstration of improved performance on an unencoded hydrogen molecule.