In this paper, we propose quantum [[10, 10, 2]] stabilizer codes constructed from the existing quantum [[5, 5, 1]] Hamming codes. The proposed quantum [[10, 10, 2]] codes satisfy the symplectic inner product (SIP) to ensure stabilizer generators commute each other for successful detections and corrections. A comprehensive end-to-end construction from parity-check matrix (PCM) of classical codes is provided to ensure quantum error correction codes (QECC) with higher information. We perform a series of computer simulations to evaluate proposed quantum [[10, 10, 2]] codes in terms of quantum word error rate (QWER) performances under the depolarizing channels. The results confirm that the proposed quantum [[10, 10, 2]] codes can successfully detect and correct all single quantum bit (qubit) errors, which is validated by the curve of QWER performances that coincide with the theoretical QWER performances. The results are expected to contribute to the development of QECC in quantum computing and communication systems.
In this paper, we propose quantum [[10, 10, 2]] stabilizer codes constructed from the existing quantum [[5, 5, 1]] Hamming codes. The proposed quantum [[10, 10, 2]] codes satisfy the symplectic inner product (SIP) to ensure stabilizer generators commute each other for successful detections and corrections. A comprehensive end-to-end construction from parity-check matrix (PCM) of classical codes is provided to ensure quantum error correction codes (QECC) with higher information. We perform a series of computer simulations to evaluate proposed quantum [[10, 10, 2]] codes in terms of quantum word error rate (QWER) performances under the depolarizing channels. The results confirm that the proposed quantum [[10, 10, 2]] codes can successfully detect and correct all single quantum bit (qubit) errors, which is validated by the curve of QWER performances that coincide with the theoretical QWER performances. The results are expected to contribute to the development of QECC in quantum computing and communication systems.
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.
We design classes of quantum low-density paritycheck (LDPC) codes, called quasi-cyclic stabilizer (QCS) codes, from conventional QC-LDPC codes. The proposed QCS codes belong to the family of non-Calderbank-Shor-Steane stabilizer codes. The QC-LDPC codes are self-orthogonal with respect to the symplectic inner product (SIP) and are constructed from submatrices of nonorthogonal Latin squares via array dispersion. The Latin squares are constructed using quadratic (non)-residue sets of prime modulus p, where p = 4n ± 1. For p = 4n - 1, two constructions, namely, Type-I-A and Type-I-B QCS codes, are proposed based on matrix superposition and matrix concatenation, respectively. For p = 4n +1, Type-II QCS codes are proposed based on permutations of a base matrix. We show that the parity-check matrix for Type-I-B and Type-II QCS codes is self-orthogonal with respect to the SIP for all orders of circulant permutation matrix. This resulting in ensembles of QCS codes characterized by a single base matrix. We show that the minimum distance of Type-II QCS codes can be lower bounded by the minimum distance of the QC-LDPC codes. Simulation results show that the proposed QCS codes outperform some codes in the literature with a noteworthy low-error floor, below 10-7, over quantum depolarizing channels.
Stabilizer codes are often constructed within the Calderbank--Shor--Steane (CSS) framework, where two mutually orthogonal binary classical codes define X and Z-type stabilizer generators. While this structure is algebraically convenient, additional non-CSS constraints may help suppress low-weight logical operators and improve decoding performance in the finite-length regime. We thus introduce quantum XYZ stabilizer codes, whose parity-check matrix (PCM) is built from three pairwise orthogonal binary PCMs associated with X-, Y-, and Z-type stabilizer generators. A nontrivial point is that an XYZ code instance is not automatically genuinely non-CSS: the same stabilizer group may admit a CSS generating set. We characterize this collapse, obtaining algebraic and rank conditions for deciding when the Y-type checks are redundant and when they define genuinely non-CSS stabilizer constraints. We also derive upper and lower bounds on the quantum minimum distance, including bounds for mixed Pauli logical operators. The novel framework includes a known non-CSS topological code, namely the XYZ2 hexagonal code, and yields also sparse finite-length quantum low-density parity-check (qLDPC) constructions from intersecting-subset and quasi-dyadic code families. Simulations under depolarizing code-capacity noise and quaternary belief propagation decoding show that the proposed XYZ qLDPC instances can outperform representative CSS qLDPC instances with similar finite-length parameters.
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.
We construct a new family of Calderbank-Shor-Steane (CSS) codes using the generator and parity-check matrices of Low-Density Generator Matrix (LDGM) codes, with row operations applied to both matrices in order to achieve the desired quantum rate. Decoding is performed in an iterative manner, by applying message passing over the associated graph, and discrete Density Evolution (DDE) is used to optimize performance in the depolarizing channel. The proposed construction offers high flexibility and easiness in the design, producing quantum codes that possess excellent error correction capabilities. By properly designing the structure of the code, we are able to control and bound the weight of the stabilizer generators to a small value, which results in codes particularly well suited for fault-tolerant quantum computation. At the same time, these codes achieve very good performance in terms of error correction capability.
This paper proposes simple quantum [[9],[1],[3]] error correction codes for quantum systems beyond qubits, focusing on quantum ternary digits (qutrits) that represent the simplest higher-dimensional systems for future quantum technologies. Although qutrit-based systems offer larger capacity information over 2-dimensional systems, the challenge of identifying commuting stabilizers for constructing error correction codes becomes more difficult as the dimensionality increases or when working with a large number of qutrits. We design qutrit codes based on the generalized Pauli group and investigate the underlying structure, including shift, phase, and rotation operators. We successfully create suitable stabilizer codes that detect and correct qutrit errors. Furthermore, we (i) identify operators in qutrit, which are selected from various matrix interpretations for higher-dimensional systems, (ii) propose stabilizers for 9-qutrit error correction codes, and (iii) design a new 9-qutrit quantum circuit for communications. Our study is expected to contribute a foundational basis for possible future theoretical and experimental advancements in quantum computing, communications, and code-based cryptography in higher dimensional systems.