Abstract Recent advances in quantum communication demand the development of a robust quantum infrastructure for secure data transmission across multimodal networks. Quantum communication transmits information using quantum states (qubits), which act as fundamental units and are vulnerable to noise, decoherence, and hardware imperfections in optical networks. This paper presents a fault-tolerant framework for quantum communication. It integrates Clifford hierarchy-based diagonal gates with Calderbank-Shor-Steane (CSS) codes to ensure error-resilient operation. The proposed scheme uses recursive algebraic structures, including transversal Clifford gates (T, CZ, CCZ). It also employs concatenated Reed-Muller code families to build logical operators that maintain fault tolerance under realistic channel conditions. The recursive algebraic structure of the Clifford hierarchy and the application of quantum Reed-Muller code families in this study provide reliable fault tolerant links and quantum key distribution (QKD) mechanisms for effective noise mitigation. Channel modeling is performed for both fiber-based and free-space optical links, incorporating depolarizing noise, photon loss, and detector dark counts to evaluate system reliability. The Steane and Reed-Muller codes effectively depolarize noise and minimize the logical error rate to an optimum level. Simulation results demonstrate that the proposed coding scheme reduces logical error rates by nearly an order of magnitude compared to unencoded qubits, maintains state fidelity above 0.9 under moderate loss conditions, and achieves secure key rates well beyond the 7% QBER threshold of the conventional BB84 protocol. The results extend the operational noise tolerance of QKD systems under realistic fiber-based conditions and highlight the framework’s adaptability to free-space optical channels. Overall, the developed framework bridges the gap between mathematical fault-tolerant code design and practical quantum communication architectures.
Abstract Recent advances in quantum communication demand the development of a robust quantum infrastructure for secure data transmission across multimodal networks. Quantum communication transmits information using quantum states (qubits), which act as fundamental units and are vulnerable to noise, decoherence, and hardware imperfections in optical networks. This paper presents a fault-tolerant framework for quantum communication. It integrates Clifford hierarchy-based diagonal gates with Calderbank-Shor-Steane (CSS) codes to ensure error-resilient operation. The proposed scheme uses recursive algebraic structures, including transversal Clifford gates (T, CZ, CCZ). It also employs concatenated Reed-Muller code families to build logical operators that maintain fault tolerance under realistic channel conditions. The recursive algebraic structure of the Clifford hierarchy and the application of quantum Reed-Muller code families in this study provide reliable fault tolerant links and quantum key distribution (QKD) mechanisms for effective noise mitigation. Channel modeling is performed for both fiber-based and free-space optical links, incorporating depolarizing noise, photon loss, and detector dark counts to evaluate system reliability. The Steane and Reed-Muller codes effectively depolarize noise and minimize the logical error rate to an optimum level. Simulation results demonstrate that the proposed coding scheme reduces logical error rates by nearly an order of magnitude compared to unencoded qubits, maintains state fidelity above 0.9 under moderate loss conditions, and achieves secure key rates well beyond the 7% QBER threshold of the conventional BB84 protocol. The results extend the operational noise tolerance of QKD systems under realistic fiber-based conditions and highlight the framework’s adaptability to free-space optical channels. Overall, the developed framework bridges the gap between mathematical fault-tolerant code design and practical quantum communication architectures.
Abstract Storing quantum information in a quantum error correction code can protect it from errors, but the ability to transform the stored quantum information in a fault tolerant way is equally important. Logical Pauli group operators can be implemented on Calderbank-Shor-Steane (CSS) codes, a commonly-studied category of codes, by applying a series of physical Pauli X and Z gates. Logical operators of this form are fault-tolerant because each qubit is acted upon by at most one gate, limiting the spread of errors, and are referred to as transversal logical operators. Identifying transversal logical operators outside the Pauli group is less well understood. Pauli operators are the first level of the Clifford hierarchy which is deeply connected to fault-tolerance and universality. In this work, we study transversal logical operators composed of single- and multi-qubit diagonal Clifford hierarchy gates. We demonstrate algorithms for identifying all transversal diagonal logical operators on a CSS code that are more general or have lower computational complexity than previous methods. We also show a method for constructing CSS codes that have a desired diagonal logical Clifford hierarchy operator implemented using single qubit phase gates. Our methods rely on representing operators composed of diagonal Clifford hierarchy gates as diagonal XP operators and this technique may have broader applications.
Fault tolerant quantum computation is a critical step in the development of practical quantum computers. Unfortunately, not every quantum error correcting code can be used for fault tolerant computation. Rengaswamy et. al. define CSS-T codes, which are CSS codes that admit the transversal application of the T gate, which is a key step in achieving fault tolerant computation. They then present a family of quantum Reed-Muller fault tolerant codes. Their family of codes admits a transversal T gate, but the asymptotic rate of the family is zero. We build on their work by reframing their CSS-T conditions using the concept of self-orthogonality. Using this framework, we define an alternative family of quantum Reed-Muller fault tolerant codes. Like the quantum Reed-Muller family found by Rengaswamy et. al., our family admits a transversal T gate, but also has a nonvanishing asymptotic rate.\nWe prove three key results in our search for a Reed-Muller CSS-T family with a nonvanishing rate. First, we show an equivalence between a code containing a self-dual subcode and the dual of that code being self-orthogonal. This allows us to more easily determine if a pair of codes define a CSS-T code. Next, we show that if C1 and C2 are both Reed-Muller codes that form a CSS-T code, C1 must be self-orthogonal. This limits the rate of any family that is constructed solely from Reed-Muller codes. Lastly, we define a family of CSS-T codes by choosing C1 = RM(r, 2r + 1) and C2 = RM(0, 2r + 1) for some nonnegative integer r. We show that this family has an asymptotic rate of 1/2, and show that it is the only possible CSS-T family constructed only from Reed-Muller codes where C1 is self dual.
Quantum key distribution (QKD) enables secure underwater communications essential for maritime infrastructure. Underwater optical channels introduce substantial photon loss (erasures) and ambient noise that degrade QKD performance. This paper investigates whether two four-qubit Calderbank-Shor-Steane (CSS) quantum error correction codes (QECC) mitigate these impairments in vertical underwater communication BB84 QKD protocol. After developing a comprehensive stochastic channel model incorporating photon loss, geometric spreading, and solar noise, we assess the viability of QECC through the analytical study of the quantum bit error rate (QBER) and the secure key rate (SKR) with and without security depending on the signal-to-noise ratio (SNR) validated against Monte Carlo simulations. The standard four-qubit CSS code achieves a 3 dB SNR gain at the QBER 11% security threshold; the discard code variant achieves 4.5 dB. However, QECC includes an encoding overhead that reduces the SKR. We demonstrate a crucial relationship between the SKR and the probability of arrival of the sent photon. Analysis shows that QECC is beneficial exclusively in marginal SNR regimes; at high SNR, raw BB84 dominates. For an ocean Type III Jerlov water scenario with sunlight coming from the sun located at the top of the atmosphere, we identify operational depth-range windows where QECC enables communication otherwise infeasible. This analysis establishes that error correction deployment must be scenario-dependent: extend operational range at the cost of throughput when SNR is marginal, or prioritize key generation rates at high SNR.
This paper introduces a framework for constructing Calderbank-Shor-Steane (CSS) codes that support fault-tolerant logical transversal Z-rotations. Using this framework, we obtain asymptotically good CSS codes that fault-tolerantly realize the logical transversal Clifford group (i.e., transversal single-qubit Clifford gates are realized within a single code block, while transversal two-qubit Clifford gates are realized across two identical code blocks). Furthermore, investigating CSS-T codes, we: (a) demonstrate asymptotically good CSS-T codes wherein the transversal T realizes the logical transversal S†; (b) show that the condition C2∗C1⊆C1⊥ is necessary but not sufficient for CSS-T codes; and (c) revise the characterizations of CSS-T codes wherein the transversal T implements the logical identity and the logical transversal T, respectively.
Storing quantum information in a quantum error correction code can protect it from errors, but the ability to transform the stored quantum information in a fault tolerant way is equally important. Logical Pauli group operators can be implemented on Calderbank-Shor-Steane (CSS) codes, a commonly-studied category of codes, by applying a series of physical Pauli X and Z gates. Logical operators of this form are fault-tolerant because each qubit is acted upon by at most one gate, limiting the spread of errors, and are referred to as transversal logical operators. Identifying transversal logical operators outside the Pauli group is less well understood. Pauli operators are the first level of the Clifford hierarchy which is deeply connected to fault-tolerance and universality. In this work, we study transversal logical operators composed of single- and multi-qubit diagonal Clifford hierarchy gates. We demonstrate algorithms for identifying all transversal diagonal logical operators on a CSS code that are more general or have lower computational complexity than previous methods. We also show a method for constructing CSS codes that have a desired diagonal logical Clifford hierarchy operator implemented using single qubit phase gates. Our methods rely on representing operators composed of diagonal Clifford hierarchy gates as diagonal XP operators and this technique may have broader applications.
Fault-tolerant quantum computation requires error-correcting codes that not only protect quantum information but also support useful logical operations with practical physical realizations. Here we develop a general framework in which the stabilizer syndromes of commuting Pauli rotations directly organize logical-gate design. We derive an exact codespace-preservation condition valid at arbitrary rotation angles, providing a common stabilizer-level description for CSS and non-CSS codes and for both Clifford and non-Clifford operations. A particularly useful equal-syndrome specialization reduces Clifford gate design to stabilizer factorization, yields a general generator weight bound of ⌈d/2⌉ for distance-d codes, and systematically extends to logical rotations higher in the Clifford hierarchy. Applied to bivariate-bicycle (BB) and non-CSS Mirror qLDPC codes, the framework gives explicit low-weight Clifford and controlled-phase gates. Our implementations yield new low-weight gates, including an addressed T gate on the [[16,4,4]] BB code, T-type rotations on the [[64,8,8]] BB code, and CCZ-, CS-, and T-type operations on the [[36,6,6]] Mirror code. We also develop analytic constructions using Clifford algebra bases, which turn the syndrome constraints into explicit algebraic relations and provide a systematic route to logical addressability. We use this approach to analytically construct addressed CNOT, SWAP, controlled, and CCZ-type gates on recursive hypercube codes. An independent construction on middle quantum Reed--Muller codes yields CrZ gates that saturate the equal-syndrome weight bound. These results establish stabilizer syndromes as a constructive link between the stabilizer structure of a quantum code, the logical computations it can directly support, and the fault-tolerance properties of their physical realizations.