Quantum communication channels benefit from nonbinary entanglement-assisted stabilizer codes using preshared entangled states for achieving better error correction capability compared to those that do not use preshared entanglement, making them indispensable for realizing large-scale quantum computing and communication systems over qudits. We provide a previously unreported design architecture of the syndrome computation unit for qudit stabilizer codes based on classical additive codes using discrete Fourier transform gates, $\mathrm{ADD}$ gates, and multiplication gates. The proposed syndrome computation circuit architectures are necessary toward the implementable realization of such entanglement-assisted and -unassisted qudit stabilizer codes within the quantum transceiver system. We further provide an equivalent design architecture of the syndrome computation unit that decomposes into two syndrome computation units based on X errors and Z errors separately for entanglement-assisted and -unassisted qudit CSS codes. The proposed quantum error correction architectures are useful for building high-density coded quantum memories for archival quantum storage.
Quantum communication channels benefit from nonbinary entanglement-assisted stabilizer codes using preshared entangled states for achieving better error correction capability compared to those that do not use preshared entanglement, making them indispensable for realizing large-scale quantum computing and communication systems over qudits. We provide a previously unreported design architecture of the syndrome computation unit for qudit stabilizer codes based on classical additive codes using discrete Fourier transform gates, ADD gates, and multiplication gates. The proposed syndrome computation circuit architectures are necessary toward the implementable realization of such entanglement-assisted and -unassisted qudit stabilizer codes within the quantum transceiver system. We further provide an equivalent design architecture of the syndrome computation unit that decomposes into two syndrome computation units based on X errors and Z errors separately for entanglement-assisted and -unassisted qudit CSS codes. The proposed quantum error correction architectures are useful for building high-density coded quantum memories for archival quantum storage.
Quantum coding schemes over qudits using preshared entanglement between the encoder and decoder can provide better error correction capability than without it. In this article, we develop procedures for constructing encoding operators for entanglement-unassisted and entanglement-assisted qudit stabilizer codes over Fpk, with p prime and k ≥ 1 from first principles, generalizing prior works on qubit-based codes and codes that work over Fp. We also provide quantum encoding architectures based on the proposed encoding procedures using one and two qudit gates, useful toward realizing coded quantum computing and communication systems using qudits.
Achieving scalable, fault-tolerant quantum computation requires quantum memory architectures that minimize error correction overhead while preserving coherence. This work presents a framework for high-dimensional qudit memory in 153Eu:Y2SiO5, integrating three core mechanisms: (i) non-destructive syndrome extraction, using spin-echo sequences to encode error syndromes without direct measurement; (ii) adaptive quantum Fourier transform (QFT) for error identification, leveraging frequency-space transformations to reduce gate complexity; and (iii) coset-based fault-tolerant correction, factorizing large stabilizer-like unitaries into modular operations to confine error propagation. By combining generalized stabilizer formalism, Weyl-Heisenberg operators, and finite-group coset decompositions, we develop a qudit error correction scheme optimized for solid-state quantum memory. This approach circumvents resource-intensive multi-qubit concatenation, enabling scalable, long-lived quantum storage with efficient state retrieval and computational redundancy. These results provide a pathway toward practical fault-tolerant architectures for rare-earth-ion-doped quantum memories.
Quantum computers are a revolutionary class of computational platforms with applications in combinatorial and global optimization, machine learning, and other domains involving computationally hard problems. While these machines typically operate on qubits—quantum information elements that can occupy superpositions of the basis∣0⟩and∣1⟩states—recent advances have demonstrated the practical implementation of higher-dimensional quantum systems (qudits) across various hardware platforms. In these hardware realizations, the higher-order states are less stable and thus remain coherent for a shorter duration than the basis∣0⟩and∣1⟩states. Moreover, formal methods for designing efficient encoder circuits for these systems remain underexplored. This limitation motivates the development of efficient circuit techniques for qudit systems (d-level quantum systems). Previous works have typically established generating gate sets for higher-dimensional codes by generalizing the methods used for qubits. In this work, we introduce a systematic framework for optimizing encoder circuits for prime-dimension stabilizer codes. This framework is based on novel generating gate sets whose elements map directly to efficient Clifford gate sequences. We demonstrate the effectiveness of this method on key codes, achieving a 13–44% reduction in encoder circuit gate count for the qutrit (d = 3)[[9,5,3]]3,[[5,1,3]]3, and[[7,1,3]]3codes, and a 9–21% reduction for the ququint (d = 5)[[10,6,3]]5code when compared to prior work. We also achieved circuit depth reductions up to 42%.
We propose fault-tolerant quantum encoders for Calderbank-Shor-Steane (CSS) and non-CSS codes over qudits by utilizing g-partite maximally entangled states. The qudits are grouped into g contiguous blocks such that entangled states are shared across the blocks, and multiqudit gates are applied locally to restrict the error propagation within each block. We also propose a transversal fault-tolerant block decoder to decode the blocks individually and recover the complete encoded quantum information. The optimum value of g to obtain an improvement in the probability of error propagation of the proposed encoders against non-fault-tolerant schemes is analytically determined for application in the encoding circuit. The proposed fault-tolerant scheme with entanglement assistance scales linearly with the number of g-partite entangled states and overcomes the limitations of diminishing coding rates for higher levels of concatenation within the concatenated coding scheme. We validate the proposed ideas by constructing fault-tolerant encoding circuits for the non-CSS [[5,1,3]]3 code and the [[7,1,3]]3 CSS code through illustrative examples. We reduced the probability of error propagation from 0.248 to 0.212 for the [[5,1,3]]3 code and from 0.207 to 0.174 for the [[7,1,3]]3 code. For example, over [[100,70]]2 quantum code without any fault tolerance, using three-partite entangled pairs, one can get two orders of improvement (1e-2) in the probability of error propagation compared to the non-fault-tolerant encoder by encoding over 190 physical qubits with the same number of stabilizers.
Quantum computing’s seemingly perpetual promise of nearness still has a few hurdles \nto surmount before it can become a reality. Among these hurdles is that of protection of \ninformation from random errors. A potential solution for this challenge is stabilizer codes, \nwhich are the analog of classical linear error-correcting codes, however, with an additional \naxis of error possibility. By and large, quantum computing is discussed in the standard \nqubit, or two-level, language, however, it is worth considering the case of qudits, or more \nthan two-level systems. Often times results follow simply from some form of algebraic \nextension: typically group theory or linear algebra. In this work we consider some features \nthat are not immediately apparent from that approach, more often appealing to physical \nintuition to guide our mathematical ideas, then proving these ideas using the language of \nqudit operators. \n \nHere we consider two particular previously unexplored ideas. The first idea is that of \nembedding and inscribing of codes into spaces of different sizes than the stabilizer code \nwas originally designed for. Here, we show that all codes can be embedded, and that for \ninfinitely many primes we can in fact guarantee that the distance is at least preserved–a \nsomewhat surprising result. The second idea is, in a way an application of those presented \nin the first idea, that of turning many stabilizer codes into hybrid codes by taking advantage \nof relabelling of syndromes and the rapid increase in syndrome space upon using codes in \nlarge spaces. Both of these are somewhat useful in their current forms, however, with some \nadditional mathematical work could be turned into potentially very powerful tools for the \nprotection of quantum information. We finish off by, and along the way, mentioning various \nfuture directions to carry work on this.
We develop the theory of entanglement-assisted quantum error-correcting (EAQEC) codes, a generalization of the stabilizer formalism to the setting in which the sender and receiver have access to preshared entanglement. Conventional stabilizer codes are equivalent to self-orthogonal symplectic codes. In contrast, EAQEC codes do not require self-orthogonality, which greatly simplifies their construction. We show how any classical binary or quaternary block code can be made into an EAQEC code. We provide a table of best known EAQEC codes with code length up to 10. With the self-orthogonality constraint removed, we see that the distance of an EAQEC code can be better than any standard quantum error-correcting code with the same fixed net yield. In a quantum computation setting, EAQEC codes give rise to catalytic quantum codes, which assume a subset of the qubits are noiseless. We also give an alternative construction of EAQEC codes by making classical entanglement-assisted codes coherent.