Qudits naturally correspond to multilevel quantum systems, which have the potential to enable more efficient quantum information processing, but their reliability is contingent upon quantum error-correction capabilities. In this paper, we present a general procedure for constructing error-correcting qudit codes through the irreducible representations of SU ( d ) for any odd integer d ≥ 3 . Using the Weyl character formula and inner product of characters, we deduce the relevant branching rules, through which we identify the physical Hilbert spaces that contain valid code spaces. We then discuss how two forms of permutation invariance and the Heisenberg-Weyl symmetry of su ( d ) can be exploited to simplify the construction of error-correcting codes. Finally, we use our procedure to construct an infinite class of error-correcting codes encoding a logical qudit into ( d − 1 ) 2 physical qudits.
Qudits naturally correspond to multilevel quantum systems, which have the potential to enable more efficient quantum information processing, but their reliability is contingent upon quantum error-correction capabilities. In this paper, we present a general procedure for constructing error-correcting qudit codes through the irreducible representations of SU ( d ) for any odd integer d ≥ 3 . Using the Weyl character formula and inner product of characters, we deduce the relevant branching rules, through which we identify the physical Hilbert spaces that contain valid code spaces. We then discuss how two forms of permutation invariance and the Heisenberg-Weyl symmetry of su ( d ) can be exploited to simplify the construction of error-correcting codes. Finally, we use our procedure to construct an infinite class of error-correcting codes encoding a logical qudit into ( d − 1 ) 2 physical qudits.
Universal quantum computers require fault-tolerant logical qudits, as qudits naturally align with the simulation of multi-level physical systems. Here, we present a general framework and working examples for encoding fault-tolerant logical qudits in finite-dimensional spin systems. We construct distance-3, distance-5 codewords, and general 2t+1-distance codes that can be implemented using a single physical qudit or a small number of coupled qudits for higher distances, while requiring a Hilbert space dimension significantly smaller than conventional constructions based on multiple logical qubits. Logical operations and error correction protocols can be implemented with polynomial scaling in the number of elementary operations. We further discuss schematic designs for physical implementation and required single-gate fidelities, which are compatible with current spin qudit platforms. This strategy provides a resource-efficient path toward realizing fault-tolerant logical qudits in finite multi-level physical systems.
Dynamical decoupling is a long-established and effective way to suppress unwanted interactions in qubit systems, enabling advances in fields ranging from quantum metrology to quantum computing. For general qudit systems, however, comparable protocols remain rare, mainly because Hamiltonian engineering in higher dimensions lacks the geometric intuition available for qubits. Here we present a general framework for dynamical decoupling in qudit systems, based on Lie group representation theory. By extending the group theory approach to dynamical decoupling, we show how decoupling groups can be systematically identified among the finite subgroups of SU(d) by analyzing their access to the irreducible components of the operator space. As an application, we construct new pulse sequences for interacting qutrit systems based on finite subgroups of SU(3), and show how subgroup factorizations and group orientations can be exploited to obtain shorter and more experimentally practical protocols for spin-1 systems with large zero-field splitting. We further show that the same symmetry-based framework yields quantum error-correcting codes: whenever a finite subgroup of SU(d) acts as a decoupling group for the relevant set of operators, the associated one-dimensional symmetry sectors define codespaces satisfying the Knill-Laflamme conditions, thereby unifying dynamical decoupling and quantum error correction in multi-level quantum systems.
A quantum code is a subspace of a Hilbert space of a physical system chosen to be correctable against a given class of errors, where information can be encoded. Ideally, the quantum code lies within the ground space of the physical system. When the physical model is the Heisenberg ferromagnet in the absence of an external magnetic field, the corresponding ground space contains all permutation-invariant states. We use techniques from combinatorics and operator theory to construct families of permutation-invariant quantum codes. These codes have length proportional to t2; one family of codes perfectly corrects arbitrary weight t errors, while the other family of codes approximately correct t spontaneous decay errors. The analysis of our codes' performance with respect to spontaneous decay errors utilizes elementary matrix analysis, where we revisit and extend the quantum error correction criterion of Knill and Laflamme, and Leung, Chuang, Nielsen and Yamamoto.
We investigate Permutation-Invariant (PI) quantum error-correcting codes encoding a logical qudit of dimension dL in PI states using physical qudits of dimension dP. We extend the Knill--Laflamme (KL) conditions for d−1 deletion errors from qubits to qudits and investigate numerically both qubit (dL=dP=2) and qudit (dL>2 or dP>2) PI codes. We analyze the scaling of the block length n in terms of the code distance d, and compare to existing families of PI codes due to Ouyang, Aydin--Alekseyev--Barg (AAB) and Pollatsek--Ruskai (PR). Our three main findings are: (i) We conjecture that qubit PI codes correcting up to d−1 deletion errors have block length n(d)≥(3d2+1)/4, which implies an upper bound d≤12n−3/3 on their code distance, and that PR codes can saturate this bound. (ii) For qudit PI codes encoding a single qudit we numerically observe that increasing dP results in n monotonically decreasing and approaching the quantum Singleton bound n(d)≥2d−1. (iii) We propose a semi-analytic extension of the qubit AAB construction to qudits that finds explicit solutions by solving a linear program. Our results therefore provide key insights into lower bounds on the block length scaling of both qubit and qudit PI codes, and demonstrate the benefit of increased physical local dimension in the context of PI codes.
The implementation of practical error correction protocols is essential for deployment of quantum information technologies. Ways of exploiting high-spin nuclei, which have multilevel quantum resources, have attracted interest in this context because they offer additional Hilbert space dimensions in a spatially compact and theoretically efficient structure. We present a quantitative analysis of the performance of a spin-qudit-based error-correctable quantum memory, with reference to the actual Hamiltonians of several potential candidate systems. First, the ideal code-word implemented on a spin-7/2 nucleus, which provides first-order Pauli- X , Y and Z error correction, has intrinsic infidelity due to mixed eigenstates under realistic conditions. We confirm that expansion to a spin-9/2 system with tailored code words can compensate for this infidelity. Second, we claim that electric field fluctuations, which are inevitable in real systems, should also be considered as a noise source, and we illustrate an encoding and decoding scheme for a multi-spin-qudit-based error correction code that can simultaneously compensate for both electric- and magnetic-field perturbations. Such strategies are important as we move beyond the current noisy-intermediate quantum era, and fidelities above two or three nines become crucial for the implementation of quantum technologies.
We connect the structure theorem for quasiprobability representations to quantum error correction. For a code space invariant under a subgroup of an extended Weyl-Heisenberg group, the Brif-Mann construction gives a semi-functorial representation whenever its kernel satisfies the Stratonovich-Weyl axioms, and composes on channels covering full error-correction cycles. With phase-space parity available, a kernel other than the ordinary Wigner one exists iff the code space is invariant under a lattice of displacements: among compact-phase-space bosonic codes, this is only the ideal Gottesman-Kitaev-Preskill family; in finite dimension, this is every qudit stabiliser code, whose negative volume for odd prime dimension is the syndrome-averaged logical magic. For rotation-symmetric, one-photon, non-Pauli qudit and spin codes only the ordinary kernel remains, its negativity reflecting the physical carrier, not logical content; a sector-graded representation built from recovery isometries restores a resource reading. We add a channel atlas, a closed-form magic lifetime under displacement noise, a Kirkwood-Dirac layer, and finite-energy corrections.