We study permutation-invariant quantum codes in the symmetric subspace $\mathrm{Sym}^n(\mathbb{C}^q) $ of $n$ qudits of local dimension $q$. For every integer $q\geq 2$, we construct a permutation-invariant code with parameters $((4,q,2))_q$. Thus four physical qudits suffice to encode one logical qudit with distance two in the symmetric sector for every local dimension. We also show, using linear-programming constraints for permutation-invariant quantum codes, that no permutation-invariant code of dimension $q$ and distance at least $2$ exists in $\mathrm{Sym}^n(\mathbb{C}^q)$ for $n\leq 3$. Hence four qudits are necessary and sufficient.
The construction has a simple representation-theoretic and combinatorial description. In the irreducible $\mathrm{SU}(q)$-module $\mathrm{Sym}^4(\mathbb{C}^q)$, the distance-two Knill-Laflamme conditions split into root and Cartan parts. By restricting supports to the even-entry occupation layer, all root-error conditions vanish automatically. The remaining Cartan conditions reduce to linear balancing constraints on packets of occupation vectors. These packets admit a natural graph-theoretic interpretation in terms of the vertices and edges of the complete graph $K_q$: for odd $q$, they are organized by the midpoint rule, while for even $q$, they are organized by a decomposition of $K_q$ into perfect matchings. In this way, the existence of minimal $((4,q,2))_q$ permutation-invariant codes is reduced to a parity-dependent edge-coloring problem on $K_q$.
We study permutation-invariant quantum codes in the symmetric subspace Symn(Cq) of n qudits of local dimension q. For every integer q≥2, we construct a permutation-invariant code with parameters ((4,q,2))q. Thus four physical qudits suffice to encode one logical qudit with distance two in the symmetric sector for every local dimension. We also show, using linear-programming constraints for permutation-invariant quantum codes, that no permutation-invariant code of dimension q and distance at least 2 exists in Symn(Cq) for n≤3. Hence four qudits are necessary and sufficient.
The construction has a simple representation-theoretic and combinatorial description. In the irreducible SU(q)-module Sym4(Cq), the distance-two Knill-Laflamme conditions split into root and Cartan parts. By restricting supports to the even-entry occupation layer, all root-error conditions vanish automatically. The remaining Cartan conditions reduce to linear balancing constraints on packets of occupation vectors. These packets admit a natural graph-theoretic interpretation in terms of the vertices and edges of the complete graph Kq: for odd q, they are organized by the midpoint rule, while for even q, they are organized by a decomposition of Kq into perfect matchings. In this way, the existence of minimal ((4,q,2))q permutation-invariant codes is reduced to a parity-dependent edge-coloring problem on Kq.
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.
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.
We derive an explicit formula for the intrinsic MacWilliams transform for permutation-invariant qudit codes. Such codes naturally live in symmetric power representations, where the relevant error sectors are determined by the irreducible decomposition of the conjugation action on the associated operator space. Using the multiplicity-free structure of this decomposition and the corresponding intertwiner algebra, we identify the intrinsic MacWilliams matrix with a finite Racah transform. The entries are given by a terminating hypergeometric series, and the rows of the matrix are Racah orthogonal polynomials with parameters determined explicitly by the block length and local dimension. Computing the spectrum of the degree-one twirl reveals that this spectrum lies on an affine quadratic lattice. Then we derive a tridiagonal multiplication rule from the representation theory of the adjoint sector. As consequences, we obtain closed-form orthogonality, detailed-balance, and involutivity identities for the transform. The resulting formula supplies an explicit MacWilliams matrix for computing linear programming bounds on permutation-invariant qudit codes.
We construct a two-parameter family of single-error-correcting seven-ququart codes with transversal A7~ symmetry, realizing the finite component of a two-qubit super-golden gate set. These ((7,4,3))4 codes encode two logical qubits and support non-Clifford operations by applying the same gate to each physical ququart.
Our method constructs invariant parent states rather than searching directly for a codespace. For n systems of local dimension q, suppose the local symmetry is a unitary t-group and the permutation symmetry is t-transitive. Maximal mixing of all marginals on at most t sites is then equivalent to linear equations in Schur--Weyl sector weights. Every feasible solution yields parents of pure distance at least t+1; puncturing one site produces a q-dimensional code of distance at least t with the prescribed local symmetry acting transversally.
Steane and the nonstabilizer Fake Steane code reveal why the construction leaves continuous freedom. They share the same prescribed transversal Clifford and permutation symmetries, and their eight-qubit purifications lie on a circle: error correction fixes two invariant-sector norms but leaves the relative phase free. Keeping the same permutation group and replacing Clifford symmetry by A7~≤SU(4) gives three fixed sector norms and two free relative phases, producing the new ququart family.
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.
This paper develops a MacWilliams-type enumerator theory and semidefinite programming bounds for intrinsic quantum codes. An intrinsic code is a subspace C⊂V of a finite-dimensional unitary representation V of a group G, with errors organized into the isotypic components of L(V) under conjugation. The Knill-Laflamme conditions persist under every G-equivariant realization of V, aallowing intrinsic bounds to apply across diverse physical encodings. Without a group action, we first construct projector and twirl quadratic enumerators for an arbitrary orthogonal decomposition of L(V), analogous respectively to the A and B enumerators of Shor-Laflamme theory. These enumerators satisfy positivity, normalization, and Knill-Laflamme inequalities, but the two superoperator families need not span the same space and therefore admit no canonical duality. A group action places their generalized versions in the same equivariant intertwiner algebra, and the change of basis between them defines the intrinsic MacWilliams transform. When the conjugation representation has multiplicities, this algebra is noncommutative and the enumerators are matrix-valued; positivity and detection are encoded by Loewner-order constraints, making the natural programming bound a semidefinite program. The multiplicity-free specialization recovers the known scalar MacWilliams transform and linear-programming framework. For symmetric powers, intrinsic depth equals ordinary qudit distance, yielding extremality certificates for permutation-invariant qudit codes. Finally, for the 27-dimensional SU(3) representation V=(2,2), an exact dual certificate proves the sharp bound K≤5 for codes detecting the full adjoint isotypic component. An explicit 5-dimensional code attains the bound and, under the canonical embedding, gives a four-qutrit distance-2 code.