Permutation-Invariant Quantum Codes

• Permutation-Invariant Quantum Codes are quantum error-correcting codes residing in symmetric subspaces, naturally addressing deletion and synchronization errors.
• They leverage Dicke states and representation theory, applying Schur–Weyl duality and intrinsic weight enumerators to simplify error analysis.
• Explicit constructions, including gnu and polynomial methods, support transversal logical gate implementations and optimized decoding strategies.

Searching arXiv for papers on permutation-invariant quantum codes to ground the article. Permutation-invariant quantum codes are quantum error-correcting codes whose codewords lie in the symmetric subspace and are unchanged, or whose codespace is preserved, under arbitrary permutations of the physical subsystems. In the qubit setting this means the code lives in the Dicke-state sector of $(\mathbb C^2)^{\otimes N}$; in the qudit setting it lies in the symmetric subspace of $(\mathbb C^q)^{\otimes N}$, equivalently $\mathrm{Sym}^N(\mathbb C^q)$ (Ouyang, 2013, Ouyang, 2016). This symmetry makes PI codes natural for deletion, erasure, and synchronization errors, because the decoder can depend on how many subsystems were lost rather than on which positions were affected (Ouyang, 2021, Bulled et al., 9 Feb 2026). It also places PI codes at the intersection of several technical frameworks: Dicke-state combinatorics, Knill–Laflamme theory, symmetric-group and $SU(q)$ representation theory, Schur–Weyl duality, linear programming bounds, and symmetry-reduced optimization (Teixeira, 14 May 2026, Bergh et al., 29 Apr 2026).

1. Definition and state-space structure

A permutation-invariant code on $N$ qudits is any subspace stabilized by the action of the symmetric group $S_N$ that permutes the $N$ subsystems, so every codeword lies in the symmetric subspace of $(\mathbb C^q)^{\otimes N}$ (Ouyang, 2016). In the qubit case, the symmetric basis is given by Dicke states $\{|D_w^N\rangle\}_{w=0}^N$, where $|D_w^N\rangle$ is the normalized symmetric superposition of computational basis states of Hamming weight $(\mathbb C^q)^{\otimes N}$0 (Ouyang et al., 2015, Ouyang, 2021). In the qudit case, Dicke states are indexed by compositions $(\mathbb C^q)^{\otimes N}$1, and a logical basis can be written as

$(\mathbb C^q)^{\otimes N}$2

This identifies PI codes as subspaces spanned by symmetric superpositions of computational basis strings (Bond et al., 11 Mar 2026).

The symmetric subspace has dimension

$(\mathbb C^q)^{\otimes N}$3

and in the qudit representation-theoretic description it is the irreducible $(\mathbb C^q)^{\otimes N}$4-representation of highest weight $(\mathbb C^q)^{\otimes N}$5, namely

$(\mathbb C^q)^{\otimes N}$6

(Ouyang, 2016, Teixeira, 14 May 2026). This representation-theoretic perspective is especially important for recent work on intrinsic weight enumerators and MacWilliams theory, where the code is analyzed inside $(\mathbb C^q)^{\otimes N}$7 rather than in the full tensor-power space (Teixeira, 14 May 2026).

Several papers use slightly different invariance conventions. Some require every codeword to be fixed by every permutation, while others require only that the codespace be preserved by the permutation action (Ouyang et al., 2015, Ouyang, 2016). Both viewpoints place the code in the symmetric sector and lead to Dicke-state expansions. This suggests that the essential structural datum is support inside the symmetric subspace, with the precise formulation adapted to the intended error model or representation-theoretic framework.

2. Error models and Knill–Laflamme conditions in the symmetric sector

The standard error-correction criterion is the Knill–Laflamme condition

$(\mathbb C^q)^{\otimes N}$8

specialized in the PI setting to Dicke-basis coefficient constraints (Ouyang, 2016, Bond et al., 11 Mar 2026). For qudit PI codes, a generalized deletion-error operator indexed by a composition $(\mathbb C^q)^{\otimes N}$9 acts on Dicke states as

$\mathrm{Sym}^N(\mathbb C^q)$0

which yields qudit analogues of the known qubit KL deletion conditions (Bond et al., 11 Mar 2026). In a broader simplex-based formalism, the same type of expression appears as the action of $\mathrm{Sym}^N(\mathbb C^q)$1-deletion Kraus operators on symmetric occupation-number states, leading to coefficient conditions $\mathrm{Sym}^N(\mathbb C^q)$2–$\mathrm{Sym}^N(\mathbb C^q)$3 for PI, spin, and bosonic codes (Aydin et al., 24 Sep 2025).

A central simplification is that permutation symmetry removes location dependence. For deletion errors, losing an unknown set of $\mathrm{Sym}^N(\mathbb C^q)$4 particles is equivalent to erasure of any fixed $\mathrm{Sym}^N(\mathbb C^q)$5 particles, because any physical location can be permuted into any other location without changing the codeword (Ouyang, 2021). This equivalence underlies several results: any permutation-invariant code with distance $\mathrm{Sym}^N(\mathbb C^q)$6 can correct $\mathrm{Sym}^N(\mathbb C^q)$7 quantum deletions in both the qubit and qudit settings (Ouyang, 2021), and a PI code correcting $\mathrm{Sym}^N(\mathbb C^q)$8 deletions also corrects all combinations of $\mathrm{Sym}^N(\mathbb C^q)$9 Pauli errors (Aydin et al., 2023).

The deletion-channel viewpoint has also been extended to synchronization errors. For binary PI codes with logical codewords

$SU(q)$0

exact correctability conditions were derived for $SU(q)$1-insertion errors, semi-insdel errors, and full-insdel errors as explicit coefficient identities $SU(q)$2–$SU(q)$3 (Bulled et al., 9 Feb 2026). The paper proves that, for PI codes, correcting $SU(q)$4 deletions is equivalent to correcting $SU(q)$5 insertions, giving a quantum PI analogue of Levenshtein’s classical insertion/deletion equivalence (Bulled et al., 9 Feb 2026). The result is specific to the PI setting; it is not claimed as a universal theorem for arbitrary quantum codes.

A recurring misconception is that PI symmetry only helps with erasure-like noise. The literature shows a broader scope: PI codes have been analyzed for arbitrary Pauli errors (Ouyang, 2013, Aydin et al., 2023), amplitude damping and spontaneous decay (Ouyang, 2013, Ouyang et al., 2015), deletions (Ouyang, 2021, Shibayama et al., 2021), insertions and insdel errors (Bulled et al., 9 Feb 2026), and correlated collective amplitude damping (Chandra et al., 2 Jul 2026). The symmetry does not trivialize these channels, but it does reorganize the analysis into Dicke-weight, deletion-syndrome, or representation-theoretic sectors.

3. Explicit constructions and canonical code families

A foundational family is the gnu code construction, parameterized by integers $SU(q)$6 with total number of qubits $SU(q)$7. Its logical states are binomially weighted superpositions of Dicke states supported on weights $SU(q)$8: $SU(q)$9 For $N$0, these codes perfectly correct arbitrary $N$1-qubit errors; for $N$2, $N$3, and suitable $N$4, they approximately correct $N$5 spontaneous decay errors with worst-case error scaling as $N$6 (Ouyang, 2013). Shifted gnu codes introduce a shift parameter $N$7 while preserving distance $N$8 (Ouyang, 2021).

Deletion-correcting PI constructions were later organized through weight sets $N$9 and coefficient functions satisfying three conditions, denoted (D1), (D2), and (D3). Under these conditions, the induced code $S_N$0 is an $S_N$1 $S_N$2-deletion error-correcting code (Shibayama et al., 2021). A notable consequence is the first example of quantum codes correcting two or more deletion errors, and the first example correcting both multiple-qubit errors and multiple-deletion errors (Shibayama et al., 2021). Subsequent work introduced a larger explicit family $S_N$3, with length

$S_N$4

that corrects $S_N$5 arbitrary Pauli errors, $S_N$6 deletions, and $S_N$7 amplitude-damping errors in appropriate parameter regimes, and contains Ouyang’s gnu codes as special cases (Aydin et al., 2023).

The polynomial-construction approach gives a different general mechanism. For $S_N$8-qudit PI codes, one chooses a polynomial $S_N$9 with multiple roots at roots of unity and a polynomially parametrized family of Dicke types $N$0. The resulting Type A and Type B theorems construct qubit and $N$1-level PI codes, respectively, correcting $N$2 errors whenever the Dicke types have separation at least $N$3 and the polynomial degrees obey the stated bounds (Ouyang, 2016). The existence theorem shows that if

$N$4

then there exists a permutation-invariant code on $N$5 qudits of dimension $N$6 correcting $N$7 errors, and if the inequality is strict then there are uncountably many such codes (Ouyang, 2016).

A separate line of work constructs multi-qubit PI codes that encode more than one logical qubit. One family uses pairwise coprime integers $N$8, defines $N$9 and $(\mathbb C^q)^{\otimes N}$0 with $(\mathbb C^q)^{\otimes N}$1, and forms logical basis states $(\mathbb C^q)^{\otimes N}$2 as Dicke-state superpositions with binomial weights (Ouyang et al., 2015). These codes encode $(\mathbb C^q)^{\otimes N}$3 logical qubits and suppress leading-order spontaneous decay so that the residual error is second order in $(\mathbb C^q)^{\otimes N}$4 (Ouyang et al., 2015).

Recent qudit work has focused on minimal-length constructions. For every integer $(\mathbb C^q)^{\otimes N}$5, there exists a permutation-invariant code with parameters

$(\mathbb C^q)^{\otimes N}$6

inside $(\mathbb C^q)^{\otimes N}$7, and no code of dimension $(\mathbb C^q)^{\otimes N}$8 and distance at least $(\mathbb C^q)^{\otimes N}$9 exists for $\{|D_w^N\rangle\}_{w=0}^N$0 (Kubischta et al., 21 May 2026). The construction splits the distance-two KL conditions into root and Cartan parts, eliminates root errors by restricting support to the even-entry occupation layer, and solves the Cartan constraints via edge-colorings of the complete graph $\{|D_w^N\rangle\}_{w=0}^N$1: midpoint packets for odd $\{|D_w^N\rangle\}_{w=0}^N$2, and a $\{|D_w^N\rangle\}_{w=0}^N$3-factorization into perfect matchings for even $\{|D_w^N\rangle\}_{w=0}^N$4 (Kubischta et al., 21 May 2026).

4. Representation theory, intrinsic enumerators, and MacWilliams transforms

The recent representation-theoretic reformulation of PI qudit codes departs from the ordinary Hamming-weight picture. For $\{|D_w^N\rangle\}_{w=0}^N$5 qudits of local dimension $\{|D_w^N\rangle\}_{w=0}^N$6, the code lives in

$\{|D_w^N\rangle\}_{w=0}^N$7

and the relevant operator space is

$\{|D_w^N\rangle\}_{w=0}^N$8

Under conjugation by $\{|D_w^N\rangle\}_{w=0}^N$9, this space decomposes multiplicity-free as

$|D_w^N\rangle$0

so the irreducible sectors $|D_w^N\rangle$1 play the role of intrinsic error sectors (Teixeira, 14 May 2026). In this framework, intrinsic weight enumerators are indexed by $|D_w^N\rangle$2, not by ordinary Pauli or Hamming weight (Teixeira, 14 May 2026).

The intrinsic MacWilliams transform is the change of basis in the intertwiner algebra

$|D_w^N\rangle$3

between orthogonal projectors $|D_w^N\rangle$4 onto $|D_w^N\rangle$5 and twirling operators

$|D_w^N\rangle$6

built from orthonormal bases of $|D_w^N\rangle$7. Because the decomposition is multiplicity-free, $|D_w^N\rangle$8 is commutative of dimension $|D_w^N\rangle$9, and the transform is an $(\mathbb C^q)^{\otimes N}$00 scalar matrix $(\mathbb C^q)^{\otimes N}$01 with

$(\mathbb C^q)^{\otimes N}$02

(Teixeira, 14 May 2026).

A central calculation is the spectral analysis of the degree-one twirl $(\mathbb C^q)^{\otimes N}$03. Its eigenvalues on $(\mathbb C^q)^{\otimes N}$04 form an affine image of the quadratic lattice

$(\mathbb C^q)^{\otimes N}$05

and the distinctness of these eigenvalues implies that $(\mathbb C^q)^{\otimes N}$06 generates the entire intertwiner algebra (Teixeira, 14 May 2026). The row polynomials $(\mathbb C^q)^{\otimes N}$07 defined by $(\mathbb C^q)^{\otimes N}$08 satisfy a three-term recurrence coming from tensoring with the adjoint sector $(\mathbb C^q)^{\otimes N}$09, and this identifies them as a finite orthogonal polynomial system (Teixeira, 14 May 2026).

The paper then identifies the system with Racah polynomials. With parameters

$(\mathbb C^q)^{\otimes N}$10

the orthogonality weight is

$(\mathbb C^q)^{\otimes N}$11

and the MacWilliams matrix becomes a finite Racah transform: $(\mathbb C^q)^{\otimes N}$12 (Teixeira, 14 May 2026). The transform satisfies weighted orthogonality $(\mathbb C^q)^{\otimes N}$13, detailed balance $(\mathbb C^q)^{\otimes N}$14, and involutivity $(\mathbb C^q)^{\otimes N}$15 (Teixeira, 14 May 2026). These formulas provide the explicit MacWilliams matrix needed for linear-programming bounds on PI qudit codes.

This intrinsic viewpoint differs sharply from older Dicke-weight constructions but is compatible with them. A plausible implication is that the traditional Dicke-basis picture is most effective for explicit codewords and channel-adapted constructions, whereas the intrinsic $(\mathbb C^q)^{\otimes N}$16-sector picture is better suited to dual enumerators, LP bounds, and structural identities.

5. Algorithms, decoding theory, and optimization frameworks

A general error-correction theory for PI codes has recently been developed using representation theory of the symmetric group. For an $(\mathbb C^q)^{\otimes N}$17-qubit PI code of distance $(\mathbb C^q)^{\otimes N}$18, any channel whose Kraus operators have weight at most $(\mathbb C^q)^{\otimes N}$19 is correctible whenever $(\mathbb C^q)^{\otimes N}$20, and the symmetrizing lemma shows that one may replace the noise by its permutation average without losing correctability (Ouyang et al., 14 Feb 2026). Decoding uses Schur–Weyl duality,

$(\mathbb C^q)^{\otimes N}$21

followed by syndrome extraction through nested total-spin measurements $(\mathbb C^q)^{\otimes N}$22, which determine a standard Young tableau syndrome $(\mathbb C^q)^{\otimes N}$23 (Ouyang et al., 14 Feb 2026). Recovery can then proceed either by inverse quantum Schur transform plus geometric phase gates, or by teleportation from the $(\mathbb C^q)^{\otimes N}$24-code to a fresh PI ancilla (Ouyang et al., 14 Feb 2026).

Deletion and erasure errors admit simpler recovery for certain PI families. For shifted gnu codes, different deletion syndromes are orthogonal when the Dicke support spacing $(\mathbb C^q)^{\otimes N}$25 exceeds the number of deletions, and recovery reduces to a modulo Dicke-weight measurement and a correcting unitary $(\mathbb C^q)^{\otimes N}$26 (Ouyang et al., 14 Feb 2026). Related work on shifted gnu codes gives explicit encoding and decoding algorithms for quantum deletion channels, with encoding in $(\mathbb C^q)^{\otimes N}$27 and decoding in $(\mathbb C^q)^{\otimes N}$28 (Ouyang, 2021).

PI symmetry has also been turned into a computational reduction for optimization. In a Schur–Weyl-based framework for permutation-invariant SDPs, operators in $(\mathbb C^q)^{\otimes N}$29 are block-diagonalized into polynomial-size blocks, PSD constraints become blockwise PSD constraints, and operations such as tensoring, partial trace, partial transpose, and blockwise relative entropy can be carried out entirely inside the symmetric subspace (Bergh et al., 29 Apr 2026). This framework yields the symmetric seesaw method for lower-bounding channel fidelity over $(\mathbb C^q)^{\otimes N}$30 channel uses, with the iterates remaining symmetric whenever the channel and initial seeds are symmetric (Bergh et al., 29 Apr 2026). The method improved lower bounds on $(\mathbb C^q)^{\otimes N}$31 and $(\mathbb C^q)^{\otimes N}$32 for $(\mathbb C^q)^{\otimes N}$33, and was used to demonstrate non-asymptotic superactivation of quantum capacity for $(\mathbb C^q)^{\otimes N}$34 in related work (Bergh et al., 29 Apr 2026).

A different optimization program uses PI states as an ansatz for coherent information. Input states of the form

$(\mathbb C^q)^{\otimes N}$35

remain permutation-invariant under i.i.d. channels, and Schur–Weyl duality yields efficient coherent-information formulas sector by sector (Bhalerao et al., 13 Aug 2025). This allowed evaluation at at least $(\mathbb C^q)^{\otimes N}$36 channel copies for qubit channels and produced improved lower bounds on quantum capacities for several channel families, including 2-Pauli, BB84, generalized amplitude damping, dephrasure, and damping-dephasing channels (Bhalerao et al., 13 Aug 2025).

6. Transversal gates, approximate codes, and applications

Permutation-invariant codes have also been studied as non-stabilizer resources for logical gates. A spin-code-to-Dicke-state correspondence, the Dicke bootstrap, maps $(\mathbb C^q)^{\otimes N}$37-covariant spin-$(\mathbb C^q)^{\otimes N}$38 codes to $(\mathbb C^q)^{\otimes N}$39-transversal PI multiqubit codes on $(\mathbb C^q)^{\otimes N}$40 qubits while preserving distance (Kubischta et al., 2023). Using binary dihedral and generalized quaternion symmetry, this framework produces PI codes with transversal generalized phase gates, including a family with transversal $(\mathbb C^q)^{\otimes N}$41 at odd distances $(\mathbb C^q)^{\otimes N}$42 and parameters such as $(\mathbb C^q)^{\otimes N}$43, $(\mathbb C^q)^{\otimes N}$44, and $(\mathbb C^q)^{\otimes N}$45 (Kubischta et al., 2023). The paper argues that, with respect to the transversal gate group, these non-additive PI codes can outperform the best known stabilizer codes (Kubischta et al., 2023).

Measurement-free code switching uses PI codes as the non-Clifford side of a universal architecture. In this setting, stabilizer codes provide transversal Clifford gates and PI codes provide transversal non-Clifford rational $(\mathbb C^q)^{\otimes N}$46-rotations, with switching mediated by bosonic-mode geometric phase gates (Ouyang et al., 2024). Specific examples include a 7-qubit PI code supporting a logical $(\mathbb C^q)^{\otimes N}$47-rotation by $(\mathbb C^q)^{\otimes N}$48, and an 11-qubit PI code with transversal logical $(\mathbb C^q)^{\otimes N}$49 via $(\mathbb C^q)^{\otimes N}$50 (Ouyang et al., 2024).

Approximate PI codes appear in distributed architectures through the W-state code, where a logical state is delocalized symmetrically over all physical positions using an extra inactive level $(\mathbb C^q)^{\otimes N}$51 (Clayton et al., 29 Sep 2025). Because the code is permutation-invariant, exchanging any two qudits merely relabels terms in the same symmetric superposition (Clayton et al., 29 Sep 2025). The code is used as a concrete example of Distributed Approximate Quantum Error Correction, with explicit encoding and decoding circuits and an advantage in heterogeneous modular noise landscapes quantified by

$(\mathbb C^q)^{\otimes N}$52

via AM–GM (Clayton et al., 29 Sep 2025).

Noise-adapted recovery has also been developed for correlated amplitude damping. A coherent ancilla-assisted quantum error recovery construction was specialized to PI codes, and a new CAD family was introduced for collective amplitude-damping errors (Chandra et al., 2 Jul 2026). The explicit codes

$(\mathbb C^q)^{\otimes N}$53

for CAD4, and

$(\mathbb C^q)^{\otimes N}$54

for CAD9, are tailored to powers of the collective lowering operator $(\mathbb C^q)^{\otimes N}$55 (Chandra et al., 2 Jul 2026). Under global symmetric amplitude damping, CAD9 outperformed the tested short PI codes by more than one order of magnitude at $(\mathbb C^q)^{\otimes N}$56 and by more than two orders of magnitude at $(\mathbb C^q)^{\otimes N}$57 (Chandra et al., 2 Jul 2026).

PI codes have also entered magic-state distillation and cryptography. GNU codes were used to build a distillation protocol as small as two qubits, with a 2-qubit protocol achieving error threshold $(\mathbb C^q)^{\otimes N}$58 and distillation rate $(\mathbb C^q)^{\otimes N}$59 (Leitch et al., 4 Mar 2026). In quantum anonymous secret sharing, PI codes enable a sender-anonymous reconstruction protocol because decoding depends only on the number of missing shares, not on which shareholders participated (Sikand et al., 30 Apr 2026).

Across these applications, a persistent tension is visible. PI symmetry yields compact descriptions, collective-control implementations, and exact or approximate resilience against highly nonlocal error models, but it also imposes strong geometric constraints. Numerical evidence suggests that qubit PI codes obey the conjectural scaling

$(\mathbb C^q)^{\otimes N}$60

with Pollatsek–Ruskai codes plausibly saturating the bound (Bond et al., 11 Mar 2026). By contrast, increasing physical local dimension in the qudit setting can reduce block length and move PI codes toward the quantum Singleton bound $(\mathbb C^q)^{\otimes N}$61 for one-error correction (Bond et al., 11 Mar 2026). This suggests that the main asymptotic limitation may be tied more strongly to qubit PI geometry than to permutation invariance as such.