Permutation-Invariant Codes: Quantum & Classical

• Permutation-invariant codes are codes whose codewords remain unchanged under any permutation of subsystems, simplifying error models in both quantum and classical settings.
• They are constructed using symmetry-adapted methods such as the Dicke basis and Schur–Weyl decomposition, which enable correction of deletion, insertion, and amplitude-damping errors.
• Applications include fault-tolerant quantum computing with transversal gate implementations, anonymous secret sharing, and optimized multiset coding for classical channels.

Permutation-invariant codes are codes whose codewords are unchanged by permutations of the underlying subsystems or symbols. In quantum information, the term usually denotes a code whose code space lies in the symmetric subspace of $(\mathbb C^q)^{\otimes n}$ and is therefore describable in a Dicke basis; in classical coding, closely related notions include permutation codes in the symmetric group and multiset codes for channels that destroy symbol order (Ouyang, 2013, Janiszczak et al., 2018, Kovačević et al., 2016). Across these settings, the central effect of permutation symmetry is the same: positional information is removed from the code description and, in many models, from the error model itself.

1. Definitions, state spaces, and symmetry models

In the quantum literature, a permutation-invariant (PI) code is a subspace

$\mathcal C \subseteq \mathrm{Sym}^n(\mathbb C^q),$

the fully symmetric subspace of $n$ qudits. For qubits, a standard basis is given by Dicke states

$$|D_w^n\rangle=\binom{n}{w}^{-1/2}\sum_{|x|=w}|x\rangle,$$

and, more generally, qudit Dicke states are indexed by occupation vectors $\lambda$ with $\sum_i \lambda_i=n$ (Ouyang, 2013, Bond et al., 11 Mar 2026). Because every Dicke state is permutation-invariant, any PI codeword is a superposition of Dicke states.

The symmetric-subspace viewpoint admits several equivalent formulations. One is the spin-theoretic picture: a spin-$j$ irreducible representation of $SU(2)$ is isomorphic to the permutation-invariant subspace of $n=2j$ qubits via the Dicke bootstrap, which maps $|j,m\rangle$ to a Dicke state (Kubischta et al., 2023). Another is the Schur–Weyl perspective, where the symmetric sector is the $\mathcal C \subseteq \mathrm{Sym}^n(\mathbb C^q),$0 block in the decomposition of $\mathcal C \subseteq \mathrm{Sym}^n(\mathbb C^q),$1 into irreducible symmetric-group and $\mathcal C \subseteq \mathrm{Sym}^n(\mathbb C^q),$2 sectors (Ouyang et al., 14 Feb 2026).

The classical literature uses related but distinct ambient spaces. A permutation code may mean a subset of $\mathcal C \subseteq \mathrm{Sym}^n(\mathbb C^q),$3 endowed with the Hamming metric,

$\mathcal C \subseteq \mathrm{Sym}^n(\mathbb C^q),$4

while multiset codes live in the simplex

$\mathcal C \subseteq \mathrm{Sym}^n(\mathbb C^q),$5

which models channels where order is irrelevant or destroyed (Janiszczak et al., 2018, Kovačević et al., 2016). These classical usages do not coincide with quantum PI codes, but they share the same organizing principle: coding over order-insensitive objects.

2. Constructive paradigms and canonical families

A large part of the theory concerns explicit families of PI quantum codes built from structured Dicke-state superpositions. The earliest major family in the supplied literature is the gnu construction, with parameters $\mathcal C \subseteq \mathrm{Sym}^n(\mathbb C^q),$6 and block length $\mathcal C \subseteq \mathrm{Sym}^n(\mathbb C^q),$7. Its logical basis states are even- and odd-weight Dicke superpositions with amplitudes proportional to square roots of binomial coefficients. When $\mathcal C \subseteq \mathrm{Sym}^n(\mathbb C^q),$8, these codes perfectly correct arbitrary $\mathcal C \subseteq \mathrm{Sym}^n(\mathbb C^q),$9-sparse errors; when $n$0 and $n$1, they approximately correct $n$2 amplitude-damping errors (Ouyang, 2013).

Subsequent constructions generalized this template in several directions. Polynomial methods produce $n$3-dimensional PI qudit codes on at least $n$4 qudits correcting $n$5 errors, and when $n$6 the construction yields an uncountable number of such codes (Ouyang, 2016). Another line of work showed that PI codes are not restricted to a single encoded qubit: one can choose $n$7 logical basis states with arithmetic separation conditions so that the code encodes $n$8 qubits while still suppressing spontaneous decay to leading order (Ouyang et al., 2015).

A more recent explicit family is the permutationally invariant family $n$9 with length $$|D_w^n\rangle=\binom{n}{w}^{-1/2}\sum_{|x|=w}|x\rangle,$$0. Under the stated parameter conditions, these codes correct $$|D_w^n\rangle=\binom{n}{w}^{-1/2}\sum_{|x|=w}|x\rangle,$$1-qubit errors, $$|D_w^n\rangle=\binom{n}{w}^{-1/2}\sum_{|x|=w}|x\rangle,$$2 deletions, and $$|D_w^n\rangle=\binom{n}{w}^{-1/2}\sum_{|x|=w}|x\rangle,$$3 amplitude-damping errors; the family contains earlier gnu codes as special cases and includes a new $$|D_w^n\rangle=\binom{n}{w}^{-1/2}\sum_{|x|=w}|x\rangle,$$4 optimal single-deletion-correcting code (Aydin et al., 2023). A distinct representation-theoretic development proves that for every $$|D_w^n\rangle=\binom{n}{w}^{-1/2}\sum_{|x|=w}|x\rangle,$$5 there exists a minimal qudit PI code with parameters $$|D_w^n\rangle=\binom{n}{w}^{-1/2}\sum_{|x|=w}|x\rangle,$$6, and that no PI code of dimension $$|D_w^n\rangle=\binom{n}{w}^{-1/2}\sum_{|x|=w}|x\rangle,$$7 and distance at least $$|D_w^n\rangle=\binom{n}{w}^{-1/2}\sum_{|x|=w}|x\rangle,$$8 exists for $$|D_w^n\rangle=\binom{n}{w}^{-1/2}\sum_{|x|=w}|x\rangle,$$9 (Kubischta et al., 21 May 2026).

The current landscape also includes numerical and semi-analytic constructions. A numerical study conjectures that qubit PI codes correcting up to $\lambda$0 deletion errors satisfy

$\lambda$1

equivalently

$\lambda$2

and reports that Pollatsek–Ruskai codes can saturate this bound (Bond et al., 11 Mar 2026). The same study numerically observes that for qudit PI codes encoding a single qudit, increasing the physical local dimension monotonically decreases the required block length and moves it toward the quantum Singleton bound $\lambda$3 (Bond et al., 11 Mar 2026).

Family or method	Representative statement	Source
gnu codes	$\lambda$4; exact $\lambda$5-error correction for $\lambda$6	(Ouyang, 2013)
Polynomial PI qudit codes	$\lambda$7-dimensional codes on at least $\lambda$8 qudits	(Ouyang, 2016)
Multi-logical PI codes	Encode $\lambda$9 qubits while suppressing spontaneous decay to leading order	(Ouyang et al., 2015)
$\sum_i \lambda_i=n$0 family	Corrects Pauli, deletion, and amplitude-damping errors under explicit conditions	(Aydin et al., 2023)
Minimal qudit PI codes	Existence of $\sum_i \lambda_i=n$1 for every $\sum_i \lambda_i=n$2; optimality of length $\sum_i \lambda_i=n$3	(Kubischta et al., 21 May 2026)

A recurring misconception is that permutation invariance forces very small logical dimension. The constructions above show otherwise: the literature contains single-qubit, multi-qubit, single-qudit, and higher-dimensional logical encodings (Ouyang et al., 2015, Ouyang, 2016, Kubischta et al., 21 May 2026).

3. Error models, distance, and general correction theory

Permutation symmetry has a particularly strong effect on deletion-type noise. For PI codes, deletion errors behave like erasures because the code is insensitive to which subsystems were lost. One paper states this as a general principle: any PI code of distance $\sum_i \lambda_i=n$4 can correct $\sum_i \lambda_i=n$5 quantum deletions in both qubit and qudit settings (Ouyang, 2021). Another develops a direct weight-shell construction $\sum_i \lambda_i=n$6 and gives the first explicit quantum codes correcting two or more deletions, as well as the first examples that correct both multiple-qubit errors and multiple deletions (Shibayama et al., 2021).

This deletion/erasure equivalence extends further. For PI codes, insertion and deletion are equivalent in the same spirit as classical Levenshtein equivalence: exact algebraic criteria $\sum_i \lambda_i=n$7 and $\sum_i \lambda_i=n$8 characterize $\sum_i \lambda_i=n$9-insertion correction, and analogous conditions $j$0–$j$1 characterize semi-insdel and full-insdel correction (Bulled et al., 9 Feb 2026). This resolves a longstanding synchronization-error question in the PI setting.

For Pauli-type noise, the Knill–Laflamme conditions can be specialized sharply. The paper introducing the $j$2 family generalizes the Pollatsek–Ruskai $j$3 conditions to arbitrary $j$4, giving explicit coefficient constraints $j$5–$j$6 on Dicke amplitudes $j$7 for a real two-dimensional PI code to correct all $j$8-qubit errors (Aydin et al., 2023). This provides a concrete algebraic test for exact correctability inside the symmetric sector.

A broader conceptual milestone is the first general theory of quantum error correction for PI codes. Using representation theory of the symmetric group, it proves a symmetrizing lemma: if a PI code has distance $j$9, then a weight-$SU(2)$0 noise channel and its symmetrized version are both correctible. The resulting recovery algorithms use measurements of nested total angular momentum, quantum Schur transforms, logical-state teleportation, and geometric phase gates; for erasure and deletion errors on certain PI codes, a simpler modulo-Dicke-weight procedure is available (Ouyang et al., 14 Feb 2026).

Two common misunderstandings are explicitly contradicted by this literature. First, deletion errors are not generically equivalent to erasures, but they become equivalent on PI codes because location information is immaterial (Ouyang, 2021). Second, PI codes are not merely ad hoc symmetric ansätze; they admit full Knill–Laflamme-style correction theory and explicit recovery algorithms (Ouyang et al., 14 Feb 2026).

4. Transversal symmetry, non-additivity, and fault-tolerant structure

One of the most distinctive modern themes is the interaction between permutation invariance and transversal gate structure. Via the Dicke bootstrap, a $SU(2)$1-covariant spin-$SU(2)$2 code of dimension $SU(2)$3 and spin distance $SU(2)$4 maps to a $SU(2)$5-transversal permutation-invariant $SU(2)$6 multiqubit code with $SU(2)$7 (Kubischta et al., 2023). This turns a spin-symmetry problem into a highly structured multiqubit PI-code construction.

The same work argues that the transversal gate group should be treated as a code invariant alongside $SU(2)$8. It focuses on binary dihedral groups

$SU(2)$9

and, for powers of two, $n=2j$0. Within this framework, the paper constructs PI non-additive codes implementing generalized phase gates transversally, including a family of $n=2j$1 codes and, most notably, an $n=2j$2 PI code with transversal $n=2j$3 gate (Kubischta et al., 2023). It also reports higher-distance transversal-$n=2j$4 PI codes $n=2j$5, $n=2j$6, $n=2j$7, $n=2j$8, and $n=2j$9, all using fewer qubits than the listed best known stabilizer codes with transversal $|j,m\rangle$0 at the same or lower distance (Kubischta et al., 2023).

This is where non-additivity becomes central. Apart from the repetition family, the only permutation-invariant stabilizer codes cited in the data are the $|j,m\rangle$1 repetition codes, so most PI codes obtained from the Dicke bootstrap are non-additive. The paper’s explicit comparisons therefore support the claim that non-additive codes can outperform stabilizer codes once transversal-gate structure is included in the comparison (Kubischta et al., 2023).

The newer $|j,m\rangle$2 family intersects this fault-tolerant picture. Some of its members reproduce previously known PI codes with nontrivial transversal groups: $|j,m\rangle$3 matches a code with logical action of the binary icosahedral group $|j,m\rangle$4, while codes of the form $|j,m\rangle$5 implement $|j,m\rangle$6 transversally when $|j,m\rangle$7, or $|j,m\rangle$8 when $|j,m\rangle$9; in particular, $\mathcal C \subseteq \mathrm{Sym}^n(\mathbb C^q),$00 is an 11-qubit one-error-correcting code with transversal $\mathcal C \subseteq \mathrm{Sym}^n(\mathbb C^q),$01 (Aydin et al., 2023).

A recurring misconception is that symmetric codes must be stabilizer-like because of their high symmetry. The cited results indicate the opposite: the symmetry often coexists with non-additivity, and that non-additivity can be operationally advantageous when transversal gates are part of the performance metric (Kubischta et al., 2023).

5. Recovery algorithms, approximate correction, and distributed architectures

Recovery for PI codes need not follow the stabilizer-syndrome template. For correlated amplitude-damping noise, a quantum error recovery framework based on coherent CPTP-map implementation constructs optimized recovery channels directly in the Dicke subspace. The recovery uses either an SDP-optimal map or the Petz/Barnum–Knill map and is compiled through ancilla-assisted branch splitting in the style of Ticozzi–Viola (Chandra et al., 2 Jul 2026). Within this framework, the paper introduces CAD codes for collective amplitude damping, including a 4-qubit CAD4 code that perfectly corrects one global symmetric amplitude-damping error and a 9-qubit CAD9 code that, under the reported benchmarks, beats several existing short PI codes by more than one order of magnitude at $\mathcal C \subseteq \mathrm{Sym}^n(\mathbb C^q),$02 and more than two orders of magnitude at $\mathcal C \subseteq \mathrm{Sym}^n(\mathbb C^q),$03 (Chandra et al., 2 Jul 2026).

The same work emphasizes a hardware consequence of permutation symmetry: reduced addressability requirements. Because encoding and recovery operate in the symmetric Dicke sector, system-only and system–ancilla primitives can be compiled with geometric phase gates, and the explicit CAD4 recovery circuit uses 10 system/system-ancilla gates realizable from linear geometric phase gates (Chandra et al., 2 Jul 2026). This suggests a direct route from symmetry-adapted channel recovery to experimentally implementable circuits.

A separate line develops approximate PI codes for modular quantum computing. The W-state code stores the logical state as a single excitation delocalized over all subsystems, for example

$\mathcal C \subseteq \mathrm{Sym}^n(\mathbb C^q),$04

and is used as the outer layer in Distributed Approximate Quantum Error Correction (DAQEC) (Clayton et al., 29 Sep 2025). The proposal is motivated by noisy inter-processor links and heterogeneous processor error rates: permutation invariance spreads the logical state across modules, supports primarily local transversal logical operations, and permits approximate decoding strategies such as measurement-based location readout or measurement-free elective decoding (Clayton et al., 29 Sep 2025).

This approximate setting is important because it changes the tradeoff imposed by exact-code theorems. The DAQEC proposal explicitly frames approximate, permutation-invariant outer codes as a way to reduce nonlocal gates, mitigate processor-dependent noise, and avoid magic-state distillation or code switching in distributed hardware (Clayton et al., 29 Sep 2025). A plausible implication is that PI symmetry is not merely a code-space restriction; it can also be an architectural abstraction for modular fault tolerance.

6. Cryptographic and communication applications

Permutation invariance is also useful beyond error suppression. In quantum anonymous secret sharing, a quantum secret is encoded into PI shares so that recovery depends only on how many shares are missing, not on which parties are absent. This symmetry is the mechanism that enables sender-anonymous recovery: shareholders can send their shares to Bob through anonymous transmission subroutines, and because the code is permutation-invariant, Bob’s decoder cannot infer which shareholders participated (Sikand et al., 30 Apr 2026).

That protocol is formulated in standard quantum error-correction language. The paper relates Knill–Laflamme conditions, a recovery-channel criterion, and a mutual-information formulation, then quantifies leakage in ramp secret sharing by the quantum conditional min-entropy

$\mathcal C \subseteq \mathrm{Sym}^n(\mathbb C^q),$05

It evaluates several PI families—including 4-, 7-, 9-, 11-, and 13-qubit codes—and reports that intermediate-share leakage profiles differ substantially across families even when the distance is the same (Sikand et al., 30 Apr 2026). This suggests that PI symmetry supports not only anonymity but also a controlled, code-dependent notion of ramp leakage.

Permutation-invariant codes also appear as an ansatz for quantum communication over i.i.d. channels. Because i.i.d. channels preserve permutation symmetry, symmetric input states admit a Schur–Weyl block decomposition that makes coherent information tractable at much larger blocklengths than generic inputs. One paper exploits this to evaluate coherent information for at least 100 channel copies in qubit-output cases and obtains improved lower bounds on quantum capacities for several families, including general Pauli channels, the dephrasure channel, the generalized amplitude damping channel, and the damping-dephasing channel (Bhalerao et al., 13 Aug 2025).

The strongest threshold improvements in that study come from repetition-code-like PI input states built from non-orthogonal pure states rather than orthodox orthogonal repetition codes. For the 2-Pauli and BB84 channel families, the resulting thresholds improve substantially over the Fern–Whaley 2008 benchmarks, while for the generalized amplitude damping channel the reported PI ansatz improves over both single-letter coherent information and the cited neural-network lower bounds (Bhalerao et al., 13 Aug 2025). This use of PI codes is operationally different from quantum memory or fault tolerance, but it relies on the same structural fact: permutation symmetry compresses an otherwise exponential multipartite optimization.

7. Related classical notions: permutation arrays, perfect deletion codes, and multiset coding

Outside the symmetric-subspace quantum literature, permutation-invariant coding appears in several classical forms. One is the theory of permutation arrays in $\mathcal C \subseteq \mathrm{Sym}^n(\mathbb C^q),$06 under Hamming distance. A recent example constructs isometry-invariant $\mathcal C \subseteq \mathrm{Sym}^n(\mathbb C^q),$07-permutation arrays for $\mathcal C \subseteq \mathrm{Sym}^n(\mathbb C^q),$08 and interprets them as separable permutation codes corresponding to mutually orthogonal Latin squares (MOLS). The resulting codes establish the lower bounds

$\mathcal C \subseteq \mathrm{Sym}^n(\mathbb C^q),$09

for the number of MOLS of those orders (Janiszczak et al., 2018).

Another direction studies stable deletions in permutation codes. A direct proof of Levenshtein’s 1992 theorem defines a family of perfect single-deletion-correcting permutation codes $\mathcal C \subseteq \mathrm{Sym}^n(\mathbb C^q),$10 using a modular sum condition on a special representation $\mathcal C \subseteq \mathrm{Sym}^n(\mathbb C^q),$11 of permutations. Each code has size $\mathcal C \subseteq \mathrm{Sym}^n(\mathbb C^q),$12, the full symmetric group $\mathcal C \subseteq \mathrm{Sym}^n(\mathbb C^q),$13 is partitioned into $\mathcal C \subseteq \mathrm{Sym}^n(\mathbb C^q),$14 such perfect codes, and the paper gives encoding in $\mathcal C \subseteq \mathrm{Sym}^n(\mathbb C^q),$15 for any fixed $\mathcal C \subseteq \mathrm{Sym}^n(\mathbb C^q),$16 and decoding in $\mathcal C \subseteq \mathrm{Sym}^n(\mathbb C^q),$17 (Gao et al., 2024).

A broader order-insensitive classical model is coding in the space of multisets. There the channel randomly permutes or completely ignores order and may additionally introduce insertions, deletions, substitutions, or erasures. Codewords are multiplicity vectors in the simplex $\mathcal C \subseteq \mathrm{Sym}^n(\mathbb C^q),$18, and the exact correction condition is minimum distance greater than the error radius in the normalized $\mathcal C \subseteq \mathrm{Sym}^n(\mathbb C^q),$19 metric

$\mathcal C \subseteq \mathrm{Sym}^n(\mathbb C^q),$20

In this setting, constructions based on Sidon sets in finite Abelian groups are shown to be asymptotically optimal for any fixed error radius and alphabet size, and in several cases optimal in the stronger maximal-cardinality sense (Kovačević et al., 2016).

These classical objects are distinct from quantum PI codes, but they illuminate the same structural principle. When order is removed from the physical or logical model, successful code design shifts from positional combinatorics to symmetry-adapted algebra: orbit decompositions, lattice packings, Sidon sets, and group actions in the classical case; Dicke sectors, Schur–Weyl blocks, spin irreducibles, and transversal symmetry groups in the quantum case.

Taken together, the literature presents permutation-invariant codes as a broad symmetry-based coding paradigm rather than a single family. In quantum error correction they provide explicit exact, approximate, qubit, and qudit codes; deletion-, insertion-, and amplitude-damping correction; transversal generalized phase gates; and implementations based on global control (Ouyang, 2013, Ouyang, 2016, Kubischta et al., 2023, Ouyang et al., 14 Feb 2026). In classical coding they organize the design of order-insensitive codes for permutation and multiset channels (Janiszczak et al., 2018, Gao et al., 2024, Kovačević et al., 2016). The unifying theme is that permutation symmetry does not merely simplify the code space; it reshapes the notion of error itself.