Quantum Quadratic Residue Codes Overview

• Quantum quadratic residue codes are quantum stabilizer codes derived from classical expurgated quadratic residue codes, yielding a one-logical-qudit construction with prime block lengths.
• The construction uses Hermitian self-orthogonality based on quadratic residue and nonresidue sets, ensuring error correction distances that often meet rigorous bounds.
• These codes underpin advanced magic-state distillation protocols and inspire diverse extensions including cyclic, quasi-cyclic, and asymmetric quantum constructions.

Searching arXiv for recent and foundational papers on quantum quadratic residue codes and closely related constructions. Quantum quadratic residue codes are quantum stabilizer codes derived from the arithmetic and algebraic structure of classical quadratic residue codes. In the most precise recent usage, a quantum quadratic residue code is the image under $\iota^{-1}$ of a Hermitian self-orthogonal expurgated quadratic residue code over $\mathbb F_{d^2}$, yielding a one-logical-qudit stabilizer code of prime length $p$ (Zurel et al., 19 Mar 2026). More broadly, the literature also uses quadratic residue structure to build cyclic stabilizer codes, quasi-cyclic stabilizer codes, quantum synchronizable codes, and asymmetric quantum codes, so the subject spans both a narrow Hermitian-stabilizer definition and a wider family of residue-based quantum constructions (Xie et al., 2014).

1. Classical quadratic-residue foundations

Let $p$ be an odd prime, and let $\mathcal Q,\mathcal N\subset \mathbb F_p^\times$ denote the nonzero quadratic residues and nonresidues modulo $p$, with $|\mathcal Q|=|\mathcal N|=(p-1)/2$. A standard expurgated quadratic residue code over $\mathbb F_q$ has generator polynomial

$g(x)=(x-1)\prod_{s\in \mathcal Q}(x-\zeta^s),$

zero set $\{0\}\cup \mathcal Q$, and parameters $\mathbb F_{d^2}$0; the corresponding augmented and extended variants supply the standard odd-like/even-like QR trichotomy that underlies most quantum constructions (Zurel et al., 19 Mar 2026).

For binary QR codes of prime length $\mathbb F_{d^2}$1, the literature used in quantum constructions distinguishes the $\mathbb F_{d^2}$2 pair $\mathbb F_{d^2}$3 from the $\mathbb F_{d^2}$4 pair $\mathbb F_{d^2}$5, and exploits the square-root bound

$\mathbb F_{d^2}$6

with the refinement

$\mathbb F_{d^2}$7

when $\mathbb F_{d^2}$8 (Xie et al., 2014). In the asymmetric setting, the same QR data appear as $\mathbb F_{d^2}$9, with

$p$0

and

$p$1

(Guardia, 2013).

A distinct but related classical object is the binary quasi-quadratic residue code

$p$2

which is self-dual of length $p$3 and dimension $p$4 when $p$5 (Boston et al., 2017). This terminology is a recurring source of ambiguity: in later quantum work, “QQR” refers to quantum quadratic residue codes, whereas in this classical literature it denotes quasi-quadratic residue codes.

2. Stabilizer realization and exact quantum definition

The most explicit modern definition takes a classical expurgated QR code $p$6 and identifies $p$7 with $p$8 through

$p$9

where $p$0. A quantum quadratic residue code on $p$1-dimensional qudits is then the image under $p$2 of a Hermitian self-orthogonal expurgated quadratic residue code over $p$3 (Zurel et al., 19 Mar 2026).

The Hermitian self-orthogonality criterion is

$p$4

for $p$5. Specializing to $p$6, the existence condition becomes

$p$7

For qubits this is equivalent to $p$8 or $p$9; for qutrits it is equivalent to $\mathcal Q,\mathcal N\subset \mathbb F_p^\times$0 or $\mathcal Q,\mathcal N\subset \mathbb F_p^\times$1. The corresponding stabilizer code has parameters

$\mathcal Q,\mathcal N\subset \mathbb F_p^\times$2

with

$\mathcal Q,\mathcal N\subset \mathbb F_p^\times$3

Thus the recent QQR family is inherently a one-logical-qudit family with prime blocklength and classical distance inherited from expurgated QR codes (Zurel et al., 19 Mar 2026).

This family is not uniformly CSS. It becomes CSS when the underlying expurgated QR code already exists over $\mathcal Q,\mathcal N\subset \mathbb F_p^\times$4, in which case

$\mathcal Q,\mathcal N\subset \mathbb F_p^\times$5

for the corresponding expurgated QR code $\mathcal Q,\mathcal N\subset \mathbb F_p^\times$6 over $\mathcal Q,\mathcal N\subset \mathbb F_p^\times$7. For qubits this occurs exactly when $\mathcal Q,\mathcal N\subset \mathbb F_p^\times$8, and for qutrits exactly when $\mathcal Q,\mathcal N\subset \mathbb F_p^\times$9 (Zurel et al., 19 Mar 2026).

3. Residue-based quantum code families beyond the narrow QQR definition

Quantum synchronizable codes form one important extension of the QR paradigm. Starting from cyclic codes $p$0 with $p$1, the synchronizable-code theorem yields

$p$2

subject to

$p$3

where $p$4. For binary QR codes of prime length $p$5, one has

$p$6

hence dual-containment, and for Mersenne primes $p$7 the generator polynomial factors into $p$8 irreducible factors of degree $p$9. Deleting $|\mathcal Q|=|\mathcal N|=(p-1)/2$0 such factors produces supercodes and a family

$|\mathcal Q|=|\mathcal N|=(p-1)/2$1

with

$|\mathcal Q|=|\mathcal N|=(p-1)/2$2

which attains the synchronization upper bound in that framework (Xie et al., 2014).

Asymmetric quantum codes arise from QR codes through expansion from $|\mathcal Q|=|\mathcal N|=(p-1)/2$3 to the prime field. For $|\mathcal Q|=|\mathcal N|=(p-1)/2$4, the classical relation

$|\mathcal Q|=|\mathcal N|=(p-1)/2$5

gives $|\mathcal Q|=|\mathcal N|=(p-1)/2$6 and yields

$|\mathcal Q|=|\mathcal N|=(p-1)/2$7

For $|\mathcal Q|=|\mathcal N|=(p-1)/2$8, the dual-containing relation

$|\mathcal Q|=|\mathcal N|=(p-1)/2$9

gives

$\mathbb F_q$0

In both families the unexpanded quantum dimension is $\mathbb F_q$1, and expansion multiplies it to $\mathbb F_q$2 (Guardia, 2013).

A different construction starts directly from quadratic residue sets rather than from classical QR duality theorems. For primes $\mathbb F_q$3, the stabilizer matrix $\mathbb F_q$4 is built so that

$\mathbb F_q$5

The resulting Type-I family is cyclic of length $\mathbb F_q$6, while Type-II is quasi-cyclic of length $\mathbb F_q$7 with $\mathbb F_q$8. For $\mathbb F_q$9 and odd $g(x)=(x-1)\prod_{s\in \mathcal Q}(x-\zeta^s),$0, the Type-I family gives

$g(x)=(x-1)\prod_{s\in \mathcal Q}(x-\zeta^s),$1

including

$g(x)=(x-1)\prod_{s\in \mathcal Q}(x-\zeta^s),$2

For $g(x)=(x-1)\prod_{s\in \mathcal Q}(x-\zeta^s),$3 and even $g(x)=(x-1)\prod_{s\in \mathcal Q}(x-\zeta^s),$4, one gets high-rate cyclic codes

$g(x)=(x-1)\prod_{s\in \mathcal Q}(x-\zeta^s),$5

The Type-II quasi-cyclic constructions produce examples such as $g(x)=(x-1)\prod_{s\in \mathcal Q}(x-\zeta^s),$6, $g(x)=(x-1)\prod_{s\in \mathcal Q}(x-\zeta^s),$7, and $g(x)=(x-1)\prod_{s\in \mathcal Q}(x-\zeta^s),$8, with code dimension determined by rank formulas $g(x)=(x-1)\prod_{s\in \mathcal Q}(x-\zeta^s),$9 or $\{0\}\cup \mathcal Q$0 depending on the parity of $\{0\}\cup \mathcal Q$1 (Xie et al., 2014).

4. Representative parameter families and code equivalences

Several distinct QR-based quantum families coexist in the literature:

Family	Main hypothesis	Quantum parameters
Hermitian QQR	$\{0\}\cup \mathcal Q$2	$\{0\}\cup \mathcal Q$3
Synchronizable QR	$\{0\}\cup \mathcal Q$4, $\{0\}\cup \mathcal Q$5	$\{0\}\cup \mathcal Q$6
Expanded AQECC from QR	$\{0\}\cup \mathcal Q$7 quadratic residue mod $\{0\}\cup \mathcal Q$8	$\{0\}\cup \mathcal Q$9
QR-set stabilizer codes	$\mathbb F_{d^2}$00	cyclic $\mathbb F_{d^2}$01, QC $\mathbb F_{d^2}$02

Within the recent Hermitian framework, the paper explicitly identifies several landmark distillation codes as quantum quadratic residue codes up to permutations of qudits and local Clifford equivalence. The length-$\mathbb F_{d^2}$03 qubit QQR code is equivalent to the $\mathbb F_{d^2}$04 perfect code, the length-$\mathbb F_{d^2}$05 qubit QQR code is equivalent to the Steane code, the length-$\mathbb F_{d^2}$06 qutrit QQR code is equivalent to the $\mathbb F_{d^2}$07-qutrit Golay code, and the length-$\mathbb F_{d^2}$08 qubit QQR code is equivalent to the $\mathbb F_{d^2}$09-qubit Golay code (Zurel et al., 19 Mar 2026).

The same work lists representative classical and quantum parameter pairs. For qubits: $\mathbb F_{d^2}$10

$\mathbb F_{d^2}$11

$\mathbb F_{d^2}$12

For qutrits: $\mathbb F_{d^2}$13

$\mathbb F_{d^2}$14

$\mathbb F_{d^2}$15

These examples exhibit the characteristic one-logical-qudit structure of the modern QQR family (Zurel et al., 19 Mar 2026).

5. Magic-state distillation and current significance

Quantum quadratic residue codes acquired renewed prominence through magic-state distillation. In that setting, the family supplies $\mathbb F_{d^2}$16-to-1 stabilizer reductions for both qubit $\mathbb F_{d^2}$17-states and qutrit Strange states, and the recent synthesis argues that several of the most important known distillation protocols are unified by the QQR formalism (Zurel et al., 19 Mar 2026).

For qubits, the reported QQR codes of lengths

$\mathbb F_{d^2}$18

distill $\mathbb F_{d^2}$19 states, with thresholds

$\mathbb F_{d^2}$20

respectively. For qutrits, the reported QQR codes of lengths

$\mathbb F_{d^2}$21

distill Strange states, with thresholds

$\mathbb F_{d^2}$22

The best thresholds in that study remain the $\mathbb F_{d^2}$23-qubit code for $\mathbb F_{d^2}$24-state distillation and the $\mathbb F_{d^2}$25-qutrit Golay/QQR code for Strange-state distillation (Zurel et al., 19 Mar 2026).

The asymptotic result is particularly notable: if $\mathbb F_{d^2}$26 is prime with

$\mathbb F_{d^2}$27

then the corresponding qubit QQR code has a choice of sign $\mathbb F_{d^2}$28 such that

$\mathbb F_{d^2}$29

hence a nontrivial distillation threshold. By Dirichlet’s theorem, this yields infinitely many quantum quadratic residue codes that distill $\mathbb F_{d^2}$30 states (Zurel et al., 19 Mar 2026).

The technical reason QR structure is useful here is not only distance. The distillation analysis is reduced to classical weight enumerators, and for the qubit case the $\mathbb F_{d^2}$31-linearity of the classical QR input aligns with a transversal $\mathbb F_{d^2}$32 Clifford. Invariant-theoretic reconstruction of extended QR weight enumerators then becomes a practical tool for analyzing distillation performance (Zurel et al., 19 Mar 2026).

6. Generalizations, terminology, and related directions

The QR paradigm extends beyond prime-length cyclic codes. Duadic group algebra codes are a generalization of quadratic residue codes, and they were shown to yield degenerate quantum stabilizer codes in which many errors of small weight do not need error correction [0701060]. More generally, $\mathbb F_{d^2}$33-adic residue codes over

$\mathbb F_{d^2}$34

admit dual-containing criteria and an orthogonality-preserving Gray map; from dual-containing odd-like class-I $\mathbb F_{d^2}$35-adic residue codes, one obtains quantum codes with parameters

$\mathbb F_{d^2}$36

where $\mathbb F_{d^2}$37 and $\mathbb F_{d^2}$38 is the Gray distance (Kuruz et al., 2018). Since quadratic residue codes correspond to $\mathbb F_{d^2}$39, this places quantum QR constructions inside a broader residue-code hierarchy.

A second axis of generalization is ring lifting. Quadratic residue codes over $\mathbb F_{d^2}$40 admit Gray maps preserving self-duality; in the case $\mathbb F_{d^2}$41, the even-like QR codes are self-orthogonal and the extended QR codes are self-dual after extension, with Gray images such as a self-orthogonal nearly MDS $\mathbb F_{d^2}$42 code over $\mathbb F_{d^2}$43 and a self-dual $\mathbb F_{d^2}$44 code over $\mathbb F_{d^2}$45 (Goyal et al., 2016). Over $\mathbb F_{d^2}$46, the decomposition

$\mathbb F_{d^2}$47

and a duality-preserving Gray map to $\mathbb F_{d^2}$48 yield self-orthogonal QR codes $\mathbb F_{d^2}$49, self-dual extended QR codes $\mathbb F_{d^2}$50 for $\mathbb F_{d^2}$51, and concrete self-dual $\mathbb F_{d^2}$52-codes with parameters $\mathbb F_{d^2}$53 and $\mathbb F_{d^2}$54 (Gao et al., 2014). These are not quantum codes in themselves, but they supply classical self-orthogonal and dual-containing structures of the kind routinely used in stabilizer constructions.

A final source of ambiguity is terminological. “QQR code” may denote the classical quasi-quadratic residue code

$\mathbb F_{d^2}$55

which is a binary self-dual code of length $\mathbb F_{d^2}$56 and dimension $\mathbb F_{d^2}$57 for $\mathbb F_{d^2}$58, with an extended code carrying an action of $\mathbb F_{d^2}$59 and a weight polynomial divisible by $\mathbb F_{d^2}$60 (Boston et al., 2017). In the quantum literature, however, “quantum quadratic residue code” refers either to the Hermitian one-logical-qudit family over $\mathbb F_{d^2}$61 or, more loosely, to any stabilizer or CSS construction whose commutativity and distance properties are organized by quadratic residue sets (Zurel et al., 19 Mar 2026). The modern subject therefore combines a precise stabilizer definition with a wider algebraic design space shaped by residue/nonresidue arithmetic.