Routing Codes in Quantum and Network Applications

• Routing codes are a versatile family of quantum LDPC codes defined on toric lattices using routing vectors for optimized non-local qubit interactions.
• They achieve hardware efficiency by reducing coupling lengths and mitigating routing conflicts via parallel non-local connections and symmetric routing schedules.
• In network settings, routing codes encompass multi-path routing and adaptive FEC schemes, offering robust error correction through combinatorial coding techniques.

Routing codes is a polysemous technical term. In the 2026 quantum-error-correction literature, it denotes a family of quantum low-density parity-check codes defined to combine high rate and threshold with short, mutually parallel non-local connectivity, explicitly targeting implementability on superconducting and neutral-atom hardware (Zhang et al., 24 Jun 2026). Earlier communication-theoretic literature also used the phrase for coding schemes designed around routed networks rather than fully mixed network coding, including joint multi-path routing with adaptive FEC, subset codes for packet-switched routing, subspace codes in simplified routing-based interference models, and constant-weight lattice codes for store-and-forward routing 0611086.

1. Quantum routing codes as a qLDPC family

Routing codes are defined on an even–even torus

$$L=\mathbb Z_l\times\mathbb Z_m,\qquad l,m\ \text{even},$$

with group ring

$R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$

Each lattice site $(i,j)\in L$ corresponds to the monomial $x^i y^j$. Qubits are partitioned by the parity of $i+j$ into data qubits

$D=\{(i,j)\mid i+j\ \text{even}\},$

$X$-syndrome qubits

$X=\{(i,j)\mid i\ \text{odd},\ j\ \text{even}\},$

and $Z$-syndrome qubits

$Z=\{(i,j)\mid i\ \text{even},\ j\ \text{odd}\}.$

The construction chooses routing vectors $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$0 for $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$1-syndromes and $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$2 for $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$3-syndromes; at time step $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$4, each $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$5-syndrome at $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$6 swaps via iSWAP with data at $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$7, and each $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$8-syndrome at $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$9 swaps with data at $(i,j)\in L$0 (Zhang et al., 24 Jun 2026).

The associated involutive permutation is

$(i,j)\in L$1

With $(i,j)\in L$2, an $(i,j)\in L$3-syndrome initially at $(i,j)\in L$4 collects parity from

$(i,j)\in L$5

and similarly for $(i,j)\in L$6-syndromes. Fixing reference syndrome positions $(i,j)\in L$7 and $(i,j)\in L$8, the stabilizer supports are encoded by

$(i,j)\in L$9

All $x^i y^j$0 stabilizers are generated by even–even translations of $x^i y^j$1, and all $x^i y^j$2 stabilizers by translations of $x^i y^j$3. In CSS form,

$x^i y^j$4

Commutativity is characterized by Proposition 1: the stabilizers commute iff, in

$x^i y^j$5

every term $x^i y^j$6 with $x^i y^j$7 both even has $x^i y^j$8 in $x^i y^j$9. Proposition 2 gives a sufficient symmetry condition: if $i+j$0 for all $i+j$1 and $i+j$2, then commutativity is automatic. The explicit constructions in the report use a time-reversal symmetric sequence of length $i+j$3 for weight-7 codes, containing one non-local vector plus local steps $i+j$4 and $i+j$5. The logical dimension satisfies

$i+j$6

and $i+j$7 is computed via integer-programming on $i+j$8.

2. Connectivity reduction and parallel non-locality

The distinctive hardware feature of routing codes is not merely the presence of non-local couplings, but the simultaneous reduction of coupling length, qubit connectivity, and routing conflict. Each qubit has degree

$i+j$9

where $D=\{(i,j)\mid i+j\ \text{even}\},$0 or $D=\{(i,j)\mid i+j\ \text{even}\},$1 and $D=\{(i,j)\mid i+j\ \text{even}\},$2 equals the number of distinct non-local vectors, which is $D=\{(i,j)\mid i+j\ \text{even}\},$3 in the explicit constructions. For the $D=\{(i,j)\mid i+j\ \text{even}\},$4 instance, the non-local vector is $D=\{(i,j)\mid i+j\ \text{even}\},$5, with

$D=\{(i,j)\mid i+j\ \text{even}\},$6

The comparison given in the report is with bivariate bicycle codes using $D=\{(i,j)\mid i+j\ \text{even}\},$7, giving $D=\{(i,j)\mid i+j\ \text{even}\},$8; this reduces maximum coupling length by $D=\{(i,j)\mid i+j\ \text{even}\},$9 (Zhang et al., 24 Jun 2026).

For weight-7 routing codes, the check weight is $X$0, identical to BB codes, but qubit connectivity is only $X$1 or $X$2 instead of $X$3. Proposition 3 states that, on a sufficiently large torus that avoids path wrapping, time-reversal symmetric and non-negative $X$4 guarantee that no two CXSWAP gates conflict temporally or spatially, so all gates assigned to a given step $X$5 can be implemented in parallel. Because all non-local edges share the same direction $X$6, no wiring crossings occur in a multi-layer superconducting layout and AOD-shuttled atom trajectories never intersect in neutral-atom arrays.

This combination of short-range non-locality and mutual parallelism is the central differentiator of the family. A plausible implication is that the usual trade-off between high-rate qLDPC structure and hardware routability is being shifted from a question of whether non-locality exists to a question of how structured that non-locality is.

3. Parameters, thresholds, and overhead

The report lists selected instances

$X$7

with rates

$X$8

and distances $X$9, respectively. In these families $X=\{(i,j)\mid i\ \text{odd},\ j\ \text{even}\},$0 remains fixed at $X=\{(i,j)\mid i\ \text{odd},\ j\ \text{even}\},$1 as $X=\{(i,j)\mid i\ \text{odd},\ j\ \text{even}\},$2 grows, so $X=\{(i,j)\mid i\ \text{odd},\ j\ \text{even}\},$3; the report also notes that one may increase $X=\{(i,j)\mid i\ \text{odd},\ j\ \text{even}\},$4 by tiling multiple $X=\{(i,j)\mid i\ \text{odd},\ j\ \text{even}\},$5 orbits, although the focus is on fixed-$X=\{(i,j)\mid i\ \text{odd},\ j\ \text{even}\},$6 instances (Zhang et al., 24 Jun 2026).

Under circuit-level simulation with Stim for depolarizing noise, using a two-qubit error rate $X=\{(i,j)\mid i\ \text{odd},\ j\ \text{even}\},$7 after each iSWAP and depolarized single-qubit idles, and BP-OSD decoding for all codes, the reported threshold is approximately $X=\{(i,j)\mid i\ \text{odd},\ j\ \text{even}\},$8. For $X=\{(i,j)\mid i\ \text{odd},\ j\ \text{even}\},$9, the $Z$0 routing code achieves $Z$1. Relative to the rotated surface code with $Z$2–$Z$3 at $Z$4, routing codes require $Z$5 fewer physical qubits for the same logical error rate. The circuit-level distance $Z$6 equals the theoretical $Z$7 or exceeds the BB codes’ $Z$8 on instances with equal $Z$9.

System	Connectivity and $Z=\{(i,j)\mid i\ \text{even},\ j\ \text{odd}\}.$0	Rate / hardware score
Routing $Z=\{(i,j)\mid i\ \text{even},\ j\ \text{odd}\}.$1	$Z=\{(i,j)\mid i\ \text{even},\ j\ \text{odd}\}.$2, $Z=\{(i,j)\mid i\ \text{even},\ j\ \text{odd}\}.$3	$Z=\{(i,j)\mid i\ \text{even},\ j\ \text{odd}\}.$4, $Z=\{(i,j)\mid i\ \text{even},\ j\ \text{odd}\}.$5
BB $Z=\{(i,j)\mid i\ \text{even},\ j\ \text{odd}\}.$6	$Z=\{(i,j)\mid i\ \text{even},\ j\ \text{odd}\}.$7, $Z=\{(i,j)\mid i\ \text{even},\ j\ \text{odd}\}.$8	$Z=\{(i,j)\mid i\ \text{even},\ j\ \text{odd}\}.$9, $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$00
Surface code $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$01	$R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$02 local, $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$03	$R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$04, $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$05

The comparison given in the report summarizes the design point succinctly: routing-code rates $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$06–$R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$07, BB rates $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$08–$R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$09, surface-code rates $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$10; routing-code connectivity $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$11–$R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$12, BB connectivity $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$13, surface connectivity $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$14; routing-code $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$15–$R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$16, BB $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$17, surface $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$18; and qubit overhead reduction $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$19 relative to surface codes.

4. Hardware mapping in superconducting and neutral-atom platforms

The report provides platform-specific implementation guidance. In superconducting multi-layer layouts, using the HAL framework, routing codes require $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$20–$R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$21 routing tiers versus $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$22–$R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$23 for BB codes; average coupler length is approximately $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$24–$R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$25 unit-spacings versus approximately $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$26–$R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$27; average bump bonds per edge are $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$28 versus $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$29–$R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$30; TSVs per edge are $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$31 versus $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$32; and the composite hardware-complexity score is $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$33–$R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$34 versus $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$35–$R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$36. The implementation recommendation is to place each non-local vector on a single higher tier and exploit mutual parallelism to avoid via crossings. For neutral-atom AOD scheduling, the shuttle distance for $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$37 is $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$38, compared with $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$39 for the BB vector $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$40; conflict-graph coloring yields $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$41 time steps for routing codes versus up to $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$42 for BB codes, and the suggested schedule is a symmetric forward-and-backward routing cycle (Zhang et al., 24 Jun 2026).

These claims sit within a broader hardware-aware qLDPC-routing landscape. HAL, introduced for arbitrary non-local QECC placement and routing in multi-layer superconducting hardware, defines raw layout metrics $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$43 through $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$44 and combines them into

$R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$45

with the rotated surface code as the $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$46 baseline; it was used to compare BB, tile, and radial-code families and to quantify multi-tier cost directly (Mathews et al., 30 Jul 2025). A different hardware-software co-design for BB codes uses a programmable 2D toric oscillator network, reducing long-range couplers from $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$47 to $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$48, with symmetry-aware routing and fully parallel long-range gates within each syndrome-extraction pass (Liu et al., 20 Apr 2026).

A common misconception is that hardware-aware qLDPC implementation is exhausted by finding any long-range routing substrate. The comparison between routing codes, HAL-style placement studies, and programmable toric fabrics indicates a narrower issue: the geometry, directionality, and scheduling structure of non-local interactions materially affect tiers, crossings, cycle duration, and transport conflicts.

5. Earlier communication-theoretic meanings

Before the qLDPC usage, “routing codes” appeared in networked communication settings where coding was designed explicitly for routed traffic. In real-time streaming, the term denoted the joint design of multi-path routing and adaptive FEC. The key scalar was the Redundancy Overall Requirement,

$R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$49

which rates a multi-path routing topology by the total adaptive redundancy required under link failures. The capillary routing algorithm constructed multi-path patterns and the reported trend was that further diversity, when achieved by capillary routing, reduced the overall requirement in FEC codes rather than increasing it [0611086].

For routed packet networks, subset codes provided a coding-theoretic framework in which a transmission is a subset $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$50 of packets and the receiver sees $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$51 under permutation, deletions, insertions, and errors. The defining metric is the symmetric-difference distance

$R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$52

and the correction condition is

$R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$53

This formulation was presented explicitly as the routed-network analogue of subspace coding for RLNC, with the statement that routed permutation networks preserve sets of packets rather than subspaces (Kovačević et al., 2012).

A related but distinct model studied subspace coding for error control and interference mitigation in a simplified routing-based transmission scheme with cooperative destination nodes. There, interference across parallel SISO links was interpreted as uncontrolled RLNC over $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$54, and the paper used a Koetter–Kschischang constant-dimension code $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$55 over $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$56. With minimum subspace distance $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$57, the code corrects up to

$R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$58

dimension errors, and the numerical findings reported that subspace coding remained near zero decoding-failure probability up to approximately $R=\mathbb F_2[x,y]/(x^l-1,\ y^m-1).$59 random bit-flips in the pooled matrix, whereas routing alone became highly fragile (Brahimi et al., 2022).

In random store-and-forward routing, constant-weight lattice codes were built from uniquely decomposable lattice elements, with distance induced by symmetric difference of constituent sets. The paper states that constructing such codes is equivalent to constructing a Johnson graph with appropriate parameters, and presents them as error-and-erasure codes for random networks employing SAF routing (Ghatak, 2013).

Taken together, these works show that the older communication-theoretic meaning of routing codes is not a single algebraic object. It is a family resemblance across code designs whose invariants are chosen to match routing, packet permutation, or route diversity rather than unrestricted in-network mixing.

6. Distinctions, misconceptions, and future directions

The principal contemporary misconception is terminological: routing codes in the 2026 qLDPC sense are not merely any quantum codes that require routing, and they are not simply BB codes with a better layout. Their defining properties are the routing-vector construction on the torus, the commutativity condition enforced by time-reversal symmetry, and the use of a single non-local direction that makes all non-local couplings mutually parallel (Zhang et al., 24 Jun 2026). A second misconception, inherited from older networking usage, is that routing codes are synonymous with network coding. The subset-code framework was introduced precisely for networks “employing routing in network nodes,” in contrast to the Koetter–Kschischang subspace framework for RLNC (Kovačević et al., 2012).

The outlook stated for the quantum family includes generalizing beyond toric symmetries, planar boundary layouts, custom routing on defective connectivity graphs, and fault-tolerant logical operations via dynamical lattice surgery. Related qLDPC-routing work identifies complementary research directions: programmable toric communication fabrics, erasure-aware decoders for dual-rail architectures, and automated hardware-feasibility evaluation through placement-and-routing heuristics (Liu et al., 20 Apr 2026, Mathews et al., 30 Jul 2025). This suggests that future usage of the term will likely remain hardware-centric, with routing treated not as a secondary compilation problem but as a code-design primitive.