Qutrit Random Cubic Codes: 3D CSS Models

• The paper introduces qutrit random cubic codes as 3D local CSS stabilizer Hamiltonians built on a cube-check geometry with spatially varying admissible coefficients ensuring local commutation and the no-string property.
• Numerical analyses reveal that these nontranslation-invariant models exhibit distinct logical operator structures and finite-size degeneracy—manifesting membrane-like behaviors unlike those in conventional fracton codes.
• A related approach employs random-looking qutrit CSS codes for magic-state distillation, achieving cubic noise suppression and establishing benchmarks for contextual qutrit state distillation.

Searching arXiv for papers directly relevant to qutrit random cubic codes and closely related qutrit/cubic-code literature. Searching arXiv for "qutrit random cubic codes". Qutrit random cubic codes are a family of three-dimensional local qutrit Calderbank-Shor-Steane stabilizer Hamiltonians built on the cube-check geometry of Haah’s Code 1 but with spatially varying stabilizer coefficients constrained by local commutation and a global torus relation (Yan, 18 Jun 2026). In this primary sense, they are non-translation-invariant qutrit stabilizer models on a cubic lattice that rigorously retain the no-string property and numerically exhibit logical-operator and degeneracy behavior distinct from translation-invariant fracton codes. In a looser but related usage, the same phrase also refers to random-looking large qutrit stabilizer or CSS codes whose magic-state distillation map for the qutrit strange state has cubic noise suppression (Prakash et al., 2024).

1. Definition and lattice realization

The system is defined on the periodic cubic lattice

$\Lambda_L=(\mathbb Z/L\mathbb Z)^3,$

with two qutrits per lattice site, $(r,1)$ and $(r,2)$, so the total number of physical qutrits is

$n=2L^3.$

A qutrit has computational basis $\{\lvert a\rangle:a\in\mathbb F_3\}$, with Pauli operators

$$X\lvert a\rangle=\lvert a+1\rangle,\qquad Z\lvert a\rangle=\omega^a\lvert a\rangle,\qquad \omega=e^{2\pi i/3},$$

and

$Z^zX^x=\omega^{zx}X^xZ^z,\qquad x,z\in\mathbb F_3.$

For CSS operators, commutation reduces to

$\mathbf z\cdot \mathbf x=\sum_j z_jx_j=0\pmod 3.$

The local geometry uses the eight corners

$$O=(0,0,0),\quad A=(1,0,0),\quad B=(0,1,0),\quad C=(0,0,1),$$

$$D=(0,1,1),\quad E=(1,0,1),\quad F=(1,1,0),\quad G=(1,1,1).$$

At each cube anchored at $(r,1)$0, one places a $(r,1)$1-type stabilizer $(r,1)$2 and an $(r,1)$3-type stabilizer $(r,1)$4. The $(r,1)$5-check has no support at $(r,1)$6 and two-qutrit support at $(r,1)$7, while the $(r,1)$8-check is reflected: it has two-qutrit support at $(r,1)$9 and no support at $(r,2)$0. The other six corners each carry a single-qutrit operator (Yan, 18 Jun 2026).

The stabilizers are

$(r,2)$1

with exponent vector

$(r,2)$2

and

$(r,2)$3

with

$(r,2)$4

The Hamiltonian is

$(r,2)$5

Since qutrit stabilizers are order $(r,2)$6,

$(r,2)$7

and the ground space is the common $(r,2)$8 eigenspace of all $(r,2)$9 and $n=2L^3.$0 (Yan, 18 Jun 2026).

2. Admissibility, constrained randomness, and relation to Haah-type models

The code is not defined by arbitrary spatially varying coefficients. Admissibility requires both a global $n=2L^3.$1-check relation and local CSS commutation. On the torus,

$n=2L^3.$2

At the coefficient level this becomes the sitewise constraints

$n=2L^3.$3

$n=2L^3.$4

all in $n=2L^3.$5 (Yan, 18 Jun 2026).

Local commutation requires every $n=2L^3.$6 to commute with every $n=2L^3.$7. The construction introduces the ratios

$n=2L^3.$8

$n=2L^3.$9

$\{\lvert a\rangle:a\in\mathbb F_3\}$0

Once one chooses $\{\lvert a\rangle:a\in\mathbb F_3\}$1, the $\{\lvert a\rangle:a\in\mathbb F_3\}$2-check coefficients are fixed by

$\{\lvert a\rangle:a\in\mathbb F_3\}$3

$\{\lvert a\rangle:a\in\mathbb F_3\}$4

$\{\lvert a\rangle:a\in\mathbb F_3\}$5

The remaining commutation equations reduce to constraints purely on the $\{\lvert a\rangle:a\in\mathbb F_3\}$6-field. A defining feature of the model is therefore that the randomness is highly constrained: one chooses an admissible $\{\lvert a\rangle:a\in\mathbb F_3\}$7-field, and the $\{\lvert a\rangle:a\in\mathbb F_3\}$8-field is then locally determined (Yan, 18 Jun 2026).

The family is best understood as a qutrit, spatially nonuniform analog of Haah’s Code 1. The similarities are the same cube-check support geometry, the same CSS structure, and the same local pattern of a $\{\lvert a\rangle:a\in\mathbb F_3\}$9-check and reflected $$X\lvert a\rangle=\lvert a+1\rangle,\qquad Z\lvert a\rangle=\omega^a\lvert a\rangle,\qquad \omega=e^{2\pi i/3},$$0-check. The differences are qutrits instead of qubits, coefficients in $$X\lvert a\rangle=\lvert a+1\rangle,\qquad Z\lvert a\rangle=\omega^a\lvert a\rangle,\qquad \omega=e^{2\pi i/3},$$1 with nonzero values $$X\lvert a\rangle=\lvert a+1\rangle,\qquad Z\lvert a\rangle=\omega^a\lvert a\rangle,\qquad \omega=e^{2\pi i/3},$$2 and $$X\lvert a\rangle=\lvert a+1\rangle,\qquad Z\lvert a\rangle=\omega^a\lvert a\rangle,\qquad \omega=e^{2\pi i/3},$$3, and the breaking of translation invariance through spatially varying admissible coefficients (Yan, 18 Jun 2026).

A convenient uniform benchmark is the spatially constant qutrit version with

$$X\lvert a\rangle=\lvert a+1\rangle,\qquad Z\lvert a\rangle=\omega^a\lvert a\rangle,\qquad \omega=e^{2\pi i/3},$$4

and corresponding $$X\lvert a\rangle=\lvert a+1\rangle,\qquad Z\lvert a\rangle=\omega^a\lvert a\rangle,\qquad \omega=e^{2\pi i/3},$$5-coefficients

$$X\lvert a\rangle=\lvert a+1\rangle,\qquad Z\lvert a\rangle=\omega^a\lvert a\rangle,\qquad \omega=e^{2\pi i/3},$$6

The data indicate that this uniform model behaves much more like standard Haah-type fracton codes, whereas constrained randomness alters both the logical-operator structure and the finite-size degeneracy (Yan, 18 Jun 2026).

3. No-string theorem and local proof structure

A central rigorous result is that every locally admissible qutrit random cubic code has no fixed-width logical strings. In Haah’s sense, a logical string segment consists of a finite Pauli operator $$X\lvert a\rangle=\lvert a+1\rangle,\qquad Z\lvert a\rangle=\omega^a\lvert a\rangle,\qquad \omega=e^{2\pi i/3},$$7 together with two congruent anchor cubes $$X\lvert a\rangle=\lvert a+1\rangle,\qquad Z\lvert a\rangle=\omega^a\lvert a\rangle,\qquad \omega=e^{2\pi i/3},$$8, such that all stabilizer checks away from the anchors commute with $$X\lvert a\rangle=\lvert a+1\rangle,\qquad Z\lvert a\rangle=\omega^a\lvert a\rangle,\qquad \omega=e^{2\pi i/3},$$9. The segment is nontrivial if every equivalent representative remains connected and trivial if stabilizer multiplication can disconnect it (Yan, 18 Jun 2026).

The theorem states that for any locally admissible coefficient field on $Z^zX^x=\omega^{zx}X^xZ^z,\qquad x,z\in\mathbb F_3.$0,

$Z^zX^x=\omega^{zx}X^xZ^z,\qquad x,z\in\mathbb F_3.$1

with the explicit conservative bound

$Z^zX^x=\omega^{zx}X^xZ^z,\qquad x,z\in\mathbb F_3.$2

On the periodic torus, the same holds for string segments contained in a contractible no-wraparound region. The theorem does not rule out noncontractible plane-supported logical operators (Yan, 18 Jun 2026).

The proof is a qutrit-weighted extension of Haah’s original argument for Code 1 and uses three local mechanisms: corner erasing, good-edge erasing, and a confusing constraint on exposed edges. For corner erasing, a corner with local $Z^zX^x=\omega^{zx}X^xZ^z,\qquad x,z\in\mathbb F_3.$3-vector $Z^zX^x=\omega^{zx}X^xZ^z,\qquad x,z\in\mathbb F_3.$4 imposes

$Z^zX^x=\omega^{zx}X^xZ^z,\qquad x,z\in\mathbb F_3.$5

on a local $Z^zX^x=\omega^{zx}X^xZ^z,\qquad x,z\in\mathbb F_3.$6-exponent $Z^zX^x=\omega^{zx}X^xZ^z,\qquad x,z\in\mathbb F_3.$7. For good-edge erasing, the local commutation condition on an $Z^zX^x=\omega^{zx}X^xZ^z,\qquad x,z\in\mathbb F_3.$8-type operator across a good edge is

$Z^zX^x=\omega^{zx}X^xZ^z,\qquad x,z\in\mathbb F_3.$9

for nonzero $\mathbf z\cdot \mathbf x=\sum_j z_jx_j=0\pmod 3.$0. After flattening to thickness-one rectangles, the exposed edge obeys the triangular recurrence

$\mathbf z\cdot \mathbf x=\sum_j z_jx_j=0\pmod 3.$1

with all coefficients nonzero, implying

$\mathbf z\cdot \mathbf x=\sum_j z_jx_j=0\pmod 3.$2

Iterating this row by row creates an empty interval that disconnects the support (Yan, 18 Jun 2026).

The significance of the theorem is structural rather than merely finite-size. The proof is purely local and deterministic: it does not use translation symmetry, the $\mathbf z\cdot \mathbf x=\sum_j z_jx_j=0\pmod 3.$3 line symmetry used in numerics, or statistical randomness. It uses only the cube support geometry, the reflected CSS structure, the local commutation equations, and the requirement that all coefficients are nonzero in $\mathbf z\cdot \mathbf x=\sum_j z_jx_j=0\pmod 3.$4 (Yan, 18 Jun 2026).

4. Degeneracy, membrane logicals, and charge-push diagnostics

Finite-size numerics were carried out for $\mathbf z\cdot \mathbf x=\sum_j z_jx_j=0\pmod 3.$5 in a subfamily of admissible fields invariant along the $\mathbf z\cdot \mathbf x=\sum_j z_jx_j=0\pmod 3.$6 direction,

$\mathbf z\cdot \mathbf x=\sum_j z_jx_j=0\pmod 3.$7

The ground-state degeneracy is summarized by

$\mathbf z\cdot \mathbf x=\sum_j z_jx_j=0\pmod 3.$8

Among sampled admissible models, the smallest observed degeneracy exponent follows the parity rule

$\mathbf z\cdot \mathbf x=\sum_j z_jx_j=0\pmod 3.$9

The retained models satisfy

$$O=(0,0,0),\quad A=(1,0,0),\quad B=(0,1,0),\quad C=(0,0,1),$$0

By contrast, the translation-invariant benchmark exhibits strong arithmetic finite-size effects, including

$$O=(0,0,0),\quad A=(1,0,0),\quad B=(0,1,0),\quad C=(0,0,1),$$1

This contrast is one of the clearest numerical distinctions between the random and uniform qutrit cubic models (Yan, 18 Jun 2026).

A second numerical result is that noncontractible plane-supported logical operators span the entire logical space in all curated random samples studied. For a plane support $$O=(0,0,0),\quad A=(1,0,0),\quad B=(0,1,0),\quad C=(0,0,1),$$2, the logical content in the $$O=(0,0,0),\quad A=(1,0,0),\quad B=(0,1,0),\quad C=(0,0,1),$$3-sector is

$$O=(0,0,0),\quad A=(1,0,0),\quad B=(0,1,0),\quad C=(0,0,1),$$4

and similarly in the $$O=(0,0,0),\quad A=(1,0,0),\quad B=(0,1,0),\quad C=(0,0,1),$$5-sector,

$$O=(0,0,0),\quad A=(1,0,0),\quad B=(0,1,0),\quad C=(0,0,1),$$6

The reported outcome is

$$O=(0,0,0),\quad A=(1,0,0),\quad B=(0,1,0),\quad C=(0,0,1),$$7

for all $$O=(0,0,0),\quad A=(1,0,0),\quad B=(0,1,0),\quad C=(0,0,1),$$8 curated samples. The smallest plane-supported logical operator weights obey

$$O=(0,0,0),\quad A=(1,0,0),\quad B=(0,1,0),\quad C=(0,0,1),$$9

The paper explicitly notes that this is not a full code-distance proof; it is a minimum over plane-supported representatives rather than a global optimization over all possible Pauli supports (Yan, 18 Jun 2026).

The same study searched for axis-aligned tube logical operators with support $$D=(0,1,1),\quad E=(1,0,1),\quad F=(1,1,0),\quad G=(1,1,1).$$0. The diagnostic

$$D=(0,1,1),\quad E=(1,0,1),\quad F=(1,1,0),\quad G=(1,1,1).$$1

and the analogous $$D=(0,1,1),\quad E=(1,0,1),\quad F=(1,1,0),\quad G=(1,1,1).$$2, vanish in all scanned cases: $$D=(0,1,1),\quad E=(1,0,1),\quad F=(1,1,0),\quad G=(1,1,1).$$3 This strengthens the conclusion that the logical operators observed numerically are membrane-like rather than wide-string-like (Yan, 18 Jun 2026).

Charge-push diagnostics were used to probe the presence or absence of self-similar fractal mechanisms. Starting from a single local $$D=(0,1,1),\quad E=(1,0,1),\quad F=(1,1,0),\quad G=(1,1,1).$$4 operator, one repeatedly cancels moving charges using only $$D=(0,1,1),\quad E=(1,0,1),\quad F=(1,1,0),\quad G=(1,1,1).$$5 or $$D=(0,1,1),\quad E=(1,0,1),\quad F=(1,1,0),\quad G=(1,1,1).$$6 moves, with update exponents

$$D=(0,1,1),\quad E=(1,0,1),\quad F=(1,1,0),\quad G=(1,1,1).$$7

for the first-qutrit channel and

$$D=(0,1,1),\quad E=(1,0,1),\quad F=(1,1,0),\quad G=(1,1,1).$$8

for the second-qutrit channel. In the translation-invariant qutrit Haah reference, the process shows exact power-of-three recurrence,

$$D=(0,1,1),\quad E=(1,0,1),\quad F=(1,1,0),\quad G=(1,1,1).$$9

For representative random models, this recurrence disappears. At layer $(r,1)$00, the charge count is distributed between $(r,1)$01 and $(r,1)$02, with mean

$(r,1)$03

and no sample returns to the four-charge pattern. At layer $(r,1)$04,

$(r,1)$05

whereas the uniform model again returns to $(r,1)$06 (Yan, 18 Jun 2026).

These numerical results do not prove the absence of every possible irregular fractal support. They do, however, provide the paper’s main evidence that the canonical self-similar fractal mechanism of the translation-invariant Haah-type model is absent in the representative random qutrit codes that were studied (Yan, 18 Jun 2026).

5. Distillation-oriented random cubic qutrit codes

In a distinct but related line of work, the phrase “qutrit random cubic codes” is used for random-looking large qutrit stabilizer or CSS codes whose magic-state distillation map suppresses noise to third order for the qutrit strange state (Prakash et al., 2024). The motivating question is whether distillation of a contextual qutrit state becomes more generic as code length grows, and whether thresholds can approach the contextuality boundary.

The target magic state is the qutrit strange state

$(r,1)$07

After Clifford twirling, noisy copies are described by

$(r,1)$08

and the state is contextual for $(r,1)$09. The paper treats the strange state as a canonical test case for the question of whether contextuality is sufficient for universal qutrit quantum computation (Prakash et al., 2024).

A central theorem links projection success onto a stabilizer code to the complete weight enumerator of the dual code. For a single-qudit mixed state with Wigner function $(r,1)$10, the probability for successful projection onto the trivial-syndrome eigenspace of $(r,1)$11 is

$(r,1)$12

For the strange state, the Wigner function takes only two values, reducing the complete weight enumerator to a simple weight enumerator. The output noise parameter obeys

$(r,1)$13

with

$(r,1)$14

Cubic suppression means

$(r,1)$15

for small $(r,1)$16 (Prakash et al., 2024).

The search covered all $(r,1)$17 stabilizer codes for $(r,1)$18, all $(r,1)$19 codes obtainable by shortening a $(r,1)$20 stabilizer state, and odd-length CSS codes $(r,1)$21 with a complete set of transversal Clifford gates for odd $(r,1)$22. The CSS search used maximal self-orthogonal ternary codes $(r,1)$23 via

$(r,1)$24

The restriction to codes with transversal $(r,1)$25 allows one to search only the trivial syndrome sector (Prakash et al., 2024).

The main empirical findings are sharply length dependent. For all searched codes with $(r,1)$26, no better than linear suppression was found except for the 11-qutrit Golay code. For $(r,1)$27, the study found over $(r,1)$28 CSS codes with cubic noise suppression; more precisely, $(r,1)$29 inequivalent indecomposable $(r,1)$30-qutrit CSS codes. Their thresholds range from about $(r,1)$31 to $(r,1)$32, and none surpass the Golay threshold. The 11-qutrit Golay benchmark is an $(r,1)$33 CSS code with

$(r,1)$34

for which

$(r,1)$35

The highest-threshold new $(r,1)$36-qutrit code is a $(r,1)$37 code with

$(r,1)$38

This suggests that cubic distillation of the strange state may be somewhat generic for sufficiently large structured qutrit CSS codes, while still leaving open whether any family approaches the contextuality boundary $(r,1)$39 (Prakash et al., 2024).

6. Terminology, adjacent constructions, and unresolved questions

A recurrent ambiguity is the word “cubic,” which does not have a single meaning across the relevant literature.

Usage	Meaning	Source
Qutrit random cubic codes	Cube-check 3D local qutrit CSS stabilizer Hamiltonians with spatially varying admissible coefficients	(Yan, 18 Jun 2026)
Random cubic codes in distillation	Random-looking large qutrit stabilizer/CSS codes with cubic noise suppression for $(r,1)$40	(Prakash et al., 2024)
Ternary cubic codes	Quasi-cyclic ternary codes of co-index $(r,1)$41 arising as Gray images	(Shi et al., 2016)

In coding theory, “cubic code” can mean a quasi-cyclic code of co-index $(r,1)$42. A representative construction works over

$(r,1)$43

with extension

$(r,1)$44

trace map $(r,1)$45, evaluation code

$(r,1)$46

and Gray map

$(r,1)$47

The ternary image $(r,1)$48 is quasi-cyclic of co-index $(r,1)$49, and the paper derives three-weight and two-weight families, proves optimality in the two-weight cases via the ternary Griesmer bound, and analyzes dual distance and secret sharing consequences (Shi et al., 2016). This is a different notion of “cubic” from the cube-check Hamiltonian usage.

A separate but adjacent strand constructs qutrit stabilizer codes from quasi-cyclic codes over $(r,1)$50 with large Hermitian hulls, using quantum Construction X. For an $(r,1)$51-linear code $(r,1)$52,

$(r,1)$53

and there exists an

$(r,1)$54

quantum stabilizer code with

$(r,1)$55

In the qutrit case, this systematic construction yields many best-known codes and an optimal

$(r,1)$56

code improving the previous $(r,1)$57 (Ezerman et al., 2019). This literature supplies qutrit stabilizer-code context but does not study qutrit random cubic codes in the cube-check sense.

The nearest antecedent on the Hamiltonian side is the study of 3D local qupit stabilizer codes without string logical operators. That work considers prime local dimension $(r,1)$58, one stabilizer per cube, and gives a sufficient algebraic condition for the absence of string logical operators. The abstract states that the minimal prime dimension satisfying the sufficient condition is $(r,1)$59, while for $(r,1)$60 the paper reports numerical indications that the maximum string length may still be bounded, without a proof (Kim, 2012). In this lineage, the qutrit random cubic codes are notable because they establish a fully qutrit, cube-check, no-string family with spatially varying stabilizers (Yan, 18 Jun 2026).

Several unresolved questions remain explicit in the literature. For the three-dimensional Hamiltonian family, the plane-logical and charge-push data do not constitute a full code-distance proof, and the absence of every possible irregular fractal support is not established (Yan, 18 Jun 2026). For the distillation-oriented family, the existence of many $(r,1)$61-qutrit cubic-suppressing CSS codes does not show that thresholds can approach the contextuality boundary $(r,1)$62, nor does it establish that all contextual qutrit states are distillable (Prakash et al., 2024). Taken together, these results suggest broader classes of spatially nonuniform stabilizer phases and qutrit code families than those captured by the standard translation-invariant topological or fracton templates, but they stop short of a complete classification.