Triorthogonal Quantum Codes

• Triorthogonal codes are quantum CSS codes defined by binary matrices whose rows satisfy pairwise and triplewise even-overlap constraints, enabling transversal non-Clifford logical operations.
• They underpin magic-state distillation and fault-tolerant protocols with key examples like the [[15,1,3]] and [[49,1,5]] codes demonstrating transversal T and CCZ gate implementations.
• Construction methods range from numerical searches and algebraic techniques based on self-dual codes to optimization via integer linear programming, with extensions to qudit settings and improved decoding strategies.

Triorthogonal codes are quantum Calderbank–Shor–Steane (CSS) codes defined by a binary matrix whose rows obey simultaneous pairwise and triplewise even-overlap constraints. In the binary setting, these constraints are imposed on the supports of generator rows, and they underlie several of the best-known constructions for transversal non-Clifford logical gates, especially in magic-state distillation and related fault-tolerant protocols (Bravyi et al., 2012, Paetznick et al., 2013, Camps-Moreno et al., 2024). Subsequent work has connected triorthogonal matrices to CSS-$T$ codes, characterized their equivalence classes and extremal structure, developed constructive and classification methods, and extended the framework to decoding and to non-binary qudit settings (Camps-Moreno et al., 2024, Baldelli et al., 23 May 2026).

1. Definition and structural identities

Let $T\in\mathbb F_2^{m\times n}$ have row vectors $r_1,\dots,r_m\in\mathbb F_2^n$. The matrix is triorthogonal when, for all distinct pairs and triples of rows,

$$|r_i\star r_j| \equiv 0 \pmod 2, \qquad |r_i\star r_j\star r_k| \equiv 0 \pmod 2,$$

where $\star$ denotes the Schur product and $|\cdot|$ denotes Hamming weight modulo $2$ (Camps-Moreno et al., 2024). Equivalent formulations in the literature describe the same condition as every pair of rows overlapping in an even number of coordinates and every triple of rows also overlapping in an even number of coordinates (Bravyi et al., 2012, Mishchenko et al., 2022).

A standard decomposition separates odd-weight and even-weight rows. If the top $k_1$ rows are odd and the bottom $k_0$ rows are even, so that $T=[G_1;G_0]$, then one has

$T\in\mathbb F_2^{m\times n}$0

which is a useful normal-form identity for binary triorthogonal matrices (Camps-Moreno et al., 2024). In the CSS interpretation, the even-weight rows become $T\in\mathbb F_2^{m\times n}$1-type stabilizer generators, while the odd-weight rows correspond to logical $T\in\mathbb F_2^{m\times n}$2 operators (Mishchenko et al., 2022, Bravyi et al., 2012).

This binary definition is the core combinatorial object. Many later results amount to determining which linear codes admit such a generating matrix, how these matrices behave under equivalence and puncturing, and which logical gates their associated CSS codes support.

2. CSS and CSS-$T\in\mathbb F_2^{m\times n}$3 formulations

If $T\in\mathbb F_2^{m\times n}$4 is triorthogonal and $T\in\mathbb F_2^{m\times n}$5, then

$T\in\mathbb F_2^{m\times n}$6

and the hull

$T\in\mathbb F_2^{m\times n}$7

is exactly $T\in\mathbb F_2^{m\times n}$8 (Camps-Moreno et al., 2024). This identifies triorthogonal codes as a special subclass of CSS-$T\in\mathbb F_2^{m\times n}$9 constructions. In particular, the CSS pair

$r_1,\dots,r_m\in\mathbb F_2^n$0

satisfies the CSS-$r_1,\dots,r_m\in\mathbb F_2^n$1 condition

$r_1,\dots,r_m\in\mathbb F_2^n$2

and therefore yields an $r_1,\dots,r_m\in\mathbb F_2^n$3 quantum code with $r_1,\dots,r_m\in\mathbb F_2^n$4 and $r_1,\dots,r_m\in\mathbb F_2^n$5 (Camps-Moreno et al., 2024).

This formulation makes the odd/even row split intrinsic rather than basis-dependent. A key structural consequence is that a binary triorthogonal matrix determines the parameters of the corresponding triorthogonal quantum code uniquely: if $r_1,\dots,r_m\in\mathbb F_2^n$6 and $r_1,\dots,r_m\in\mathbb F_2^n$7 are equivalent and both triorthogonal, then they give rise to the same $r_1,\dots,r_m\in\mathbb F_2^n$8 parameters (Camps-Moreno et al., 2024). More precisely, $r_1,\dots,r_m\in\mathbb F_2^n$9 is triorthogonal if and only if

$$|r_i\star r_j| \equiv 0 \pmod 2, \qquad |r_i\star r_j\star r_k| \equiv 0 \pmod 2,$$0

where $$|r_i\star r_j| \equiv 0 \pmod 2, \qquad |r_i\star r_j\star r_k| \equiv 0 \pmod 2,$$1 is a permutation matrix on the odd-weight rows, $$|r_i\star r_j| \equiv 0 \pmod 2, \qquad |r_i\star r_j\star r_k| \equiv 0 \pmod 2,$$2 is invertible on the even-weight rows, and $$|r_i\star r_j| \equiv 0 \pmod 2, \qquad |r_i\star r_j\star r_k| \equiv 0 \pmod 2,$$3 is arbitrary (Camps-Moreno et al., 2024). Thus equivalent triorthogonal generator matrices differ only by permuting odd rows, mixing in $$|r_i\star r_j| \equiv 0 \pmod 2, \qquad |r_i\star r_j\star r_k| \equiv 0 \pmod 2,$$4, or changing the basis of $$|r_i\star r_j| \equiv 0 \pmod 2, \qquad |r_i\star r_j\star r_k| \equiv 0 \pmod 2,$$5.

The same paper introduces a partial order on triorthogonal codes,

$$|r_i\star r_j| \equiv 0 \pmod 2, \qquad |r_i\star r_j\star r_k| \equiv 0 \pmod 2,$$6

and characterizes extremal elements in this CSS-$$|r_i\star r_j| \equiv 0 \pmod 2, \qquad |r_i\star r_j\star r_k| \equiv 0 \pmod 2,$$7 poset. A code is minimal if and only if it is one-dimensional and even, so $$|r_i\star r_j| \equiv 0 \pmod 2, \qquad |r_i\star r_j\star r_k| \equiv 0 \pmod 2,$$8; it is maximal if and only if

$$|r_i\star r_j| \equiv 0 \pmod 2, \qquad |r_i\star r_j\star r_k| \equiv 0 \pmod 2,$$9

(Camps-Moreno et al., 2024). These results make precise when a triorthogonal code can no longer be reduced or enlarged without leaving the triorthogonal class.

3. Transversal gates and logical action

Triorthogonal codes were introduced in connection with magic-state distillation, where the central operational property is a transversal realization of non-Clifford gates (Bravyi et al., 2012). In the Bravyi–Haah construction, any triorthogonal matrix gives a stabilizer code for which there exists a diagonal Clifford $\star$0, composed only of $\star$1 and $\star$2 gates, such that

$\star$3

acts as a logical $\star$4 on all encoded qubits simultaneously (Bravyi et al., 2012). Later formulations state the same operational fact as a transversal $\star$5 followed by prescribed Clifford fixups, typically single-qubit $\star$6 and two-qubit $\star$7 corrections (Jain et al., 2024).

The logical action can be sharper under additional conditions. In the CSS framework $\star$8, a full generator matrix of $\star$9 realizes physical transversal $|\cdot|$0 as logical $|\cdot|$1 without further Clifford correction if and only if the matrix is triorthogonal and an additional Hamming-weight condition modulo $|\cdot|$2 holds: $|\cdot|$3 for the appropriate logical representative $|\cdot|$4, stabilizer element $|\cdot|$5, and logical bit pattern $|\cdot|$6 (Rengaswamy et al., 2020, Rengaswamy et al., 2019). The same line of work proves that triorthogonal codes are, essentially, the most general CSS codes that realize logical transversal $|\cdot|$7 via physical transversal $|\cdot|$8 (Rengaswamy et al., 2019, Rengaswamy et al., 2020).

A second major gate-theoretic property is transversal $|\cdot|$9. Any triorthogonal code admits

$2$0

as a logical $2$1 on three encoded blocks, because the triple-overlap condition cancels all unwanted phase terms except the logical cubic phase (Paetznick et al., 2013). Together with transversal Hadamard plus fault-tolerant error correction, this yields a universal distillation-free gate set (Paetznick et al., 2013). More recently, for $2$2-triorthogonal CSS codes $2$3, parallel physical $2$4 gates between corresponding qubits of two blocks were shown to implement logical $2$5, and this property was used to construct lower-overhead logical $2$6 gadgets and teleportation protocols between a triorthogonal code and a companion symmetric CSS code (Jiao et al., 7 Oct 2025).

The gate-theoretic picture therefore has two layers: triorthogonality guarantees level-3 Clifford-hierarchy structure, while additional constraints determine whether the implementation is exact, corrected by a Clifford, or embedded into a larger code-switching architecture.

4. Construction methods and propagation rules

Several distinct construction paradigms are now known. The earliest systematic family is the Bravyi–Haah $2$7 family, which exists for every even $2$8 and gives encoding rate $2$9 (Paetznick et al., 2013, Bravyi et al., 2012). The same work introduced a numerical search method based on column types and linear equations over $k_1$0, yielding, among other examples, a distance-5 $k_1$1 code (Bravyi et al., 2012).

A more algebraic route starts from binary self-dual codes. An algorithm in "Triorthogonal Codes and Self-dual Codes" constructs a maximal triorthogonal subspace $k_1$2 inside a self-dual $k_1$3 code $k_1$4 by iteratively adding codewords that satisfy pairwise orthogonality and triplewise orthogonality with the current generating set (Shi et al., 2024). That paper also generalizes classical shortening and extending to the triorthogonal setting. Shortening on a coordinate preserves triorthogonality, and extending by prepending a standard basis column $k_1$5 likewise preserves triorthogonality (Shi et al., 2024). Additional propagation rules include direct concatenation, block-diagonal direct sum, Plotkin-style sums, and a building-up construction by a single extra row (Shi et al., 2024).

The CSS-$k_1$6 viewpoint yields another propagation rule. If $k_1$7 is triorthogonal and $k_1$8, then

$k_1$9

is again triorthogonal (Camps-Moreno et al., 2024). If $k_0$0 has odd weight, the associated quantum code gains one logical qubit, producing parameters $k_0$1; if $k_0$2 has even weight, the logical dimension is unchanged, but the distance bound $k_0$3 may improve (Camps-Moreno et al., 2024).

A third construction framework is optimization-based. "On Constructing and Decoding Quantum Triorthogonal Codes" formulates the search for even-weight triorthogonal generator matrices with prescribed dual distance as an integer linear program combining triorthogonality constraints, MacWilliams identities, and Krawtchouk-polynomial conditions (Baldelli et al., 23 May 2026). For odd $k_0$4, dual distance $k_0$5, and $k_0$6, this approach finds nontrivial $k_0$7 codes at lengths

$k_0$8

with some marked as triply-even and others explicitly not necessarily generated by classical triply-even codes (Baldelli et al., 23 May 2026).

5. Representative families, examples, and finite-length classification

The finite-length landscape is anchored by a small set of canonical examples. The $k_0$9 punctured Reed–Muller code is the first standard nontrivial case and remains the prototypical triorthogonal $T=[G_1;G_0]$0-code (Bravyi et al., 2012, Jain et al., 2024). The $T=[G_1;G_0]$1 code can be viewed as a doubling of the $T=[G_1;G_0]$2 self-dual color code and the $T=[G_1;G_0]$3 Reed–Muller code, and the $T=[G_1;G_0]$4 code is obtained by doubling the $T=[G_1;G_0]$5 Golay code with the $T=[G_1;G_0]$6 triorthogonal code (Sullivan, 2023).

Code	Construction	Role noted in the literature
$T=[G_1;G_0]$7	punctured Reed–Muller	standard small triorthogonal $T=[G_1;G_0]$8-code
$T=[G_1;G_0]$9	doubling $T\in\mathbb F_2^{m\times n}$00 and $T\in\mathbb F_2^{m\times n}$01	crucial ingredient in Golay conversion
$T\in\mathbb F_2^{m\times n}$02	doubling $T\in\mathbb F_2^{m\times n}$03 and $T\in\mathbb F_2^{m\times n}$04	transversal $T\in\mathbb F_2^{m\times n}$05 and code-conversion target

For the doubling construction, the distance formula is explicit. The $T\in\mathbb F_2^{m\times n}$06 code satisfies

$T\in\mathbb F_2^{m\times n}$07

and the $T\in\mathbb F_2^{m\times n}$08 code satisfies

$T\in\mathbb F_2^{m\times n}$09

(Sullivan, 2023). These are not isolated examples: a 2024 compilation lists triorthogonal $T\in\mathbb F_2^{m\times n}$10 codes for odd distances $T\in\mathbb F_2^{m\times n}$11 with lengths

$T\in\mathbb F_2^{m\times n}$12

and states, to the authors’ knowledge, that these are the smallest known triorthogonal codes for their respective distances (Jain et al., 2024). In that family the qubit overhead scales as $T\in\mathbb F_2^{m\times n}$13, and the stabilizer generator weights scale roughly as $T\in\mathbb F_2^{m\times n}$14 (Jain et al., 2024).

At very small sizes, the classification problem is essentially complete. There are exactly $T\in\mathbb F_2^{m\times n}$15 distinguished triorthogonal subspaces with $T\in\mathbb F_2^{m\times n}$16, and every triorthogonal code in that range descends from one of them through puncturing and deleting qubits (Nezami et al., 2021). This classification is phrased in terms of Reed–Muller indicator polynomials and provides a finite-length taxonomy of all small examples (Nezami et al., 2021).

A useful caution appears already in the smallest worked examples. For the $T\in\mathbb F_2^{m\times n}$17 matrix with three odd rows and no even rows,

$T\in\mathbb F_2^{m\times n}$18

the formal CSS-$T\in\mathbb F_2^{m\times n}$19 lower bound gives $T\in\mathbb F_2^{m\times n}$20, but in fact $T\in\mathbb F_2^{m\times n}$21 and $T\in\mathbb F_2^{m\times n}$22 has distance $T\in\mathbb F_2^{m\times n}$23 (Camps-Moreno et al., 2024). This clarifies a recurring point: formal lower bounds arising from the hull construction need not coincide with the true $T\in\mathbb F_2^{m\times n}$24-distance.

6. Decoding, applications, limitations, and non-binary extensions

Triorthogonal codes are the workhorse for low-overhead magic-state distillation because their pairwise and triplewise even-overlap structure guarantees that a transversal $T\in\mathbb F_2^{m\times n}$25-gate on physical qubits implements a logical $T\in\mathbb F_2^{m\times n}$26 on the distilled subspace (Baldelli et al., 23 May 2026). In the original Bravyi–Haah framework, recursive distillation based on an $T\in\mathbb F_2^{m\times n}$27 triorthogonal code gives overhead

$T\in\mathbb F_2^{m\times n}$28

and the $T\in\mathbb F_2^{m\times n}$29 family approaches $T\in\mathbb F_2^{m\times n}$30 (Bravyi et al., 2012). Generalized triorthogonal constructions later pushed asymptotic distillation efficiency toward $T\in\mathbb F_2^{m\times n}$31 (Haah et al., 2017).

On the decoding side, recent work evaluates high-distance triorthogonal codes over the dephasing channel using bounded-distance decoding, belief propagation plus ordered-statistics post-processing, and an adapted GRAND decoder (Baldelli et al., 23 May 2026). For the $T\in\mathbb F_2^{m\times n}$32 and $T\in\mathbb F_2^{m\times n}$33 codes, qGRAND with large query budget is reported as slightly better at low $T\in\mathbb F_2^{m\times n}$34 than the best BP+OSD variants on the $T\in\mathbb F_2^{m\times n}$35 code, and competitive or better on the denser $T\in\mathbb F_2^{m\times n}$36 code; the paper concludes that the GRAND-based decoder is a promising option (Baldelli et al., 23 May 2026).

Triorthogonal codes also appear in code-conversion architectures. The $T\in\mathbb F_2^{m\times n}$37 Golay code can be fault-tolerantly converted to a $T\in\mathbb F_2^{m\times n}$38 triorthogonal code with a transversal $T\in\mathbb F_2^{m\times n}$39 gate, using only transversal Clifford operations, single-qubit measurements, and an ancilla construction based on the $T\in\mathbb F_2^{m\times n}$40 code (Sullivan, 2023). A different universal-computation route pairs a triorthogonal code with a symmetric CSS code so that $T\in\mathbb F_2^{m\times n}$41 and $T\in\mathbb F_2^{m\times n}$42 are transversal in the triorthogonal block while Clifford gates are transversal in the symmetric block, with logical teleportation between the two (Jiao et al., 7 Oct 2025).

In secure multi-party quantum computation, triorthogonal CSS codes broaden the set of feasible node counts relative to triply-even codes. Reported examples include $T\in\mathbb F_2^{m\times n}$43, $T\in\mathbb F_2^{m\times n}$44, and $T\in\mathbb F_2^{m\times n}$45, whereas the small triply-even examples are $T\in\mathbb F_2^{m\times n}$46 and $T\in\mathbb F_2^{m\times n}$47 (Mishchenko et al., 2022). This supports the narrower but important distinction that triply-even codes form a subclass of triorthogonal codes, while triorthogonality permits additional small-length instances (Mishchenko et al., 2022, Baldelli et al., 23 May 2026).

Several limitations recur across the literature. Known binary triorthogonal codes often have relatively low distance, frequently $T\in\mathbb F_2^{m\times n}$48 or $T\in\mathbb F_2^{m\times n}$49, and $T\in\mathbb F_2^{m\times n}$50 is often small, commonly $T\in\mathbb F_2^{m\times n}$51 (Paetznick et al., 2013, Mishchenko et al., 2022). Logical Hadamard is generally not transversal in a single triorthogonal code and must be supplied by error correction, teleportation, or code switching (Paetznick et al., 2013, Mishchenko et al., 2022, Jiao et al., 7 Oct 2025). These constraints motivate current work on higher-distance constructions, better decoding, and hybrid architectures.

The framework has also been generalized beyond binary qubits. For qutrits, there is a family of $T\in\mathbb F_2^{m\times n}$52 triorthogonal codes with

$T\in\mathbb F_2^{m\times n}$53

which tends to $T\in\mathbb F_2^{m\times n}$54 as $T\in\mathbb F_2^{m\times n}$55; the $T\in\mathbb F_2^{m\times n}$56 example has $T\in\mathbb F_2^{m\times n}$57 (Prakash et al., 2024). Over $T\in\mathbb F_2^{m\times n}$58, asymptotically good triorthogonal codes with transversal $T\in\mathbb F_2^{m\times n}$59 have been constructed, including a $T\in\mathbb F_2^{m\times n}$60 example (Cervia et al., 26 Dec 2025). For all prime dimensions, punctured Reed–Muller constructions yield sublogarithmic overhead, and a $T\in\mathbb F_2^{m\times n}$61 code with $T\in\mathbb F_2^{m\times n}$62 is explicitly reported (Saha et al., 12 Oct 2025). These non-binary results suggest that triorthogonality is not merely a qubit-specific device but a broader coding-theoretic mechanism for transversal level-3 Clifford-hierarchy gates.