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Classification of Generalised Triorthogonal Codes through Length 54

This paper significantly expands the classification of generalised triorthogonal codes for magic state distillation from length 38 to 54, identifying 74 optimal protocols (65 of which are new) by extending the unital triorthogonal space classification using a directional derivative method.

Original authors: Adam Wills, Shubham P. Jain, Shraddha Singh

Published 2026-09-28

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CC BY 4.0

Original authors: Adam Wills, Shubham P. Jain, Shraddha Singh

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Quantum computers promise to solve problems that would take today's machines thousands of years, but they are incredibly fragile. To work, they must perform calculations using special operations that are not part of the standard toolkit available to most quantum systems. These special operations require a resource known as a "magic state," a highly precise quantum condition that is difficult to create and even harder to keep clean. In the real world, every attempt to make these states introduces errors, much like trying to pour water from a leaky bucket. To fix this, scientists use a process called distillation, where they take many noisy, imperfect copies of a magic state and combine them to produce a single, high-quality version. This is a critical bottleneck; without efficient ways to clean these states, the powerful algorithms that quantum computers could run remain out of reach.

The challenge lies in finding the most efficient way to perform this cleaning. Scientists have long used a specific mathematical framework to design these distillation recipes, known as generalised triorthogonal codes. These codes act as blueprints that tell a quantum computer how to arrange its qubits and operations to filter out errors. For years, researchers have been searching through the vast space of possible blueprints to find the ones that use the fewest resources. Previous efforts had mapped out the most efficient options for protocols involving up to thirty-eight input states, but the landscape beyond that remained largely uncharted. The search was difficult because the number of possibilities grows explosively, and finding the absolute best solution required checking a space that was too large for previous methods to handle.

In a new study, a team of researchers has pushed this boundary significantly further, mapping out the most efficient distillation protocols for systems using up to fifty-four input states. They focused on protocols that are robust enough to catch errors, a requirement that keeps the list of candidates manageable while ensuring the results are useful for real-world machines. By developing a new mathematical technique to navigate this complex space, they identified seventy-four distinct protocols that represent the best possible trade-offs between the number of inputs needed, the physical space required to run the process, and the ability to detect errors. Of these seventy-four optimal solutions, sixty-five were entirely new discoveries, expanding the known toolkit for quantum engineers by a wide margin.

The researchers achieved this by refining the way they break down the problem. Instead of trying to build every possible code from scratch, they first identified a core set of mathematical structures, which they call unital triorthogonal spaces. These structures serve as the stable foundation for the codes. The team then systematically added the necessary logical components to these foundations to create full protocols. To handle the sheer volume of possibilities, they employed a method that analyzes how these mathematical structures change when viewed from different angles, allowing them to reconstruct the full list of possibilities from smaller, simpler pieces. This approach allowed them to extend the classification of these codes from the previous limit of thirty-eight inputs all the way to fifty-four.

The results reveal a rich variety of efficient protocols that were previously unknown. The team found that for many different types of output states, there are now proven ways to distill them using fewer inputs or less physical space than ever before. For instance, they identified new methods for creating specific multi-qubit states that are essential for complex quantum algorithms. While some of the best-known protocols from earlier work still hold up, the new list offers many alternatives that are better suited for different hardware constraints. The researchers also noted that as the size of the system grows, the number of possible structures increases so rapidly that simply listing them all becomes impractical without new theoretical breakthroughs. This suggests that while the current map is the most complete ever made, the journey to find even better methods will require fresh ideas rather than just more computing power.

The study provides a definitive catalog for the quantum computing community, offering a clear set of options for building the next generation of fault-tolerant machines. By knowing exactly which protocols are optimal for a given number of inputs, engineers can stop guessing and start building with confidence. The work also highlights the complementary nature of different approaches; while other methods exist for creating these states, they often work best at a lower level of error protection. The new classification fills a crucial gap by providing high-performance options that are robust enough for the most demanding applications. With these new blueprints in hand, the path toward reliable, large-scale quantum computation becomes a little less uncertain and a little more concrete.

Technical Summary: Classification of Generalised Triorthogonal Codes through Length 54

Problem Statement
Magic state distillation (MSD) is a critical primitive for fault-tolerant quantum computation, enabling the implementation of non-Clifford gates (specifically third-level diagonal gates like $T$ , $CS$ , and $CCZ$ ) using only stabilizer operations and noisy input states. While stabilizer operations are efficient, the production of high-fidelity non-Clifford resources remains a key bottleneck. The most widely studied framework for MSD involves generalised triorthogonal codes, which distill $n$ noisy input $|T\rangle$ states into $k$ purified output magic states.

Prior classification efforts, notably by Nezami and Haah, were limited to protocols with total length $n + k \le 38$ and distance $d \ge 3$ . Existing searches often relied on heuristic methods without guaranteed optimality. Furthermore, while alternative methods like "synthillation" can construct protocols for specific magic states, they are restricted to distance $d=2$ . The authors identify a need for a comprehensive, optimal classification of generalised triorthogonal protocols at higher distances ( $d \ge 3$ ) and longer lengths to better understand the trade-offs between input count ( $n$ ), spatial footprint ( $N$ ), and error suppression distance ( $d$ ).

Methodology
The authors extend the classification of generalised triorthogonal codes to length $n \le 54$ with distance $d_Z \ge 3$ . Their approach involves three main technical components:

Reduction to Unital Triorthogonal Spaces:
The authors establish a reduction from generalised triorthogonal matrices to unital triorthogonal spaces. A generalised triorthogonal matrix $G$ is decomposed into logical rows ( $G_1$ ) and stabiliser rows ( $G_0$ ). By analyzing the stabiliser columns, they map the problem to finding a linear subspace of $\mathbb{F}_2^c$ containing the all-ones vector (unitality) where the overlap of any three vectors is zero (triorthogonality).
- They demonstrate that a generalised triorthogonal matrix of length $n$ corresponds to a unital triorthogonal space of length $c = n$ (if $n$ is even) or $c = n+1$ (if $n$ is odd).
- Once a valid stabiliser space is identified, they systematically add compatible logical rows. This involves solving for $k$ -dimensional subspaces that are totally isotropic with respect to specific bilinear forms derived from the stabiliser overlaps.
Directional Derivative Method for Reed-Muller Classification:
To classify the required unital triorthogonal spaces up to length 54, the authors build upon the correspondence between these spaces and Reed-Muller (RM) codewords. Specifically, unital triorthogonal spaces of dimension $1+m$ and length $c$ correspond to affine equivalence classes of RM codewords in $RM(m-4, m)$ with weight $c$ .
- While Nezami and Haah used classical results to classify up to length 38, the authors develop a directional derivative method to push this to length 54.
- This recursive method decomposes a polynomial $f(z, x_m)$ into $f_0(z) + x_m g(z)$ , where $g$ is a derivative with fewer variables and lower degree. By bounding the number of "double pairs" (points where $f(z,0)=f(z,1)=1$ ), they reconstruct all possible polynomials $f$ from lower-variable catalogues, significantly expanding the search space beyond previous capabilities.
Pareto Optimality Search:
The authors define a protocol as Pareto optimal if no other protocol exists that is at least as good in both input count ( $n$ ) and spatial footprint ( $N$ ) while being strictly better in one, for a fixed output class and distance. They filter out protocols with duplicate columns or zero columns, as these can be removed to improve efficiency.

Key Contributions and Results

Expanded Classification: The paper presents the first complete classification of generalised triorthogonal protocols for lengths up to $n=54$ at distance $d_Z \ge 3$ .
New Protocols: The authors identify 74 Pareto-optimal protocols spanning 62 distinct output magic state classes. Of these, 65 protocols are new to the literature, even when accounting for simple modifications of existing protocols (such as basis changes or restricting logical rows).
Distance and Output Diversity:
- 67 protocols have distance $d_Z = 3$ .
- 5 protocols have distance $d_Z = 4$ .
- 2 protocols have distance $d_Z = 5$ .
- The output classes include various combinations of $T$ , $CS$ , and $CCZ$ gates, extending beyond simple $T$ -state distillation to exotic magic states.
Complementarity to Synthillation: The results are complementary to "synthillation," which can distill third-level states but is limited to distance 2. This work provides the optimal distance-3+ alternatives for the same class of gates.
Data Availability: The authors provide a complete dataset of the classified protocols, unital triorthogonal spaces, and verification evidence, along with the code used for regeneration.

Significance and Limitations
The paper claims that this classification significantly expands the known landscape of efficient magic state distillation protocols. By identifying 65 new optimal protocols, it offers concrete options for fault-tolerant architectures that may reduce resource overhead (input $T$ states and qubit count) for specific non-Clifford operations.

However, the authors are modest regarding the scalability of their method. They note that the number of unital triorthogonal spaces grows exponentially with length (e.g., from 28 at length 48 to over 293 million at length 54). They estimate that a classification up to length 56 would require $\sim 10^8$ core-hours, suggesting that further exhaustive classification via this specific method is computationally impractical without new theoretical breakthroughs. Consequently, they point to companion work utilizing non-exhaustive, AI-assisted methods for discovering protocols beyond this exhaustive frontier.

The work does not propose new experimental implementations but rather provides a theoretical catalog of optimal code structures. The equivalence of magic states is defined strictly under CNOT and $S$ gates (a subset of full Clifford equivalence), acknowledging that full stabilizer equivalence might group some distinct classes together, a topic left for future understanding.

Technical Summary: Classification of Generalised Triorthogonal Codes through Length 54

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