Reed–Muller Distillation Factory

• Reed–Muller distillation factories are magic-state distillation architectures that use Reed–Muller, CSS-T, and triorthogonal codes to enable efficient transversal non-Clifford operations.
• They integrate recursive concatenation and geometric unfolding techniques to convert hypercube code structures into 2D/3D local layouts for stabilizer measurements.
• Representative modules, such as the 15-T, 64-T, and 127-T routines, achieve precise error suppression and highlight tradeoffs in code parameters and resource overhead.

A Reed–Muller distillation factory is a magic-state distillation construction whose coding-theoretic core is supplied by Reed–Muller codes, quantum Reed–Muller codes, or punctured/shortened Reed–Muller-derived CSS and triorthogonal codes. In the literature, the term encompasses at least three closely related objects: Reed–Muller code families that admit transversal non-Clifford action and therefore serve as candidate distillation ingredients; explicit distillation modules such as $15\ket T\to\ket T$, $64\ket T\to\ket{CCZ}$, and $127\ket T\to\ket T$ routines; and architectural realizations in which the relevant stabilizer measurements are made geometrically local in 2D or 3D. The subject therefore sits at the intersection of algebraic coding theory, transversal-gate constructions, and factory-level resource optimization (Andrade et al., 2023, Londe, 7 May 2026, Saha et al., 12 Oct 2025, Campbell et al., 2012).

1. Conceptual scope and factory model

In the code-theoretic sense, Reed–Muller constructions are relevant because transversal $T$-type operations act nontrivially on suitable Reed–Muller codes, so the codes can be used as distillation or checking gadgets for noisy $T$ states, or for producing $CCZ$ states (Londe, 7 May 2026). In the stabilizer-code sense, a CSS-T code is a CSS code $CSS(C_1,C_2)$ built from binary linear codes $C_2\subseteq C_1$ such that the code space is preserved by transversal physical $T$ and $T^\dagger$, making it a natural ingredient for implementing logical non-Clifford resources (Andrade et al., 2023).

In the protocol sense, a factory is a recursively concatenated family of triorthogonal-code modules. For a triorthogonal code with parameters $64\ket T\to\ket{CCZ}$0, the routine consumes $64\ket T\to\ket{CCZ}$1 noisy input magic states and outputs $64\ket T\to\ket{CCZ}$2 improved ones, with leading-order suppression

$64\ket T\to\ket{CCZ}$3

where $64\ket T\to\ket{CCZ}$4 is the number of weight-$64\ket T\to\ket{CCZ}$5 logical $64\ket T\to\ket{CCZ}$6-type operators (Saha et al., 12 Oct 2025). After $64\ket T\to\ket{CCZ}$7 concatenation levels,

$64\ket T\to\ket{CCZ}$8

from

$64\ket T\to\ket{CCZ}$9

noisy magic states, and the asymptotic overhead scales as

$127\ket T\to\ket T$0

This gives a precise operational meaning to “factory”: an iterated family of distillation blocks whose performance is governed by $127\ket T\to\ket T$1, $127\ket T\to\ket T$2, $127\ket T\to\ket T$3, $127\ket T\to\ket T$4, and postselection (Saha et al., 12 Oct 2025).

A common misconception is to identify every Reed–Muller transversal-$127\ket T\to\ket T$5 code with a complete factory protocol. That identification is too strong. Some works provide explicit distillation maps and leading-order error formulas, whereas others provide only the code-family input that such a factory would require. In particular, the CSS-T Reed–Muller analysis of (Andrade et al., 2023) is explicitly coding-theoretic rather than protocol-theoretic.

2. Reed–Muller algebraic foundations

The binary Reed–Muller code used in the CSS-T setting is

$127\ket T\to\ket T$6

with parameters

$127\ket T\to\ket T$7

nesting

$127\ket T\to\ket T$8

and duality

$127\ket T\to\ket T$9

(Andrade et al., 2023).

The local-unfolding literature uses a quantum Reed–Muller notation

$T$0

defined on $T$1 physical qubits, with $T$2-stabilizer group $T$3 and $T$4-stabilizer group $T$5. Equivalently, the $T$6 stabilizers are generated by subcubes of dimension $T$7, and the $T$8 stabilizers by subcubes of dimension $T$9. A subset of vertices is represented by

$T$0

with associated Pauli operators

$T$1

(Londe, 7 May 2026).

For prime dimension $T$2, the classical Reed–Muller code generalizes to

$T$3

whose codewords are evaluations

$T$4

Its dual is

$T$5

(Saha et al., 12 Oct 2025).

A distinct odd-prime construction uses shortened first-order $T$6-ary Reed–Muller codes to define a one-logical-qudit CSS code

$T$7

on

$T$8

physical qudits, with

$T$9

(Campbell et al., 2012). This family underlies explicit qutrit and ququint distillation protocols rather than only asymptotic code families.

3. Transversal non-Clifford structure

For binary CSS-T codes, the defining conditions are that $CCZ$0 is an even code and that for each codeword $CCZ$1, there exists a self-dual code in $CCZ$2 supported on $CCZ$3 (Andrade et al., 2023). A structural criterion used in the analysis is:

An $CCZ$4 binary linear code $CCZ$5 contains a self-dual code if and only if $CCZ$6 is even and $CCZ$7 is self-orthogonal, meaning that $CCZ$8.

Combined with puncturing/shortening duality,

$CCZ$9

this yields the equivalent support condition

$CSS(C_1,C_2)$0

(Andrade et al., 2023).

The central Reed–Muller CSS-T characterization takes

$CSS(C_1,C_2)$1

and proves that $CSS(C_1,C_2)$2 is a CSS-T code if and only if

$CSS(C_1,C_2)$3

or

$CSS(C_1,C_2)$4

The proof proceeds by translating the support condition into a puncturing/shortening inclusion and then using degree bounds for products of polynomials (Andrade et al., 2023).

In the prime-dimensional triorthogonal setting, the exact algebraic criterion is

$CSS(C_1,C_2)$5

Given a triorthogonal generator matrix $CSS(C_1,C_2)$6, one punctures $CSS(C_1,C_2)$7 coordinates to obtain $CSS(C_1,C_2)$8, shortens on the same coordinates to obtain $CSS(C_1,C_2)$9, and forms

$C_2\subseteq C_1$0

If the surviving submatrix has full rank, then the resulting CSS code has length $C_2\subseteq C_1$1, dimension $C_2\subseteq C_1$2, and in the nondegenerate case considered,

$C_2\subseteq C_1$3

(Saha et al., 12 Oct 2025).

For odd prime dimension $C_2\subseteq C_1$4, the transversal structure is expressed in terms of a diagonal non-Clifford gate $C_2\subseteq C_1$5. An $C_2\subseteq C_1$6-distillation code satisfies

$C_2\subseteq C_1$7

with logical operators

$C_2\subseteq C_1$8

(Campbell et al., 2012). This transversality is the mechanism that turns a Reed–Muller code into a distillation primitive.

The 3D local “rubik’s code” $C_2\subseteq C_1$9 makes this logical action particularly explicit. It encodes 15 logical qubits indexed by 2-element subsets of $T$0, and a transversal $T$1 or $T$2 on all 64 qubits acts logically as the product of all $T$3 gates on triples of logical qubits that partition $T$4: $T$5 (Londe, 7 May 2026).

4. Distillation modules and representative constructions

The Reed–Muller factory literature contains both one-output and multi-output modules. In odd prime dimension, one round of the protocol takes $T$6 copies of a noisy single-qudit state, twirls each under the Clifford $T$7, measures the $T$8-type stabilizers, applies a Clifford correction conditioned on those outcomes, measures the $T$9-type stabilizers and postselects on the trivial syndrome, then decodes to one qudit (Campbell et al., 2012). In the punctured-triorthogonal setting, the design recipe is: choose a triorthogonal $T^\dagger$0, puncture a set $T^\dagger$1, shorten on the same coordinates, form the CSS code, and use its transversal $T^\dagger$2 to distill $T^\dagger$3 magic states from $T^\dagger$4 noisy ones (Saha et al., 12 Oct 2025).

Module	Map or parameters	Reported leading performance
Punctured $T^\dagger$5	$T^\dagger$6-type Reed–Muller $T^\dagger$7-state factory	$T^\dagger$8
$T^\dagger$9	$64\ket T\to\ket{CCZ}$00	$64\ket T\to\ket{CCZ}$01
Big unfolded code	$64\ket T\to\ket{CCZ}$02	$64\ket T\to\ket{CCZ}$03
Punctured $64\ket T\to\ket{CCZ}$04	$64\ket T\to\ket{CCZ}$05	$64\ket T\to\ket{CCZ}$06
$64\ket T\to\ket{CCZ}$07	$64\ket T\to\ket{CCZ}$08 qutrit code	$64\ket T\to\ket{CCZ}$09
$64\ket T\to\ket{CCZ}$10	$64\ket T\to\ket{CCZ}$11 ququint code	$64\ket T\to\ket{CCZ}$12

For the local binary factories, the leading-order rule is

$64\ket T\to\ket{CCZ}$13

where $64\ket T\to\ket{CCZ}$14 is the $64\ket T\to\ket{CCZ}$15-distance and $64\ket T\to\ket{CCZ}$16 is the number of minimum-weight nontrivial $64\ket T\to\ket{CCZ}$17 logical operators (Londe, 7 May 2026). Specific values reported are: punctured $64\ket T\to\ket{CCZ}$18, $64\ket T\to\ket{CCZ}$19, $64\ket T\to\ket{CCZ}$20; $64\ket T\to\ket{CCZ}$21, $64\ket T\to\ket{CCZ}$22, $64\ket T\to\ket{CCZ}$23; the big unfolded code, $64\ket T\to\ket{CCZ}$24, $64\ket T\to\ket{CCZ}$25; and punctured $64\ket T\to\ket{CCZ}$26, $64\ket T\to\ket{CCZ}$27, $64\ket T\to\ket{CCZ}$28 (Londe, 7 May 2026).

For odd-prime protocols, the qutrit code $64\ket T\to\ket{CCZ}$29 has threshold

$64\ket T\to\ket{CCZ}$30

for all noise types and

$64\ket T\to\ket{CCZ}$31

for depolarizing noise, while the ququint code $64\ket T\to\ket{CCZ}$32 has

$64\ket T\to\ket{CCZ}$33

for generic noise and

$64\ket T\to\ket{CCZ}$34

for depolarizing noise (Campbell et al., 2012). The ququint update under depolarizing noise is given exactly by

$64\ket T\to\ket{CCZ}$35

The prime-$64\ket T\to\ket{CCZ}$36 punctured Reed–Muller program emphasizes multi-output modules and asymptotic overhead rather than only one-output routines. Its most practically striking searched example is the ququint code

$64\ket T\to\ket{CCZ}$37

with

$64\ket T\to\ket{CCZ}$38

The same table gives

$64\ket T\to\ket{CCZ}$39

and with $64\ket T\to\ket{CCZ}$40, one round gives

$64\ket T\to\ket{CCZ}$41

with distillation cost

$64\ket T\to\ket{CCZ}$42

(Saha et al., 12 Oct 2025). This suggests that search-optimized puncturing can materially outperform analytically convenient puncturing rules at finite size.

5. Geometric locality and unfolding

A major recent development is the conversion of Reed–Muller distillation constructions into geometrically local factories by “unfolding” the hypercube description of the code into 2D or 3D layouts in which a basis of the $64\ket T\to\ket{CCZ}$43-stabilizer group becomes local (Londe, 7 May 2026). The underlying algebraic mechanism is a product decomposition of polynomial subspaces associated with subcube types, rather than an ad hoc layout search.

For $64\ket T\to\ket{CCZ}$44, two cube coordinates are grouped into one planar axis and the other two into the second planar axis. For $64\ket T\to\ket{CCZ}$45, the coordinates are split as $64\ket T\to\ket{CCZ}$46 and $64\ket T\to\ket{CCZ}$47, producing a planar $64\ket T\to\ket{CCZ}$48 structure; 49 local square checks are obtained from the product grid, and 8 more checks are appended to span the full 57-dimensional $64\ket T\to\ket{CCZ}$49-stabilizer group (Londe, 7 May 2026). The Gray-code basis used for $64\ket T\to\ket{CCZ}$50 is

$64\ket T\to\ket{CCZ}$51

For the 64-qubit “rubik’s code” $64\ket T\to\ket{CCZ}$52, the coordinates are grouped as

$64\ket T\to\ket{CCZ}$53

The 27 bulk cubes correspond to the little cubes of a Rubik’s cube, and 15 more local cubes are placed on the boundary faces, giving 42 $64\ket T\to\ket{CCZ}$54-stabilizer generators in total, matching

$64\ket T\to\ket{CCZ}$55

According to the introduction, the 3D layout uses 64 data qubits plus 42 additional qubits (Londe, 7 May 2026).

The “big unfolded code” interpolates between $64\ket T\to\ket{CCZ}$56 and $64\ket T\to\ket{CCZ}$57: it has the same $64\ket T\to\ket{CCZ}$58-stabilizer group as $64\ket T\to\ket{CCZ}$59, but its $64\ket T\to\ket{CCZ}$60-stabilizer generators are chosen as a subset of the square checks from $64\ket T\to\ket{CCZ}$61. Three omitted $64\ket T\to\ket{CCZ}$62-stabilizer generators become three logical $64\ket T\to\ket{CCZ}$63’s, corresponding to logical qubits of types $64\ket T\to\ket{CCZ}$64, and the resulting logical action is

$64\ket T\to\ket{CCZ}$65

This yields a $64\ket T\to\ket{CCZ}$66 distillation factory (Londe, 7 May 2026).

For $64\ket T\to\ket{CCZ}$67, the coordinates are grouped as

$64\ket T\to\ket{CCZ}$68

and the construction yields 99 generators total, matching

$64\ket T\to\ket{CCZ}$69

The 3D local implementation uses 127 data qubits plus 152 additional qubits (Londe, 7 May 2026).

A common misunderstanding is that unfolding makes the entire stabilizer structure local in a symmetric way. The paper is explicit that locality is achieved for a basis of the $64\ket T\to\ket{CCZ}$70-stabilizer group, while nonlocal $64\ket T\to\ket{CCZ}$71-stabilizers may remain acceptable, especially for biased-noise hardware such as cat qubits (Londe, 7 May 2026).

6. Asymptotic behavior, overhead tradeoffs, and limitations

The asymptotic behavior of Reed–Muller factory families depends strongly on which Reed–Muller formalism is used. For the binary CSS-T construction, the exact CSS dimension is

$64\ket T\to\ket{CCZ}$72

the block length is

$64\ket T\to\ket{CCZ}$73

and the CSS lower bound on distance simplifies to

$64\ket T\to\ket{CCZ}$74

(Andrade et al., 2023). The asymptotic Reed–Muller rate satisfies

$64\ket T\to\ket{CCZ}$75

and for

$64\ket T\to\ket{CCZ}$76

the CSS-T family rate tends to

$64\ket T\to\ket{CCZ}$77

with

$64\ket T\to\ket{CCZ}$78

The maximum rate of a CSS-T code defined by Reed–Muller codes is therefore

$64\ket T\to\ket{CCZ}$79

(Andrade et al., 2023).

This same work proves an important limitation: a family of classical Reed–Muller codes cannot have both nonvanishing rate and nonvanishing relative distance, and CSS codes built only from nested Reed–Muller codes therefore have vanishing quantum relative distance (Andrade et al., 2023). For distillation-factory interpretation, that limitation does not eliminate usefulness, because the minimum distance can still diverge. If $64\ket T\to\ket{CCZ}$80, then the asymptotic rate remains nonzero and $64\ket T\to\ket{CCZ}$81 can scale like $64\ket T\to\ket{CCZ}$82, yielding a diverging distance lower bound while $64\ket T\to\ket{CCZ}$83 stays bounded away from zero (Andrade et al., 2023). A plausible implication is that these families are best viewed as candidates for constant-overhead magic-state distillation rather than asymptotically good quantum memories.

For punctured prime-dimensional Reed–Muller factories, the central yield parameter is

$64\ket T\to\ket{CCZ}$84

and the paper derives sublogarithmic magic-state cost

$64\ket T\to\ket{CCZ}$85

for all prime dimensions $64\ket T\to\ket{CCZ}$86 (Saha et al., 12 Oct 2025). In the analytically tractable Manhattan-weight puncturing family,

$64\ket T\to\ket{CCZ}$87

with

$64\ket T\to\ket{CCZ}$88

(Saha et al., 12 Oct 2025). The optimized asymptotic values reported include

$64\ket T\to\ket{CCZ}$89

$64\ket T\to\ket{CCZ}$90

and the large-$64\ket T\to\ket{CCZ}$91 summary is

$64\ket T\to\ket{CCZ}$92

(Saha et al., 12 Oct 2025).

Several limitations recur across the literature. The CSS-T Reed–Muller paper does not give a full $64\ket T\to\ket{CCZ}$93-state factory protocol, explicit MSD thresholds, acceptance probabilities, or routing architecture (Andrade et al., 2023). The local-unfolding work does not provide a full circuit-level resource table for the new factories and states that detailed protocol verification and numerical simulation beyond the earlier small-factory case are left to future work (Londe, 7 May 2026). The Manhattan-weight puncturing scheme is explicitly not optimal and is chosen because it makes the distance analytically computable (Saha et al., 12 Oct 2025). The odd-prime Reed–Muller distillation protocols assume protected stabilizer operations and do not include layout, routing, ancilla scheduling, or a complete architecture-level threshold for a higher-dimensional fault-tolerant stack (Campbell et al., 2012).

Taken together, these results define the modern Reed–Muller distillation factory as a family of non-Clifford magic-state distillation architectures whose core resource is the algebraic structure of Reed–Muller codes: binary CSS-T families with nonvanishing asymptotic rate up to $64\ket T\to\ket{CCZ}$94, explicit local qubit factories such as $64\ket T\to\ket{CCZ}$95 and $64\ket T\to\ket{CCZ}$96, odd-prime one-output protocols such as the 8-qutrit and 4-ququint schemes, and prime-$64\ket T\to\ket{CCZ}$97 punctured-triorthogonal families with sublogarithmic asymptotic overhead (Andrade et al., 2023, Londe, 7 May 2026, Saha et al., 12 Oct 2025, Campbell et al., 2012).