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This paper constructs the full logical Clifford group of high-rate quantum Reed-Muller codes using only transversal and fold-transversal gates, yielding addressable Clifford gates without ancilla qubits.

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2026-08-03 02:41 UTC pith:AA4VPQWT

load-bearing objection Ancilla-free full Clifford group for high-rate Reed-Muller codes: solid, honest, and worth a serious referee. the 1 major comments →

arxiv 2602.09788 v3 pith:AA4VPQWT submitted 2026-02-10 quant-ph

Construction of the full logical Clifford group for high-rate quantum Reed-Muller codes using only transversal and fold-transversal gates

Theerapat Tansuwannont , Tim Chan , Ryuji Takagi This is my paper
classification quant-ph
keywords quantum Reed-Muller codesClifford groupfold-transversal gatestransversal gatesaddressable logical gateshigh-rate quantum error correctionself-dual CSS codesconstant-depth circuits
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Quantum Reed–Muller codes QRM(m) encode about n/√(π log n) logical qubits into n=2^m physical qubits, a rate that grows near-linearly in n. The paper proves that for every even m, the full logical Clifford group of QRM(m) is generated by the transversal Hadamard gate together with a family of phase-type fold-transversal gates built from automorphisms of the underlying classical Reed–Muller code. From these generators the paper explicitly constructs addressable S, H, C00Z (controlled-Z with minus on |00⟩) and swap gates on any logical qubit, meaning any Clifford operation on any subset of logical qubits can be run without ancilla blocks. A second result shows no code of this type can realize every Clifford gate in constant depth once k grows faster than √n log n, and for these codes some gate requires depth Ω(n/(log n)²). If correct, this removes a major overhead for fault-tolerant computation with high-rate codes and offers a concrete testbed for near-term demonstrations.

Core claim

The paper's central discovery is Theorem 9: for the self-dual quantum Reed–Muller code QRM(m) with m even, the full logical Clifford group C_k is generated by H^{⊗n} together with the phase-type fold-transversal gates U_P(Q(K)) for all automorphism products Q(K) with |K| ≤ m/2. Using these generators the authors construct addressable phase gates S(B) and controlled-Z gates C00Z(B,B′) for logical qubit pairs that differ in one basis vector; combining these with H^{⊗n} yields addressable H and swap gates, and then C00Z on any pair. Since H, S, and C11Z generate the Clifford group, the addressable gates generate all of C_k. The proof works by computing the logical action of each fold-transversa

What carries the argument

The load-bearing objects are the phase-type fold-transversal gates U_P(Q(K)), constant-depth circuits of physical S and C11Z gates whose qubit pairing comes from an automorphism Q(K) of the classical Reed–Muller code RM(m/2−1,m). Each such automorphism is a product of commuting elementary maps Q(i,j) that send v_i to v_i+v_j; the associated replacement operator R(K)=M_{Q(K)}+I tracks which logical Pauli operators get extended by Z factors. The key computational tool is the identity that the product over all subsets L⊆K of U_P(Q(L)) cancels every non-addressable term, leaving either a single addressable S(F1(K)) (when |K|=m/2) or a single addressable C00Z between a uniquely determined pair (w

Load-bearing premise

The construction rests on a parity lemma — that the overlap between a stabilizer-sized wedge vector and its automorphic image is always a multiple of four — so that no spurious phases leak into the extracted gates; if that parity ever failed, the so-called addressable S would be entangled with an extra C11Z.

What would settle it

On QRM(6) (n=64, k=20), compute the logical action of the product of fold-transversal gates for a K of size |K|=m/2−1=2 on a logical qubit B with |B∩F1(K)|=m/2−1; the theorem predicts a pure C00Z on the pair (B,B′) with no phase error. Directly simulating the compiled circuit and checking that X(B) maps to −X(B)Z(B′) and Z(B) is fixed — versus acquiring any extra C11Z factor — would falsify Theorem 5 if the phase −1 were off. Similarly, checking that the sequence for S(B) returns exactly i on that logical qubit and identity on all others for all 20 logical qubits would settle the addressabilit

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Any addressable Clifford gate on any logical qubit or pair can be implemented by a sequence of constant-depth transversal and fold-transversal gates, with no ancilla qubits.
  • Addressable H and S run in depth O(√n); addressable C00Z and SW run in depth O(√n log n).
  • No constant-depth implementation of the full Clifford group is possible for these codes: some logical Clifford gate requires depth Ω(n/(log n)²), by a counting argument that applies to any code with k = ω(√n log n).
  • The family includes the [[4,2,2]] code and the [[16,6,4]] tesseract code, so the construction generalizes known small-case addressable gates to all even m.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The subset-product cancellation trick — multiplying U_P(Q(L)) over all L⊆K to kill non-addressable terms — is a generic combinatorial mechanism that could be ported to any self-dual CSS code whose automorphism group is rich enough; the authors leave this generalization implicit.
  • The lower bound of Corollary 2 is a counting bound and likely not tight; a more refined circuit compiler that exploits non-addressable constant-depth gates such as the multi-pair C11Z implemented by U_P(e) could push practical Clifford circuits closer to the bound or reveal a stronger one.
  • A natural testable extension is to combine this ancilla-free Clifford layer with magic state cultivation or code switching to reach universality, since the Clifford layer is now resource-cheap for a high-rate code.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 4 minor

Summary. The paper studies the family of self-dual quantum Reed–Muller codes QRM(m) with parameters [[2^m, C(m,m/2), 2^{m/2}]] and proposes that the full logical Clifford group can be generated by the transversal gate H^{⊗n} together with phase-type fold-transversal gates U_P(Q(K)) built from classical Reed–Muller automorphisms. The main constructive claims are that addressable logical S, H, C00Z, and SW gates can be synthesized from these constant-depth, ancilla-free ingredients, with depths ranging from O(√n) to O(√n log n); the paper also proves a counting lower bound on the depth of arbitrary logical Clifford gates and provides an open-source Python package for explicit gate construction and numerical verification.

Significance. If correct, the main theorem would be a notable advance: it would give the first ancilla-free construction of the full logical Clifford group from transversal and fold-transversal gates for a code family whose rate grows near-linearly in n (up to a 1/√log n factor), and the depth lower bound in Theorem 10 is a useful structural result. The automorphism-based construction, the explicit logical-action proofs, and the accompanying numerical package are strengths. However, the proof contains a serious, load-bearing error in the construction of addressable H gates (Theorem 7), which invalidates the paper's derivation of Theorem 9 as written.

major comments (1)
  1. [Section 3.2.2, Theorem 7(2)] The asserted construction of an addressable H(B) from H^{⊗n}, S(B), and S(B^c) is false already for the smallest code QRM(2) (the [[4,2,2]] code). For m=2, H^{⊗n} acts on the two logical qubits as U0=(H⊗H)SW, and Corollary 1 provides S^† gates on each logical qubit. The unitary group generated by U0, S1, S2 does not contain H1: in the symplectic representation, S1,S2 have order 2, U0 swaps the two qubits, and the image has order at most 8, while H1 (X1↔Z1, X2,Z2 fixed) is not in that subgroup. Equivalently, the displayed sequence with S before each H^{⊗n} reduces, because S1S2 commutes with U0, to (S1S2)^3 U0^3 = (S^†S^†)U0 (or (SS)U0 when the available gate is S^†), which maps X1 to Z2 rather than Z1. Thus no sequence of H^{⊗n} and addressable S gates can implement H(B). Since Theorems 8 and 9 rely on this H(B) construction, the claimed synthesis of addressable SW and arbitrary C00Z gat
minor comments (4)
  1. [Section 4, Eq. (15)] The combinatorial factor in the bound for N_{l,n} is hard to parse as printed: the numerator appears to be [C(n,l) l]! rather than the expected [ceil(n/l) l]! times a choice of qubits. The asymptotic Θ(n log n) and the resulting depth bound are unaffected, but the formula should be corrected or stated more carefully.
  2. [Section 2.3, distance discussion] The text says the distance is determined by the classical code distance of Cx=Cz=RM(m/2,2); this should presumably read RM(m/2,m).
  3. [Section 3.3, UP(Q(1,2)) discussion] After Proposition 3, the text says 'the fold-transversal gate US(P(1,2)) maps the logical Pauli operators as follows' but then lists the action of US(Q(1,2)). The gate name should be corrected.
  4. [Section 5, numerical verification claim] The statement that Theorems 7–8 have been numerically verified up to m=6 is in tension with the m=2 counterexample to Theorem 7. If the package verifies a different circuit from the one stated in the manuscript, the discrepancy should be explained and the theorem statement should match the implemented construction.

Circularity Check

0 steps flagged

No significant circularity; the Clifford-group construction is self-contained and its inputs are standard external facts.

full rationale

The claimed derivation is self-contained rather than circular. The central construction defines fold-transversal gates UP(Q(K)) from automorphisms of classical Reed–Muller codes, proves code-space preservation in Theorem 3 and Theorem 4, computes their logical action in Theorem 5, and then forms products over all L ⊆ K in Theorem 6. The cancellation of proper-subset terms is a counting argument (2^{|K\M|} is even for M⊊K), not an assumption of the desired conclusion. Addressable S and C00Z gates are then derived in Corollary 1 from this proven logical action, and addressable H, SW, and further C00Z gates are constructed in Theorems 7 and 8. Theorem 9 then invokes the standard fact that H, S, and C11Z generate the Clifford group; this does not reduce to the construction. The lower-bound result in Theorem 10 is an independent counting argument on the size of the Clifford group and the number of depth-D circuits. The only mildly self-referential point is the basis-independence step, which cites the authors' prior work [76] for the fact that any symplectic basis is related to another by a logical Clifford operation. That fact is a standard transitive-action property of the Clifford group and is not equivalent to the theorem being proved; moreover, the explicit generation in the chosen basis already constitutes the substantive construction. No fitted parameter is relabeled as a prediction, and no load-bearing conclusion is obtained by definition or by self-citation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No free parameters or invented entities are introduced. The construction uses standard Reed–Muller, CSS, and Clifford-group facts; the main cited-but-unproved facts are symplectic-basis conversion [76] and Clifford-count scaling [77]. New lemmas and theorems are proved in the appendix, with numerical checks up to m=10.

axioms (5)
  • standard math RM(r,m)^⊥ = RM(m−r−1,m)
    Used in Definition 4 to construct QRM(m) via the CSS construction from RM(m/2−1,m); standard Reed–Muller duality.
  • standard math Any two symplectic bases of the logical Pauli group are related by a logical Clifford operation
    Invoked in Section 3.2.3 to extend Theorem 9 beyond the canonical basis; cited to [76], not proved in the paper.
  • standard math The n-qubit Clifford group has size Θ(2^{n^2})
    Used in the counting lower bound of Theorem 10; cited to [77].
  • standard math The k-qubit Clifford group is generated by H, S, and C11Z on arbitrary qubits and pairs
    Used in the proof of Theorem 9 to conclude that addressable H, S, and C00Z gates generate C_k.
  • domain assumption Fold-transversal gates are fault tolerant under the noise models considered
    Motivates the ancilla-free FTQC claim; the paper explicitly notes that strict fault-tolerance conditions of [9] are not satisfied and relies on numerical evidence from [25–28].

pith-pipeline@v1.3.0-alltime-deepseek · 40204 in / 18923 out tokens · 178607 ms · 2026-08-03T02:41:16.417260+00:00 · methodology

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read the original abstract

To build large-scale quantum computers while minimizing resource requirements, one may want to use high-rate quantum error-correcting codes that can efficiently encode information. However, realizing an addressable gate$\unicode{x2014}$a logical gate on a subset of logical qubits within a high-rate code$\unicode{x2014}$in a fault-tolerant manner can be challenging and may require ancilla qubits. Transversal and fold-transversal gates could provide a means to fault-tolerantly implement logical gates using a constant-depth circuit without ancilla qubits, but available gates of these types could be limited depending on the code and might not be addressable. In this work, we study a family of $[\![n=2^m,k={m \choose m/2}\approx n/\sqrt{\pi \log_2(n)/2},d=2^{m/2}=\sqrt{n}]\!]$ self-dual quantum Reed$\unicode{x2013}$Muller codes, where $m$ is a positive even number. For any code in this family, we construct a generating set of the full logical Clifford group comprising only transversal and fold-transversal gates, thus enabling the implementation of any addressable Clifford gate. To our knowledge, this is the first known construction of the full logical Clifford group using only transversal and fold-transversal gates without requiring ancilla qubits for a family of codes in which $k$ grows near-linearly in $n$ up to a $1/\sqrt{\log n}$ factor.

Figures

Figures reproduced from arXiv: 2602.09788 by Ryuji Takagi, Theerapat Tansuwannont, Tim Chan.

Figure 1
Figure 1. Figure 1: Circuit diagrams for the Hadamard gate H, the phase gate S, the swap gate SW, the controlled-NOT gate CX, and two types of controlled-Z gates C00Z and C11Z. H, S, SW, CX, C00Z and C11Z transform the Pauli X and Z operators as follows: H: X 7→ Z, Z 7→ X, S: X 7→ iXZ, Z 7→ Z, SW: X ⊗ I 7→ I ⊗ X, Z ⊗ I 7→ I ⊗ Z, CX: X ⊗ I 7→ X ⊗ X, Z ⊗ I 7→ Z ⊗ I, I ⊗ X 7→ I ⊗ X, I ⊗ Z 7→ Z ⊗ Z, C00Z: X ⊗ I 7→ −X ⊗ Z, Z ⊗ I 7… view at source ↗
Figure 2
Figure 2. Figure 2: Examples of swap-type fold-transversal gates for the quantum Reed–Muller code [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Examples of phase-type fold-transversal gates for the quantum Reed–Muller code [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

discussion (0)

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Homological origin of transversal implementability of logical diagonal gates in quantum CSS codes

    quant-ph 2026-02 unverdicted novelty 6.0

    A homological framework identifies necessary and sufficient obstruction conditions for transversal logical diagonal gates in quantum CSS codes.

  2. No-Go Theorem on Fault Tolerant Gadgets for Multiple Logical Qubits

    quant-ph 2026-02 reject novelty 6.0

    No stabilizer code can implement the full logical Clifford group on multiple logical qubits using transversal gates, fold-transversal gates beyond two qubits, or code automorphisms.

Reference graph

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