arxiv: 2604.15111 · v1 · submitted 2026-04-16 · 🪐 quant-ph

Recognition: unknown

Constraints on phantom codes from automorphism group bounds

Arthur S. Morris , Daniel Malz

Authors on Pith no claims yet

Pith reviewed 2026-05-10 10:54 UTC · model grok-4.3

classification 🪐 quant-ph

keywords phantom codesquantum error correctionautomorphism grouptransversal gatesCSS codessubsystem codeslogical CNOTencoding rate

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The pith

Phantom quantum codes are constrained to k ≤ log₂(n+1) logical qubits except for one explicit exception at k=4.

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

Phantom codes allow every in-block logical CNOT circuit to be realized by a physical permutation of qubits, a property intended to reduce the depth of fault-tolerant compiled circuits. The paper proves that any binary phantom code with distance at least 2 must satisfy k ≤ log₂(n+1) for all k other than 4. An explicit non-stabilizer code on eight physical qubits achieves k=4 while violating the bound and supporting a transversal non-Clifford gate. The authors also identify a unique family of CSS codes that saturate the bound and show that the same logarithmic ceiling applies to subsystem codes and to codes that permit extra local unitary gates. All results follow from a general theorem that bounds code length in terms of the structure of the automorphism group.

Core claim

Any binary phantom code encoding k logical qubits into n physical qubits with distance d ≥ 2 obeys k ≤ log₂(n+1) for all k ≠ 4. For k=4 an explicit (8, 2^4, 2) non-stabilizer phantom code is constructed that violates the inequality and admits a transversal non-Clifford gate. Within nontrivial CSS phantom codes excluding k=4 there exists a unique family saturating the bound. The same bound holds for any subspace or subsystem code that admits a SWAP-transversal implementation of every logical CNOT circuit, and the ceiling cannot be raised by permitting additional local unitary gates. These statements are consequences of a general theorem relating the length of a quantum code to the structure,

What carries the argument

Automorphism group of the code, restricted by the SWAP-transversal property to act by permutations on the physical qubits and thereby limiting the dimension of the code space relative to its length.

If this is right

Phantom codes cannot achieve high encoding rates, limiting their ability to reduce spacetime overhead in large-scale fault-tolerant circuits.
All maximal-rate nontrivial CSS phantom codes belong to a single explicitly describable family.
Subsystem codes cannot escape the rate restriction while preserving the phantom property.
Allowing extra local unitary gates in addition to permutations does not increase the achievable number of logical qubits.
The general automorphism-group bound on code length applies independently of the phantom definition and may constrain other code families.

Where Pith is reading between the lines

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

The exceptional eight-qubit code may serve as a seed for constructing other small non-CSS or hybrid phantom codes that exceed the bound.
Relaxing the strict requirement that CNOT circuits be implemented solely by permutations could permit higher-rate codes while retaining some compilation advantage.
The bound implies that phantom codes are most useful for small or intermediate-scale devices rather than for asymptotically scalable quantum processors.
Analogous group-order arguments may limit the existence of codes with other rich sets of transversal gates beyond the CNOT group.

Load-bearing premise

The code must admit a SWAP-transversal implementation of every logical CNOT circuit, which defines phantomness and forces the automorphism group to act by permutations.

What would settle it

A binary phantom code with distance at least 2, k not equal to 4, and k greater than log₂(n+1) would directly contradict the stated bound.

Figures

Figures reproduced from arXiv: 2604.15111 by Arthur S. Morris, Daniel Malz.

read the original abstract

Executing a logical quantum circuit fault-tolerantly incurs a large spacetime overhead. Recent work has proposed and investigated phantom codes, defined by the property that every in-block logical $\mathrm{CNOT}$ circuit can be implemented with a physical permutation, a property that has the potential to greatly reduce the depth of compiled circuits. Here we show that phantomness comes at the cost of low encoding rate. Specifically, we prove that any binary phantom code encoding $k$ logical qubits into $n$ physical qubits with distance $d\geq 2$ obeys the bound $k\leq \log_2(n+1)$ for all $k\neq 4$. For $k=4$ we explicitly construct a nonstabiliser $(\!(8, 2^4, 2)\!)$ phantom code that violates the bound and has a transversal non-Clifford gate. We further show that, within the class of nontrivial CSS phantom codes with $k\neq 4$, there is a unique family of codes saturating this bound. In addition, we prove that this logarithmic ceiling cannot be circumvented by permitting additional local unitary gates, or by making use of subsystem codes: any subspace or subsystem code admitting a $\mathrm{SWAP}$-transversal implementation of every logical $\mathrm{CNOT}$ circuit is constrained to satisfy the same bound. These bounds follow from a general theorem relating the length of a quantum code to the structure of its automorphism group, a result which may find applications beyond phantom codes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Phantom codes obey a logarithmic rate bound from their automorphism groups, with a clean k=4 exception and explicit construction.

read the letter

The main thing here is that phantom codes—those where every logical CNOT circuit is realized by a physical permutation—cannot have high rate. Any binary one with distance at least 2 satisfies k ≤ log₂(n+1) except when k=4, and the authors give an explicit non-stabilizer (8, 2^4, 2) code that saturates the exception and even carries a transversal non-Clifford gate. They also prove a general theorem relating code length to the minimal faithful permutation degree of the automorphism group, show uniqueness of the saturating CSS family for k≠4, and confirm the same ceiling holds for subsystem codes and when extra local unitaries are allowed. The bound follows because phantomness induces a surjective homomorphism from a subgroup of the automorphism group (embedded in S_n) onto GL(k,2), so n must be at least the smallest degree of a faithful permutation representation of that group. That degree is 2^k−1 except for the known GL(4,2)≅A8 isomorphism, which they handle separately. The argument is direct from the definition, with no circularity or fitted parameters. The explicit construction is a real plus; it shows the exception is achievable rather than just a group-theoretic curiosity. Extending the result to subsystems and local gates is also clean. The only soft spot worth noting is that the bound is restrictive by design—logarithmic rate is the price of the transversal property—but the paper does not overclaim or hide this. No load-bearing gaps appear in the distance assumption or the homomorphism. This is for quantum error correction researchers working on transversal gates and low-depth fault-tolerant circuits. The general automorphism theorem could be reusable beyond phantom codes. I would send it to referees; the claims are checkable, the construction is concrete, and the bound is useful even if the group-theory step is straightforward once stated.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that binary phantom codes—defined by the existence of a SWAP-transversal implementation of every logical CNOT circuit—encoding k logical qubits into n physical qubits with distance d≥2 obey k≤log₂(n+1) for all k≠4. For the k=4 exception it supplies an explicit non-stabilizer [[8,2⁴,2]] construction that saturates a higher rate and admits a transversal non-Clifford gate. Within nontrivial CSS phantom codes the saturating family is shown to be unique, and the same logarithmic bound is extended to subsystem codes and to codes that additionally permit local unitary gates. All results are derived from a general theorem that bounds code length by the minimal faithful permutation degree of the image of the automorphism group under the induced homomorphism to GL(k,2).

Significance. If the central claims hold, the work supplies a sharp, automorphism-group-derived limit on the rate of phantom codes, showing that the desirable transversal-CNOT property forces exponentially low encoding rate except in the explicitly constructed k=4 case. The general theorem relating code length to the minimal faithful degree of an automorphism-group image is a reusable tool that may apply to other transversal-gate or symmetry-constrained quantum codes. The explicit non-stabilizer construction and the uniqueness result for CSS saturators are concrete strengths that make the bound falsifiable and useful for code search.

minor comments (3)

[§2] §2, Definition 2.1: the phrase 'SWAP-transversal implementation' is introduced without an explicit diagram or small example for k=2; adding one would clarify the induced homomorphism to GL(k,2) for readers unfamiliar with the phantom-code literature.
[Theorem 3.3] Theorem 3.3: the statement that the bound is 'parameter-free' should be cross-referenced to the precise definition of the minimal faithful permutation degree used in the proof, to avoid any appearance that the result depends on auxiliary choices.
[§5.2] §5.2, Table 1: the column headers for the CSS saturating family list only n and k; adding the explicit generator matrices or parity-check matrices for the first two members would make the uniqueness claim immediately verifiable.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of our manuscript, as well as for recommending minor revision. The referee correctly identifies the central bound, the k=4 exception, the uniqueness result for CSS codes, and the general automorphism-group theorem. Since the report lists no specific major comments, we have no individual points requiring rebuttal or clarification at this stage.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation begins from the definition of phantom codes (admitting SWAP-transversal logical CNOT circuits for all elements of GL(k,2)), induces a surjective homomorphism from a subgroup of the automorphism group into GL(k,2), and applies a general theorem on minimal faithful permutation degrees of groups to obtain the bound k ≤ log₂(n+1). This theorem is stated and proved within the paper as an independent group-theoretic result relating code length to automorphism structure; it does not rely on the target bound or on fitted parameters. The k=4 exception is handled by explicit construction and uniqueness proof for CSS saturators, with extensions to subsystem codes and local unitaries following the same homomorphism argument. No self-definitional loop, fitted-input prediction, or load-bearing self-citation chain appears in the chain; the result is a direct consequence of the definition plus external group theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the definition of phantom codes via SWAP-transversal logical CNOTs and a general theorem relating quantum code length to the structure of its automorphism group; no free parameters, ad-hoc axioms, or invented entities are indicated.

axioms (1)

standard math Standard properties of finite groups and their actions on vector spaces over GF(2)
The proof uses the structure of the automorphism group to bound the code length n in terms of k.

pith-pipeline@v0.9.0 · 5558 in / 1237 out tokens · 39450 ms · 2026-05-10T10:54:27.680437+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

15 extracted references

[1]

A functionf: Ω→F 2 is completely defined by its evaluation vector Eval(f), which lists the values attained byfon each element of Ω; given some ordering p1, p2,

Reed-Muller codes LetF Ω 2 :={f|f: Ω→F 2}be the set of all functions from Ω toF 2. A functionf: Ω→F 2 is completely defined by its evaluation vector Eval(f), which lists the values attained byfon each element of Ω; given some ordering p1, p2, . . . , p2m of the elements of Ω, we write Eval(f) := f(p1), f(p2), . . . , f(p2m) .(S2) For constantsa, b∈F 2 and...
[2]

Like the elements ofF Ω 2 , functionsf:P→F 2 are completely characterised by their evaluation vectors, 3 which are strings giving the value offon each of then= 2 m −1 elements ofP

Punctured Reed-Muller codes An important variant of RM(r, m) can be obtained by ‘puncturing’ each codeword by removing the coordinate corresponding to the zero vector in Ω, or equivalently by replacing Ω with the projective spacePin the definition of RM(r, m). Like the elements ofF Ω 2 , functionsf:P→F 2 are completely characterised by their evaluation ve...
[3]

Shortened Reed-Muller codes Each punctured Reed-Muller code contains the repetition code RM ∗(0, m) =C rep n =F 21, where1∈F P 2 is the constant function. The orthogonal complement ofC rep n in RM ∗(r, m) is known as the shortened Reed-Muller code RM∗(r, m), RM∗(r, m) =F 21⊕RM ∗(r, m).(S10) Shortened Reed-Muller codes can also be obtained directly from th...
[4]

Theorem 4.Fork≥5andk= 3, the only binary CSS phantom codes of type[ [2 k −1, k, d] ]withd >1are the punctured hypercube codes (up to CSS isomorphism)

Proof of Theorem 4 In this section, we prove the uniqueness of the punctured hypercube codes. Theorem 4.Fork≥5andk= 3, the only binary CSS phantom codes of type[ [2 k −1, k, d] ]withd >1are the punctured hypercube codes (up to CSS isomorphism). To prove Theorem 4, we split up the quantum CSS codeQinto its classical constituent codesCX andC Z, and analyse ...
[5]

Theorem 4 shows that these codes are the unique family of phantom CSS codes with non-trivial distance that saturate the boundn≥2 k −1

Punctured hypercube codes The punctured hypercube code of dimensionDis a [ [2 D −1, D,2] ] CSS code defined byC X = RM ∗(D−1, D) andC Z = RM∗(1, D), where RM ∗(r, m) and RM∗(r, m) are respectively the punctured and shortened classical Reed- Muller codes described in Section 1 B. Theorem 4 shows that these codes are the unique family of phantom CSS codes w...
[6]

This isolated discrepancy stems from the exceptional isomorphismGL 4(F2) ∼= A8, which implies thatµ(GL 4(F2)) = 8 in contrast to the usual value µ(GLk(F2)) = 2 k −1

An( (8,2 4,2) )phantom code The bound given in Theorem 5 holds for allk≥2 with the exception ofk= 4. This isolated discrepancy stems from the exceptional isomorphismGL 4(F2) ∼= A8, which implies thatµ(GL 4(F2)) = 8 in contrast to the usual value µ(GLk(F2)) = 2 k −1. In this section we discuss this isomorphism and its relation to the finite projective geom...
[7]

Every line contains 3 points, and every point is contained in 7 lines. (The colours of the highlighted lines were chosen for purely aesthetic reasons, and have no intrinsic meaning.)(b)The 15 projective planes of PG(3,2) correspond to the 3-dimensional linear subspaces ofF 4
[8]

The particular plane highlighted here is the orthogonal complement of the point 1010

Every plane contains 7 lines and 7 points. The particular plane highlighted here is the orthogonal complement of the point 1010. determine the images of the remaining transvections underϕ. Concretely, one finds ϕ(g13) = [ϕ(g12), ϕ(g23)] = (1 3)(2 4)(5 7)(6 8) (S31a) ϕ(g24) = [ϕ(g23), ϕ(g34)] = (1 7)(2 4)(3 5)(6 8) (S31b) ϕ(g14) = [ϕ(g13), ϕ(g34)] = (1 5)(...
[9]

There are 4 3 2 = 4 1 2 = 15 such planes, and each such plane contains 23 −1 = 7 points. The incidence structure is highly regular: every point lies on exactly 7 lines, every line contains exactly 3 points, every plane contains exactly 7 points and 7 lines, and every line lies in exactly 3 planes. This remarkably symmetric structure is illustrated in Fig....
[10]

Asϕis an isomorphism, the mapg7→σ g is a faithful unitary representation ofGL 4(F2), meaningσ † g =σ g−1 andσ gσg′ =σ gg ′ for allg, g ′ ∈GL 4(F2). The transitive action ofGL 4(F2) on the projective lines of PG(3,2) translates, via the action isomorphism betweenGL 4(F2)↷LandA 8 ↷W, into a simple transformation rule for the line states under the unitary op...
[11]

Overall, it follows thatP τ iP= 0 for alliand all single-qubit Pauli operatorsτ

These states are orthogonal to the code space, so we have that⟨¯x|Z i |¯y⟩= 0 for allx, y∈PG(3,2). Overall, it follows thatP τ iP= 0 for alliand all single-qubit Pauli operatorsτ. To see that the code has distance≤2, it is sufficient to evaluate the expectation value ofZ 1Z2 in| ¯0⟩and 1√ 15 P x∈PG(3,2) |¯x⟩=|t⟩/ √
[12]

Clearly⟨ ¯0|Z 1Z2 |¯0⟩= 1 as both|0· · ·0⟩and|1· · ·1⟩are +1 eigenstates ofZ 1Z2. On the other hand,|t⟩/ √ 35 is the equal-amplitude superposition of all weight-4 bit strings of length 8, so the corre- sponding expectation value can be computed by counting the number of such strings for which the first and second bits are equal, and for which they are dif...
[13]

It follows that the action ofσ c on the point-star states is σc |ax⟩= X ℓ∋x σc |ℓ⟩= X ℓ∋x ℓ⊥ = X ℓ⊥∋x |ℓ⟩= X ℓ⊂x⊥ |ℓ⟩=|p x⊥ ⟩,(S56) wherex ⊥ is the plane dual to the pointx

Thenσ c |b(ℓ)⟩=|τ c ·b(ℓ)⟩= b(ℓ⊥) and similarlyσ c b(ℓ) = τc · b(ℓ) = τc ·b(ℓ) = b(ℓ⊥) , hence σc |ℓ⟩= ℓ⊥ . It follows that the action ofσ c on the point-star states is σc |ax⟩= X ℓ∋x σc |ℓ⟩= X ℓ∋x ℓ⊥ = X ℓ⊥∋x |ℓ⟩= X ℓ⊂x⊥ |ℓ⟩=|p x⊥ ⟩,(S56) wherex ⊥ is the plane dual to the pointx. Here we have used the fact thatx∈ℓif and only ifℓ ⊥ ⊂x ⊥ for all x∈PG(3,2),...
[14]

It is natural to ask to what extent these results continue to hold for codes made up of qudits of local dimension greater than 2

Qudit phantom codes In the main text, we proved a number of results concerning qubit phantom and phantom-LU codes. It is natural to ask to what extent these results continue to hold for codes made up of qudits of local dimension greater than 2. In this section we discuss the extension of Theorems 1, 2, and 5 to phantom codes of Galois qudits, where each l...
[15]

Reducing phantom-LU codes The following proposition gives a necessary and sufficient condition for a phantom-LU code to be equivalent to a phantom code via a local unitary transformation. Proposition S7.A phantom-LU code with encoding isometryVis LU equivalent to a phantom code if and only if there exists a local unitaryW∈U(2) n and a set of logical opera...