Constraints on phantom codes from automorphism group bounds

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.

• The paper proves that binary phantom codes encoding k logical qubits require n ≥ 2^k − 1 physical qubits for distance d ≥ 2, except when k = 4, using automorphism-group structure and minimal permutation degrees.
• Punctured hypercube codes attain the bound for k = 3 and k ≥ 5 and are the unique saturating binary CSS family, although their checks have linear-size weight and are not qLDPC.
• The authors construct an optimal nonstabiliser ((8,2^4,2)) phantom code with full S8 permutation symmetry and a transversal non-Clifford gate, while showing local-unitary and subsystem extensions do not evade the exponential constraint.

Phantom codes, introduced by Koh et al., are quantum error-correcting codes in which every logical $CNOT$ circuit can be implemented by a mere permutation of the physical qubits. Because permutations amount to relabelling qubits and can be nearly free on platforms with long-range connectivity or shuttling (trapped ions, neutral atoms), such codes promise dramatic reductions in the depth of compiled fault-tolerant circuits. Prior work had established only weaker constraints: a CSS phantom code of distance $d\geq 3$ cannot have constant rate, and all CSS phantom codes obey a distance-dependent bound $k \leq d\log_2 n + \text{const}$. Exhaustive searches suggested, but did not prove, that the number of encoded qubits is at most logarithmic in the block length for any distance. This paper by Morris and Malz resolves that question affirmatively, proving an exponential lower bound on block length that holds for arbitrary (not necessarily stabiliser) binary codes, for phantom-LU variants allowing local unitaries, and for subsystem codes.

The automorphism group theorem

The central technical tool is a general structural result relating code length to the automorphism group $\mathrm{Aut}\,Q$, defined as the group of $SWAP$-transversal logical operators, i.e. elements of the semidirect product $M_n = \mathsf{U}(2)^n \rtimes S_n$ preserving the code space (or, for subsystem codes, acting as identity-gauge pairs). For any compact subgroup chain $N \triangleleft G \leq \mathrm{Aut}\,Q$ with quotient $S = G/N$ finite, non-Abelian, simple, and not isomorphic to $A_5$, the paper proves

$n \geq \mu(S),$

where $d\geq 3$0 denotes the minimal faithful permutation degree of $d\geq 3$1. The proof combines three ingredients: the fact that $d\geq 3$2 embeds faithfully into $d\geq 3$3, so $d\geq 3$4; the classical result of Kovács and Praeger that quotients which are simple non-Abelian cannot increase minimal permutation degree ($d\geq 3$5); and a lemma showing that any finite non-Abelian simple section of a compact subgroup of $d\geq 3$6 must be $d\geq 3$7. The latter follows from the classification of closed subgroups of $d\geq 3$8 via induction over tensor factors, using the exceptional isomorphism between the icosahedral group and $d\geq 3$9. The $k \leq d\log_2 n + \text{const}$0 exclusion is what forces the "permutation part" of the automorphism group to carry the full simple quotient, thereby transferring permutation-degree lower bounds onto the physical length $k \leq d\log_2 n + \text{const}$1. The authors note this theorem may find applications beyond phantom codes, as it constrains any code whose automorphism group contains a suitable simple section.

The logarithmic encoding-rate bound

For a phantom code, the permutations implementing logical $k \leq d\log_2 n + \text{const}$2 circuits form a group $k \leq d\log_2 n + \text{const}$3 surjecting onto $k \leq d\log_2 n + \text{const}$4 — the group of logical $k \leq d\log_2 n + \text{const}$5 circuits, faithfully represented via the symplectic representation — with kernel $k \leq d\log_2 n + \text{const}$6 consisting of permutations acting trivially on the logical space. Since $k \leq d\log_2 n + \text{const}$7 is simple and non-Abelian for $k \leq d\log_2 n + \text{const}$8, the general theorem yields $k \leq d\log_2 n + \text{const}$9 for $\mathrm{Aut}\,Q$0 and $\mathrm{Aut}\,Q$1, using Cooperstein's computation of minimal permutation degrees. The case $\mathrm{Aut}\,Q$2 follows from the quantum Singleton bound. The result is:

Theorem (phantom bound). Any phantom code of distance $\mathrm{Aut}\,Q$3 encoding $\mathrm{Aut}\,Q$4 logical qubits with $\mathrm{Aut}\,Q$5 requires $\mathrm{Aut}\,Q$6 physical qubits.

This is exponentially stronger than the previously known distance-dependent bounds, and it settles the empirical observation from prior constructions as a fundamental obstruction rather than an artefact. Notably, the bound holds without any CSS or stabiliser assumption, since only the compact Lie group structure of $\mathrm{Aut}\,Q$7 is used.

Tightness and the unique saturating family

The bound is tight. The family of $\mathrm{Aut}\,Q$8 punctured hypercube codes — CSS codes built from shortened and punctured Reed-Muller constituents $\mathrm{Aut}\,Q$9 and $SWAP$0 — achieves equality for $SWAP$1 and $SWAP$2. These admit a geometric description: physical qubits sit at the vertices of a punctured $SWAP$3-dimensional hypercube, $SWAP$4-stabilisers are square faces avoiding the origin, and logical $SWAP$5 gates are implemented by reflecting faces across symmetry axes. The paper proves a rigidity statement: for $SWAP$6 and $SWAP$7, the punctured hypercube codes are the unique binary CSS phantom codes of type $SWAP$8 with $SWAP$9, up to CSS isomorphism. The proof first shows that at saturation the kernel $M_n = \mathsf{U}(2)^n \rtimes S_n$0 is trivial, so both classical constituents must have automorphism groups containing $M_n = \mathsf{U}(2)^n \rtimes S_n$1; a classification due to Bardoe and Sin then restricts these constituents to punctured/shortened Reed-Muller codes, and a case analysis over the CSS condition leaves exactly one valid pair. A consequence is that all saturating codes have check weights scaling as $M_n = \mathsf{U}(2)^n \rtimes S_n$2 and hence are not qLDPC — a partial answer to whether phantom codes can be qLDPC, though codes with $M_n = \mathsf{U}(2)^n \rtimes S_n$3 remain unconstrained by this argument.

The $M_n = \mathsf{U}(2)^n \rtimes S_n$4 exception and a nonstabiliser code

The case $M_n = \mathsf{U}(2)^n \rtimes S_n$5 escapes the bound because of the exceptional isomorphism $M_n = \mathsf{U}(2)^n \rtimes S_n$6, giving $M_n = \mathsf{U}(2)^n \rtimes S_n$7. Exploiting the corresponding isomorphism of group actions — between $M_n = \mathsf{U}(2)^n \rtimes S_n$8 acting on the 35 projective lines of $M_n = \mathsf{U}(2)^n \rtimes S_n$9 and $N \triangleleft G \leq \mathrm{Aut}\,Q$0 acting on unordered 4–4 bipartitions of eight points — the authors construct an explicit $N \triangleleft G \leq \mathrm{Aut}\,Q$1 phantom code. Logical codewords are built from "line states" (equal superpositions of complementary weight-4 bit strings indexed by projective lines) and point-star states, with coefficients fixed by orthogonality requirements. The construction is verified to have distance exactly 2, which is optimal since no $N \triangleleft G \leq \mathrm{Aut}\,Q$2 code exists by the quantum MDS length bound. Two further properties stand out:

• Maximal symmetry: odd permutations also preserve the code space, so the permutation automorphism group is the full $N \triangleleft G \leq \mathrm{Aut}\,Q$3; a specific odd permutation realises the projective duality of $N \triangleleft G \leq \mathrm{Aut}\,Q$4 and implements a non-Clifford logical gate.
• Transversal non-Clifford gate: $N \triangleleft G \leq \mathrm{Aut}\,Q$5 acts as $N \triangleleft G \leq \mathrm{Aut}\,Q$6 on the logical space.

The code admits a compact stabiliser description via non-Pauli operators ($N \triangleleft G \leq \mathrm{Aut}\,Q$7, $N \triangleleft G \leq \mathrm{Aut}\,Q$8, and a collective-spin Casimir polynomial), and Schur-Weyl duality shows its weight-4 sector decomposes as $N \triangleleft G \leq \mathrm{Aut}\,Q$9 under $S = G/N$0. Crucially, the authors prove that no Pauli stabiliser subsystem phantom code of type $S = G/N$1 exists, using the subsystem Singleton bound together with the classification of binary linear codes invariant under $S = G/N$2. The $S = G/N$3 code is therefore genuinely nonstabiliser, explaining its absence from prior exhaustive stabiliser-code searches.

Phantom-LU and subsystem codes

A natural question is whether enriching the gate set weakens the obstruction. The answer is negative. Defining phantom-LU codes (logical $S = G/N$4s via permutations plus single-qubit gates) and their subsystem analogue ("PLUS" codes, where gauge transformations are permitted), the authors show the same bound applies: any subspace or subsystem phantom-LU code of distance $S = G/N$5 with $S = G/N$6, $S = G/N$7, requires $S = G/N$8. The proof replaces the permutation group $S = G/N$9 with the group of $A_5$0-transversal operators inducing logical $A_5$1 circuits up to gauge, establishes compactness of this group and closedness of its kernel via a continuous map into the projective unitary group of the logical space, and applies the same theorem verbatim. Thus neither local unitary freedom nor gauge degrees of freedom evade the exponential cost — the trade-off between phantom entangling gates and high rate is robust.

The paper also extends the framework to Galois qudits, where logical $A_5$2 circuits generate $A_5$3. A qudit version of the bound gives $A_5$4, with tabulated exceptional values, holding for all $A_5$5 except $A_5$6 and $A_5$7. The qudit theorem is weaker in scope than the binary one: it applies only to subgroups of the permutation automorphism group, because generalising the $A_5$8 lemma would require classifying finite simple sections of $A_5$9 for each prime power $n \geq \mu(S),$0, which is impossible to state uniformly since every finite group embeds in some $n \geq \mu(S),$1. Existence is witnessed by $n \geq \mu(S),$2 quantum Reed-Muller codes, which are phantom for all $n \geq \mu(S),$3 and $n \geq \mu(S),$4.

Limitations and open questions

Several caveats are explicit in the paper. The uniqueness theorem covers only CSS codes saturating the bound at $n \geq \mu(S),$5; it does not preclude qLDPC phantom codes with larger blocks, nor non-CSS ones, and determining whether sparse-check phantom codes exist remains open. In the qudit setting, the classification of minimal phantom codes is obstructed by two issues acknowledged directly: Bardoe–Sin's submodule classification addresses $n \geq \mu(S),$6-invariant codes while phantomness concerns $n \geq \mu(S),$7-invariance, and for extension fields $n \geq \mu(S),$8 the submodule lattice is richer than the Reed-Muller chain, leaving saturation of the qudit bound unresolved. The relationship between phantom and phantom-LU codes is also left partially open: the paper derives a necessary and sufficient condition for a phantom-LU code to be locally unitary equivalent to a phantom code, but does not determine whether the condition always holds. Finally, the analogous trade-off for automorphisms implementing gates other than $n \geq \mu(S),$9 — where a partial bound $d\geq 3$00 applies when only $d\geq 3$01 qubits enjoy permutation-implemented $d\geq 3$02 circuits — is characterised only in this restricted form.

Conclusion

This work converts an empirical pattern into a theorem: implementing the full logical $d\geq 3$03 group by physical permutations forces an exponentially large block, $d\geq 3$04 for all $d\geq 3$05, regardless of stabiliser structure, local-unitary enrichment, or gauge freedom. The bound is tight within CSS codes, saturated uniquely by punctured hypercube codes with dense checks, while the isolated $d\geq 3$06 exception yields an explicit nonstabiliser $d\geq 3$07 code with maximal $d\geq 3$08 symmetry and a transversal non-Clifford gate. Beyond phantom codes, the underlying permutation-degree theorem offers a general mechanism for bounding code length in terms of automorphism group structure, complementing the Eastin–Knill line of no-go results on shallow fault-tolerant gates.