REVIEW
Phantom Codes: Hardness, Rate Optimal qLDPC Constructions, and Distance Limits
Quantum Physics · Submitted 15 Sep 2026 · 17 references
Pith's verdict: This paper cleanly resolves three open questions about phantom codes—hardness, existence of rate-optimal qLDPC families, and a distance impossibility—and all three arguments hold up to scrutiny.
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Referee report
The paper
The paper claims
This paper constructs the first phantom qLDPC families achieving k = Θ(log n) and any fixed distance D, and proves that in any bounded-weight family with k = ω(√log n) the distance is eventually bounded by the check weight.
The reading
What carries the argument
The central objects are (1) the variable-copying degree-reduction gadget from Hastings (2021), which transforms the Hamming presentation into a sparse stabilizer code while preserving the full logical action; (2) fixed-block concatenation with a one-logical-qubit CSS code (Knill–Laflamme), which multiplies sector distances and preserves the phantom property; and (3) for the no-go theorem, the sunflower lemma (Erdős–Rado) together with orbit-stabilizer bounds on the permutation automorphism group of the code's incidence graph, combined with the structure of GL(k,F₂) as a simple group, to derive a lower bound n = exp(Ω(k²)) that contradicts k = ω(√log n).
Extended reading notes
Core claim
The paper establishes three main results. First, deciding whether a stabilizer code is phantom is GI-hard, proved via a reduction from Graph Isomorphism on CSS codes with k = 2. Second, for every fixed integer D, there exist binary phantom qLDPC CSS codes with k = Θ(log n) and distance at least D, constructed by concatenating a sparse outer family (based on a degree-split Hamming code) with a fixed one-logical-qubit CSS code. Third, a distance no-go theorem: any unbounded family of binary phantom stabilizer codes with bounded stabilizer generator weight and k = ω(√log n) satisfies d ≤ w eventually, where w is the check weight. Consequently, the constructed families are distance-optimal at the maximal logical scaling.
Load-bearing premise
The no-go theorem assumes the code is binary (over F₂) and that the stabilizer generator weight is bounded by a constant; the GI-hardness reduction assumes the input is CSS with logical dimension k = 2; the construction assumes a fixed inner CSS code of distance at least D exists (which it does for any D, e.g. a surface code).
Editorial extensions
If this is right
- Phantomness recognition is at least as hard as Graph Isomorphism, so no efficient general algorithm exists for this problem (unless P = GI).
- Phantom qLDPC families exist at the maximum logical dimension k = Θ(log n) for any chosen constant distance D, showing that sparsity and maximal logical scaling are compatible with the phantom property.
- The no-go theorem implies that to have both large logical dimension and large distance in a phantom code, one must allow the check weight to grow: any bounded-weight family with k = ω(√log n) eventually has distance no larger than its check weight.
- The constructed families are provably distance-optimal at the maximal logical scaling, because the no-go theorem rules out any bounded-weight phantom family with both k = Θ(log n) and unbounded distance.
- The results extend from CSS codes to all binary stabilizer codes, confirming that the obstruction is not an artifact of the CSS restriction.
Reading between the lines
- The GI-hardness result suggests that practical algorithms for identifying phantom codes will likely need to be heuristic or restricted to small block sizes, mirroring the difficulty of Graph Isomorphism.
- The combination of construction and no-go theorem indicates that phantom codes occupy a very narrow parameter regime: near-maximal logical dimension forces distance to be constant if checks are sparse.
- The degree-split Hamming plus concatenation method could serve as a general template for building other symmetric quantum error-correcting codes with prescribed sparsity and logical dimension.
- The sunflower-lemma-based argument relies on the code being binary; extending these results to qudit or stabilizer codes over larger fields may require new combinatorial tools.
Original abstract
An $[[n,k,d]]$ stabilizer code is phantom if every in-block logical CNOT gate can be realized by a permutation of its physical qubits. This eliminates the large and complicated physical overhead normally required for logical entangling gates. Yet this symmetry is highly restrictive: phantom codes are rare, the number of logical qubits is limited to $k=O(\log n)$, and no phantom qLDPC family with growing logical dimension was previously known. We make three contributions. 1. We prove that recognizing phantomness of a given stabilizer code is at least as hard as Graph Isomorphism, even for CSS codes encoding only $k=2$ logical qubits. 2. We propose the first rate optimal phantom qLDPC families: for every fixed $D$, our CSS families achieve the maximal logical scaling $k=\Theta(\log n)$ and distance $d\geq D$. 3. We prove a distance no-go theorem: every phantom code family with $k=\omega(\sqrt{\log n})$ and check weight $w=O(1)$ satisfies $d\leq w$ for sufficiently large $n$. Thus our fixed-distance qLDPC families are distance-scaling optimal at the maximal logical scaling $k=\Theta(\log n)$.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
An $[[n,k,d]]$ stabilizer code is phantom if every in-block logical CNOT gate can be realized by a permutation of its physical qubits. This eliminates the large and complicated physical overhead normally required for logical entangling gates. Yet this symmetry is highly restrictive: phantom codes are rare, the number of logical qubits is limited to $k=O(\log n)$, and no phantom qLDPC family with growing logical dimension was previously known. We make three contributions. 1. We prove that recognizing phantomness of a given stabilizer code is at least as hard as Graph Isomorphism, even for CSS codes encoding only $k=2$ logical qubits. 2. We propose the first rate optimal phantom qLDPC families: for every fixed $D$, our CSS families achieve the maximal logical scaling $k=\Theta(\log n)$ and distance $d\geq D$. 3. We prove a distance no-go theorem: every phantom code family with $k=\omega(\sqrt{\log n})$ and check weight $w=O(1)$ satisfies $d\leq w$ for sufficiently large $n$. Thus our fixed-distance qLDPC families are distance-scaling optimal at the maximal logical scaling $k=\Theta(\log n)$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | An $[[n,k,d]]$ stabilizer code is phantom if every in-block logical CNOT gate can be realized by a permutation of its physical qubits. | background 1.00 | 0.09 | 0.04 | asserted 1.00 |
| s1 | This eliminates the large and complicated physical overhead normally required for logical entangling gates. | interpretation 0.65 | 0.10 | 0.07 | asserted 1.00 |
| s2 | Yet this symmetry is highly restrictive: phantom codes are rare, the number of logical qubits is limited to $k=O(\log n)$, and no phantom qLDPC family with growing logical dimension was previously known. | gap 0.88 | 0.07 | 0.06 | asserted 1.00 |
| s3 | We make three contributions. 1. | contribution 0.98 | 0.65 | 0.03 | unclear 0.62 |
| s4 | We prove that recognizing phantomness of a given stabilizer code is at least as hard as Graph Isomorphism, even for CSS codes encoding only $k=2$ logical qubits. 2. | contribution 0.74 | 0.96 | 0.07 | asserted 1.00 |
| s5 | We propose the first rate optimal phantom qLDPC families: for every fixed $D$, our CSS families achieve the maximal logical scaling $k=\Theta(\log n)$ and distance $d\geq D$. 3. | contribution 0.96 | 0.95 | 0.21 | asserted 0.99 |
| s6 | We prove a distance no-go theorem: every phantom code family with $k=\omega(\sqrt{\log n})$ and check weight $w=O(1)$ satisfies $d\leq w$ for sufficiently large $n$. | result 0.50 | 0.94 | 0.23 | asserted 1.00 |
| s7 | Thus our fixed-distance qLDPC families are distance-scaling optimal at the maximal logical scaling $k=\Theta(\log n)$. | interpretation 0.92 | 0.87 | 0.53 | asserted 1.00 |
What was examined, and what was not
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References
Discussion
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