Phantom code[1]
Description
Qubit CSS code for which, in some logical basis, every ordered-pair logical \(\overline{\mathrm{CNOT}}_{ab}\) gate between logical qubits in the same code block can be implemented by a physical-qubit permutation [1].Protection
CSS phantom codes obey a Hamming-type constraint: if \(d=d_{\mu}\) for \(\mu\in\{X,Z\}\), then \(\eta(2^k-1)\leq B(n,d)\), where \(\eta\) counts weight-\(d\) logical operators in a fixed \(\mu\)-type logical equivalence class and \(B(n,d)\leq {n \choose d}\) is the maximum size of a binary length-\(n\) code whose pairwise sums have weight at least \(d\) [1]. All phantom codes identified in Ref. [1] are non-LDPC and encode \(k=O(\log n)\) logical qubits.Transversal Gates
Interblock \(\overline{\mathrm{CNOT}}\) gates are transversal because phantom codes are CSS codes. Combining transversal interblock CNOTs with in-block permutation CNOTs implements any logical CNOT circuit on \(2^a\) phantom-code blocks in physical depth at most \(4(2^a-1)\), up to a residual logical-qubit permutation; for unidirectional CNOT circuits, the bound is \(2(2^a-1)\) while preserving logical-qubit order [1].A stabilizer code supporting a logical gate by qubit permutations cannot admit any strictly transversal logical gate that does not commute with that permutation-implemented logical gate, ruling out strictly transversal implementations of several gates on phantom codes [1].Additional logical Clifford and non-Clifford gates can arise from code automorphisms combining local Cliffords and qubit permutations, from fold-diagonal gates using patterned one- and two-qubit diagonal interactions, and from non-uniform diagonal single-qubit rotations [1].Gates
Certain phantom quantum RM codes admit the full logical Clifford group via fold-\(\overline{S}_i\overline{S}_j\) gates and teleported Hadamards, and admit a distance-two magic-gate scheme by temporarily projecting into hypercube-code subspaces [1].Decoding
Spatiotemporal sliding-window correlated list and most-likely-error decoders for Steane-style error correction [1].Fault Tolerance
Preselection-based fault-tolerant state preparation and Steane-style error correction for non-LDPC phantom quantum RM codes [1].Notes
Ref. [1] exhaustively enumerates all \(2.71\times10^{10}\) inequivalent CSS codes with \(n\leq14\), identifying \(1.39\times10^5\) CSS phantom codes, and uses SAT-based search to find further examples up to \(n=21\).Cousins
- Quantum Reed-Muller (RM) code— Some quantum RM codes are phantom after selected logical qubits of a parent quantum RM code are fixed to \(\ket{\overline{0}}\) or \(\ket{\overline{+}}\), promoting the corresponding logical operators to stabilizers [1].
- Concatenated qubit code— Concatenating a phantom outer code with a one-logical-qubit inner quantum code preserves phantomness.
- Hypergraph product (HGP) code— Some hypergraph-product constructions, such as products of a classical simplex code and a repetition code, yield phantom codes, while the smallest examples have lower rates than the phantom quantum RM constructions [1].
- \([[4,2,2]]\) Four-qubit code— The \([[4,2,2]]\) code is the smallest phantom code: logical CNOT gates between its two logical qubits can be implemented by physical-qubit permutations [1]. Gluing copies of the \([[4,2,2]]\) code with \(X\)-type stabilizers yields CSS phantom codes with parameters \([[4m,2,(d_X=2,d_Z=2m)]]\), and puncturing one qubit from this construction yields \([[4m-1,2,(d_X=2,d_Z=2m-1)]]\), for \(m\geq1\) [1].
Primary Hierarchy
Parents
Phantom code
Children
The \([[2^D,D,2]]\) hypercube quantum codes are phantom codes: all ordered-pair in-block logical CNOT gates can be implemented by physical-qubit permutations [1].
Ref. [1] identifies a \([[12,2,2]]\) CSS phantom code by exhaustive enumeration.
This \([[14,3,3]]\) code is a CSS phantom code obtained from the punctured hypercube code and the two-qubit phase-flip repetition code [1].
The \([[7,2,2]]\) HGP code is the smallest member of a simplex/repetition HGP family of CSS phantom codes [1].
This code is the punctured version of the \([[8,3,2]]\) hypercube quantum code and is a phantom code [1].
References
- [1]
- J. M. Koh, A. Gong, A. C. Diaconu, D. B. Tan, A. A. Geim, M. J. Gullans, N. Y. Yao, M. D. Lukin, and S. Majidy, “Entangling logical qubits without physical operations”, (2026) arXiv:2601.20927
Page edit log
- Victor V. Albert (2026-05-20) — most recent
Cite as:
“Phantom code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/phantom