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]. Such a permutation maps the code onto itself and acts on the logical Pauli operators as the CNOT gate. It can therefore be absorbed into circuit compilation as a relabeling of the physical qubits, at zero overhead and with perfect fidelity [1]. The definition has been extended to non-CSS and non-qubit codes [2].
The smallest phantom code is the \([[4,2,2]]\) code with logical operators \(\overline{X}_1=XXII\), \(\overline{X}_2=XIXI\), \(\overline{Z}_1=ZIZI\), and \(\overline{Z}_2=ZZII\), for which swapping qubits 1 and 3 implements \(\overline{\mathrm{CNOT}}_{12}\) and swapping qubits 1 and 2 implements \(\overline{\mathrm{CNOT}}_{21}\) [1; Fig. 1]. Gluing \(m\) copies of the \([[4,2,2]]\) code into a tube, with \(X\)-type stabilizers on the faces joining consecutive copies, yields CSS phantom codes with parameters \([[4m,2,(d_X=2,d_Z=2m)]]\). Each logical CNOT of the glued code is one transposition applied in every copy. Puncturing one qubit gives \([[4m-1,2,(d_X=2,d_Z=2m-1)]]\), for \(m\geq1\) [1; Thm. 8].
Protection
CSS phantom codes obey a Hamming-type constraint [1]. If \(d=d_{\mu}\) for \(\mu\in\{X,Z\}\), then \(\eta(2^k-1)\leq B(n,d)\). Here \(\eta\) counts weight-\(d\) logical operators in a fixed \(\mu\)-type logical equivalence class. The quantity \(B(n,d)\leq {n \choose d}\) is the maximum size of a binary length-\(n\) code whose pairwise sums have weight at least \(d\) [1]. Any qubit phantom code of distance \(d\geq 2\) encoding \(k\geq 2\) logical qubits with \(k\neq 4\) obeys \(n\geq 2^k-1\), equivalently \(k\leq \log_2(n+1)\) [2]. This parameter bound also holds for non-CSS phantom codes and for qubit subspace or subsystem phantom-LU codes [2]. For qubit stabilizer codes supporting every logical CX gate by single-qubit Clifford gates and qubit permutations, the bound holds for all \(k\) and distances except the \([[2,2,1]]\) code [3; Thm. 15]. A nonstabilizer \(((8,2^4,2))\) phantom code shows why the \(k=4\) exception remains necessary for general quantum codes [2].Transversal and Permutation-Based Gates
For CSS phantom codes, interblock \(\overline{\mathrm{CNOT}}\) gates are transversal. Combining transversal interblock CNOTs with in-block permutation CNOTs implements any logical CNOT circuit on \(2^a\) phantom-code blocks. The physical depth is at most \(4(2^a-1)\), up to a residual logical-qubit permutation. For unidirectional CNOT circuits, the depth bound is \(2(2^a-1)\) and the logical-qubit order is preserved [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 gate [1]. This rules 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. Such gates can also arise 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 [1]. The same codes 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
Among the \(2.71\times10^{10}\) inequivalent CSS codes with \(n\leq14\), \(1.39\times10^5\) are CSS phantom codes [1]. Further examples are known up to \(n=21\) [1].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 inner code with a one-logical-qubit outer 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 [1]. The smallest such examples have lower rates than the phantom quantum RM constructions [1].
- \(((8,16,2))\) \(PG(3,2)\) code— This is the exceptional nonstabilizer \(k=4\) qubit phantom code of minimal length eight that violates the generic bound \(n\geq 2^k-1\) [2].
- Complete-hypergraph CSS code— The simplex phantom code is a member of this family [1]. It attains the phantom-code bound \(n\geq2^k-1\) [2].
Primary Hierarchy
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
- [2]
- A. S. Morris and D. Malz, “Constraints on phantom codes from automorphism group bounds”, (2026) arXiv:2604.15111
- [3]
- J. M. Koh, S. Majidy, A. Chakraborty, A. Gong, S. J. S. Tan, and N. Y. Yao, “Achieving the limits of automorphism gates”, (2026) arXiv:2609.19250
Page edit log
- Victor V. Albert (2026-09-26) — most recent
- Victor V. Albert (2026-09-20)
- Victor V. Albert (2026-08-24)
- Victor V. Albert (2026-06-08)
- Victor V. Albert (2026-05-20)
Cite as:
“Phantom code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/phantom, arXiv:2606.11484