Single-spin code

Single-spin code[1]

Description

An encoding into a monolithic (i.e. non-tensor-product) Hilbert space that houses an irreducible representation of \(SU(2)\) or, more generally, another Lie group. In some cases, this space can be thought of as the permutation invariant subspace of a particular tensor-product space.

The analogue of oscillator coherent states for single spins are the spin coherent states [2].

Protection

For the \(SU(2)\) case, a continuous-time single-spin noise channel akin to the depolarizing channel is the Landau-Streater channel [3]. A particular error basis of interest consists of the spherical tensors [4].

The \(SU(2)\) Lie Algebra can also be used as a noise model; it connects states whose angular momentum projections differ by at most an integer [5]. More generally, the group’s Lie algebra induces a metric on the carrying vector space, and its operators can be chosen as a noise basis [1]. Code existence is guaranteed by the Tverberg theorem [1]. There are quantum MacWilliams identities for such metric spaces [1].

Rate

There exists a family of single-spin codes for \(SU(q=N)\) with logical dimension \(K = o(2^N)\) and distance of order \(o(N/\log N)\) [6].

Cousins

PI qubit code— Single-spin codes are subspaces of a single large \(SU(2)\) spin, which can be either standalone or correspond to the PI subspace of a set of spins via the Dicke state mapping.
Bosonic code— Bosonic states are typically represented with the assumption that a common phase reference exists, and the superselection rule compliant (SSRC) framework yields expressions without this assumption [7–13]. In this framework, single-mode states can be treated as two-mode states in a fixed subspace of total occupation number \(N\) in the limit \(N \to \infty\). Passive Gaussian operations acting on the fixed-photon subspace of two modes realize \(U(2)\) transformations in the Jordan-Schwinger boson mapping [14–17].
Quantum spherical code (QSC)— Single-spin codes whose codewords are expressed in terms of discrete sets of spin-coherent states may also be interpreted as QSCs.
Permutation-invariant (PI) code— Modular-qudit PI codes can be converted to spin codes defined on the completely symmetric irrep of \(SU(q)\) via the simplex mapping [6; Prop. IV.2]. Any transversal gates are mapped to \(SU(q)\) gates on the spin codes [6].

Member of code lists

Primary Hierarchy

Quantum Domain

Spin Kingdom

Spin codeQECC Quantum

Parents

Spin code

Monolithic quantum codeQECC Quantum

Single-spin code

Children

Æ code

Since Æ codes are defined in a subspace of fixed total angular momentum and protect against errors linear in the momentum generators, so they can also be thought of a single-spin codes.

Clifford-group spin code

Landau-level spin code

The Landau-level spin code lies in a particular irrep present in the induced representation \(\text{Ind}_{U(1)}^{SU(2)} \lambda\), where \(\lambda\in \mathbb{Z}\) labels irreps of \(U(1)\) and quantifies the monopole strength [18].

References

[1]: C. Bumgardner, “Codes in W\ast-metric Spaces: Theory and Examples”, (2012) arXiv:1205.4517
[2]: J. M. Radcliffe, “Some properties of coherent spin states”, Journal of Physics A: General Physics 4, 313 (1971) DOI
[3]: L. J. Landau and R. F. Streater, “On Birkhoff’s theorem for doubly stochastic completely positive maps of matrix algebras”, Linear Algebra and its Applications 193, 107 (1993) DOI
[4]: S. Omanakuttan and J. A. Gross, “Multispin Clifford codes for angular momentum errors in spin systems”, Physical Review A 108, (2023) arXiv:2304.08611 DOI
[5]: J. A. Gross, “Designing Codes around Interactions: The Case of a Spin”, Physical Review Letters 127, (2021) arXiv:2005.10910 DOI
[6]: A. Aydin, V. V. Albert, and A. Barg, “Quantum error correction beyond \(SU(2)\): spin, bosonic, and permutation-invariant codes from convex geometry”, (2025) arXiv:2509.20545
[7]: Y. Aharonov and L. Susskind, “Charge Superselection Rule”, Physical Review 155, 1428 (1967) DOI
[8]: K. Mølmer, “Optical coherence: A convenient fiction”, Physical Review A 55, 3195 (1997) DOI
[9]: B. C. Sanders, S. D. Bartlett, T. Rudolph, and P. L. Knight, “Photon-number superselection and the entangled coherent-state representation”, Physical Review A 68, (2003) arXiv:quant-ph/0306076 DOI
[10]: S. D. BARTLETT, T. RUDOLPH, and R. W. SPEKKENS, “DIALOGUE CONCERNING TWO VIEWS ON QUANTUM COHERENCE: FACTIST AND FICTIONIST”, International Journal of Quantum Information 04, 17 (2006) arXiv:quant-ph/0507214 DOI
[11]: B. C. Sanders, “Review of entangled coherent states”, Journal of Physics A: Mathematical and Theoretical 45, 244002 (2012) arXiv:1112.1778 DOI
[12]: E. Descamps, A. Saharyan, A. Chivet, A. Keller, and P. Milman, “Unified framework for bosonic quantum information encoding, resources and universality from superselection rules”, (2025) arXiv:2501.03943
[13]: A. Saharyan, E. Descamps, A. Keller, and P. Milman, “Resources for bosonic metrology: quantum-enhanced precision from a superselection rule perspective”, (2025) arXiv:2507.13245
[14]: P. Jordan, “Der Zusammenhang der symmetrischen und linearen Gruppen und das Mehrk�rperproblem”, Zeitschrift f�r Physik 94, 531 (1935) DOI
[15]: Schwinger, Julian. On Angular Momentum. Courier Dover Publications, 2015.
[16]: J. Schwinger, “Angular Momentum”, Quantum Mechanics 149 (2001) DOI
[17]: A. Klein and E. R. Marshalek, “Boson realizations of Lie algebras with applications to nuclear physics”, Reviews of Modern Physics 63, 375 (1991) DOI
[18]: E. Kubischta and I. Teixeira, “Intrinsic Quantum Codes: One Code To Rule Them All”, (2025) arXiv:2511.14840

Page edit log

Victor V. Albert (2022-11-13) — most recent

Cite as:

“Single-spin code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/single_spin

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/spins/single_spin/single_spin.yml.