Quantum spherical code (QSC)[1]
Description
Code whose codewords are superpositions of points on an \(n\)-dimensional real or complex sphere. Such codes can in principle be defined on any configuration space housing a sphere, but the focus of this entry is on QSCs constructed out of coherent-state constellations.
More technically, a QSC is a collection \(\{\mathcal{C}_k\}_{k=1}^K\) of logical constellations, each of which yields a codeword by taking a quantum superposition of all points \(\mathbf{x}\in \mathcal{C}_k\). Taken together, the logical constellations yield the code constellation, \(\mathcal{C}=\bigcup_{k=1}^{K}\mathcal{C}_{k}\).
Codewords of coherent-state QSCs of uniform superposition are defined as \begin{align} |\mathcal{C}_{k}\rangle\sim\frac{1}{\sqrt{|{\mathcal{C}}_{k}|}}\sum_{\boldsymbol{\alpha}\in\mathcal{C}_{k}}|\sqrt{\bar{N}}\boldsymbol{\alpha}\rangle~, \tag*{(1)}\end{align} where \( |\boldsymbol{\alpha} \rangle = |\alpha_1,\alpha_2,...\alpha_n \rangle \) is an \(n\)-mode coherent state. This asymptotic expression is valid in the limit of large energy \(\bar{N}\to\infty\).
Coherent-state QSCs on \(n\) modes are denoted by \(((n,K,d_E,\langle t_{\downarrow},d_{\updownarrow},d_{\downarrow}\rangle))\), where \(K\) is codespace dimension, \(d_E\) is the squared minimum distance, i.e., the smallest Euclidean distance between pairs of distinct points across all codewords, and \( t_{\downarrow},d_{\updownarrow},d_{\downarrow} \) are the number of correctable losses (plus 1), the degree distance, and the number of detectable losses (plus 1), respectively.
Protection
The resolution \(d_E\) of the code is defined as \begin{align} d_E = \min_{\boldsymbol{\alpha},\boldsymbol{\beta}\in\mathcal{C}} \Vert\boldsymbol{\alpha}-\boldsymbol{\beta}\Vert^2~. \tag*{(2)}\end{align} The code protects against passive Gaussian transformations, which manifest as rotations on the sphere, \( |\boldsymbol{\alpha}\rangle \rightarrow |\mathbf{R}\boldsymbol{\alpha}\rangle \) for all \(\mathbf{R}\). Detectable transformations corresponds to rotations for which \begin{align} \Vert \mathbf{R}\boldsymbol{\alpha} - \boldsymbol{\alpha}\Vert^2 < d_E~, \tag*{(3)}\end{align} in the large \(\bar{N}\) limit.
The code also protects against general ladder errors, which are defined as \begin{align} \mathbf{a}^{\dagger\mathbf{p}}\mathbf{a}^{\mathbf{q}}=\prod_{j=1}^{n}a_{j}^{\dagger p_{j}}a_{j}^{q_{j}}~. \tag*{(4)}\end{align} Any AD ladder error \(\mathbf{a}^{\mathbf{q}}\) with \(|\mathbf{q}|<d_{\downarrow}\) is detectable. Any ladder error \(\mathbf{a}^{\dagger\mathbf{p}}\mathbf{a}^{\mathbf{q}}\) with \(|\mathbf{p}|,|\mathbf{q}|<t_{\downarrow}\) is detectable, implying that up to \(t_{\downarrow}-1\) losses are correctable. Any ladder error with degree \(|\mathbf{p}+\mathbf{q}|<d_{\updownarrow}\) is detectable.
Decoding
Lindbladian scheme stabilizing all points in the constellation and protecting from the AD operator \(E_{0}^{\otimes n}\) [1].Cousins
- Group-representation code— QSCs should be able to be formulated as group-representation codes whose group is that formed by the permutation representation of the code polytope symmetry group, but this representation may be reducible.
- Constant-energy code— QSCs are quantum analogues of spherical and constant-energy codes because they store information in quantum superpositions of points on a sphere in quantum phase space.
- Spherical code— QSCs are quantum analogues of spherical and constant-energy codes because they store information in quantum superpositions of points on a sphere in quantum phase space.
- Single-spin code— Single-spin codes whose codewords are expressed in terms of discrete sets of spin-coherent states may also be interpreted as QSCs.
- Qubit CSS code— CSS codes concatenated with two-component cat codes form QSCs which have a weight-based notion of distance.
- Two-component cat code— CSS codes concatenated with two-component cat codes form QSCs which have a weight-based notion of distance.
- Polytope code— QSCs can be constructed by using vertices of polytopes for logical constellations. The logical constellations form the vertices of the code constellation, a polytope compound.
- Hyperbolic sphere packing— Quantum versions [2] of hyperbolic sphere packings have been developed for hyperbolic space \(SO(n,1)/O(n)\), which is the negative-curvature analogue of the sphere.
- Tiger code— Tiger (quantum spherical) codewords consist of continuous and compact (discrete and finite) coherent-state constellations. Both codes protect against losses and gains of occupation numbers along with rotation noise stemming from modal dephasing. Protection against the latter type of noise is characterized by the minimum Euclidean distance between coherent states in different logical constellations.
- Asymmetric quantum code— QSC code parameters against loss/gain errors and Gaussian rotations can be tuned.
Primary Hierarchy
References
- [1]
- S. P. Jain, J. T. Iosue, A. Barg, and V. V. Albert, “Quantum spherical codes”, Nature Physics (2024) arXiv:2302.11593 DOI
- [2]
- Y. Wang, Y. Xu, and Z.-W. Liu, “Tessellation codes: encoded quantum gates by geometric rotation”, (2025) arXiv:2410.18713
Page edit log
- Shubham P. Jain (2023-02-23) — most recent
- Victor V. Albert (2023-02-23)
Cite as:
“Quantum spherical code (QSC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/qsc
Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/oscillators/qsc/qsc.yml.