# 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} E_{\mathbf{p},\mathbf{q}}(\mathbf{a}^{\dagger},\mathbf{a})=\prod_{j=1}^{n}a_{j}^{\dagger p_{j}}a_{j}^{q_{j}}~. \tag*{(4)}\end{align} Any AD ladder error \(E_{\mathbf{p}=\boldsymbol{0},\mathbf{q}}\) with \(|\mathbf{q}|<d_{\downarrow}\) is detectable. Any ladder error \(E_{\mathbf{p},\mathbf{q}}\) 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

## Parents

- Coherent-state constellation code — Coherent-state QSCs are coherent-state constellation codes constrained to lie on a sphere.
- Amplitude-damping (AD) code — QSC codewords are superpositions of coherent states with the same energy, but coherent states are not eigenstates of the energy Hamiltonian. The AD Kraus operator \(E_{0}^{\otimes n}\) acts identically on each coherent state by shrinking the radius of the QSC's sphere.

## Children

- 2T-qutrit code — The \(2T\)-qutrit is a QSC on the two-dimensional complex sphere whose code constellation is the \(4\{3\}4\) complex polytope.
- Cat code — Cat codes are QSCs on the one-dimensional complex sphere.
- Concatenated cat code
- Clifford subgroup-orbit QSC
- Clifford group-representation QSC — The Clifford group-representation QSC has non-uniform coefficients.
- Hessian QSC — The Hessian QSC is an example of a QSC with logical constellation built from the Hessian complex polyhedron.

## 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.
- Concatenated cat 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.
- Asymmetric quantum code — QSC code parameters against loss/gain errors and Gaussian rotations can be tuned.

## References

- [1]
- S. P. Jain et al., “Quantum spherical codes”, Nature Physics (2024) arXiv:2302.11593 DOI

## 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