Gross spin code[1]

Description

A spin code designed to realize Clifford gates using \(SU(2)\) rotations. Codewords are subspaces of a spin's Hilbert space that house irreducible representations (irreps) of the single-qubit Clifford group (i.e., the binary octahedral (\(2O\)) subgroup of \(SU(2)\)). This is done by restricting the \(SU(2)\) irrep to \(2O\), and determining the carrier spaces of any nontrivial irreps of \(2O\). Since irreps of \(2O\) do not appear in integer spins, half-integer spins are used.

A simple example of a codespace is a projection onto an instance of a particular irrep of \(2O\), referred to as either \( \varrho_4 \) or \( \varrho_5 \). In the case of only one instance of the desired irrep present in the spin, the projection is created as follows: \begin{align} P_\varrho = \frac{\text{dim} \varrho}{|2O|} \sum_{g \in 2O} \chi_\varrho (g)^* D(g)~, \end{align} where \(D(g)\) is the \(SU(2)\) Wigner matrix corresponding to group element \(g\), and the character \(\chi_\varrho (g) = \text{tr}(\varrho(g))\) is the trace of the desired irrep evaluated at a group element.

Logical Pauli matrices \(\overline{\sigma}_w\) are defined using the above projection and the angular momentum operators: \begin{align} \overline{\sigma}_w = i P_\varrho e^{-i \pi J_w} P_\varrho~. \end{align} Finally, \(|\overline{0} \rangle\) is defined as the \(+1\) eigenvalue of \(\overline{\sigma}_z\) and \(|\overline{1} \rangle = \overline{\sigma}_x |\overline{0} \rangle \).

Gates

Universal computation results from being able to prepare a single logical state, perform one measurement, and the following logical gates: the phase gate (\( \overline{S} \)), the Hadamard gate (\(\overline{H}\)), the conditional phase gate (\(\overline{CZ}\)), and the square root of the phase gate (\(\overline{T}\)). Single-qubit Cliffords can be generated using \(\overline{S}\) and \(\overline{H}\), the extension to multiple-qubit Cliffords is done using \(\overline{CZ}\), and \(\overline{T}\) is to transform to non-Clifford states. Together these gates can be used to create all logical unitaries, while preparation and measurement complete universal quantum computation.

Parent

Zoo code information

Internal code ID: j_gross

Your contribution is welcome!

on github.com (edit & pull request)

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Zoo Code ID: j_gross

Cite as:
“Gross spin code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/j_gross
BibTeX:
@incollection{eczoo_j_gross, title={Gross spin code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/j_gross} }
Permanent link:
https://errorcorrectionzoo.org/c/j_gross

References

[1]
J. A. Gross, “Designing Codes around Interactions: The Case of a Spin”, Physical Review Letters 127, (2021). DOI; 2005.10910

Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/spins/j_gross.yml.