Table 1 List of the different methods for solving the group representative problem.

Method	Cl. Mem	Q Mem	Time
Look-up	\({\mathcal{O}}(\| V\| )\)	0	\({\mathcal{O}}(1)\)
On-the-fly	\({\mathcal{O}}(1)\)	0	\({\mathcal{O}}(\| G\| C(G))\)
Divide & conquer	\({\mathcal{O}}(\| V{\| }^{\frac{1}{m}})\)	0	\({\mathcal{O}}(\| G\| C(G))-{\mathcal{O}}{(C(G))}^{{\rm{a}}}\)
Gmin	\({\mathcal{O}}(1)\)	\({\mathcal{O}}(2{\mathrm{log}}\,\| V\| +{\mathrm{log}}\,\| G\| )\)	\(\mathcal{O}(\sqrt{\| G\| }(\mathcal{C}(\hat{G})+{\mathrm{polylog}}(\| V\|)))\)

We generally expect that \({\mathrm{log}}\,| G| \ll {\mathrm{log}}\,| V| \le | G| \ll | V|\). \(C(G)\) is the classical cost to calculate the action of \(G\) on an arbitrary \(v\), while \({\mathcal{C}}(G)\) is the quantum cost to calculate the action of \(G\) on an arbitrary \(v\); \(m\) represents the number of sublattices in the divide & conquer method
\({}^{{\rm{a}}}\)In general, divide & conquer does reduce the time cost versus on-the-fly. For a general group which satisfies the necessary properties required by divide & conquer (see main text), the most we can assume is that the search is over a subgroup of size \({\mathcal{O}}(| G| )\), but with a smaller coefficient. However, in many important cases, most notably translation groups, this search for most \(v\) is of size \(O(1)\); see ref. ⁸ for details

Quick links

Search