# Hilbert space to accurately represent 3x3 Rubik's Cube

What Hilbert space of dimension greater than 4.3e19 would be most convenient for working with the Rubik's Cube verse one qudit?

The cardinality of the Rubik's Cube group is given by:

Examples

66 Qubits yields ~7.378697629484e19 states (almost more than double the number of states needed)

42 Qutrits yields ~1.094189891315e20 states (more than double the needed states)

• This question could be improved if you specify that you want a Hilbert space of dimension greater than that 4.3e19 such that particular operations from the previous question are implemented more easily. That would be a easier than a single qudit with that enormous d. – AHusain Aug 7 '18 at 3:21
• @AHusain thank you for the feedback! question updated. – meowzz Aug 7 '18 at 3:49
• A qubyte has dimension $2^8$, not 8. – DaftWullie Aug 7 '18 at 6:49

• No, it's really now. Complexity classes have nothing to do with this directly (they talk about how something scales with the size of the problem instance, and there are no problem instances here), I was just using it as an example of where counting is done in terms of $n$, not $2^n$. My reason for picking this particular example was that the relevant literature (and I mean very introductory text books) will probably discuss the $n$ vs $2^n$ issue. – DaftWullie Aug 7 '18 at 7:37
• If you ask "how hard is it to determine the mixing time or God's number of a (Cayley graph of a) group of order $O(\exp n)$, with $O(n)$ generators" that is a question about complexity classes. The number of bits/qubits to encode each element of such a group is $O(n)$. – Mark S Aug 7 '18 at 12:06