Consider the four cells on each of the six faces of the 2x2x2 Rubik's cube (the pocket cube), with front, up, and right twists being labelled as $\langle F,U,R\rangle$.

Each move uses fifteen SWAP gates. Also, each such move acts on twelve cells of the cube, and each cell can be in a superposition of six different colors. The pocket cube has a natural $k=12$-local, $d=6$ dimensional Hamiltonian.

We can Trotterize to simulate the quarter-turn Hamiltonian of the 2x2 Rubik's cube as:

$$e^{-i \mathcal{H}t}\approx\big (\sqrt{F_1}\sqrt{U_1}\sqrt{R_1}\sqrt{F_3}\sqrt{U_3}\sqrt{R_3}\big )^2=:W.$$

We have to uncompute the SWAPS in taking the roots, which doubles the number of CSWAPs. By squaring square roots, our Trotter factor is two. With six generators and thirty CSWAPs per generator, we can simulate the Hamiltonian with only $2\times 6\times 30=360$ six-dimensional CSWAP gates, or $360\times 3=1080$ binary Fredkin gates. To me, this seems surprisingly small.

Can we learn anything post-classical about the cube, with near-term NISQ era devices?

For example if we could do a Hadamard test on $W$ acting on various superpositions $\vert\psi\rangle$, would we learn anything about the connectedness of the cube relative to the superposition $\vert\psi\rangle$? The overall fidelity on near-term devices may be appreciably bounded above zero.


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