 | : Degree of connectivity in the weaved basis. This reduction in connectivity results in a circuit depth that scales polynomially in the volume. : Diagrammatic depiction of the Degree of Connectivity for a system with 16 total operators. Fig.~\ref{fig:OriginalBasis} shows the result for a Hamiltonian with the original constraint giving rise to maximal ${\rm DoC} = 16$. Fig.~\ref{fig:WeavedBasis} illustrates the DoC after the operator basis has been transformed to reduce the degree of connectivity, using the weaved operation. The degree of connectivity is now $\rm DoC = 4$, coming from the terms involving $\CO_1$, $\CO_5$, $\CO_9$ and $\CO_{13}$. Note that both Hamiltonians have the same spectrum, up to digitization effects. The operator $\CO_i$ could be $\CP_i$ or $\CQ_i$. |
