$\ket{p^{(m)}}$ with discretization size $2^{m+1}$ can be constructed from a coarser state $\ket{p^{(m)}}$ by appending one auxiliary qubit through a unitary operation $U^{(m)}.$ (b) When a state of $m$ qubits is upgraded to $m+k,$ we can study the entanglement between the old and new qubits through a Schmidt decomposition . (c) If the state has a small Schmidt number for all 1D bipartitions, it is weakly entangled and admits an efficient MPS representation .[]{data-label="fig:states"}](fig-states.pdf){width="0.9\linewidth"}

The unitary by Grover and Rudolph’s $U_p$ is a recursive construct that adds one more qubit of precision in each step. As sketched in Fig. \[fig:states\]a, the procedure reads $$\begin{aligned}
  \label{eq:GR-unitary}
   \ket{p_\text{GR}^{(m+1)}}&=u^{(m+1)}\ket{p_\text{GR}^{(m)}(s)}\ket{0}
                            = \sum_{s=0}^{2^m-1} \sqrt{p_\text{GR}^{(m)}(s)}\ket{s}(\cos(\theta_s)\ket{0}+\sin(\theta_s)\ket{1})\\
                              &= \sum_{s'=0}^{2^{m+1}-1}\sqrt{p_\text{GR}^{(m+1)}(s')}\ket{s'},
  \nonumber\end{aligned}$$ where we identify $\ket{s}\!\ket{0}=:\ket{2s}$ and $\ket{s}\!\ket{1}=:\ket{2s+1}.$ The rotation angle $0\leq\theta_i\leq\pi/2$ is obtained from two identities $$\cos(\theta_s)^2 = \frac{p_\text{GR}^{(m+1)}(2s)}{p_\text{GR}^{(m)}(s)},\; \sin(\theta_s)^2= \frac{p_\text{GR}^{(m+1)}(2s)}{p_\text{GR}^{(m)}(s)}.$$ Unfortunately, this algorithm *requires an exponentially large number of angles* and involves a highly non-local unitary with a potentially bad decomposition. The GR algorithm must be therefore considered more a proof of existence, than a practical recipe that can be used when bounding the resources of Monte Carlo analysis.

We will study the GR construct and other discretizations, demonstrating that there are efficient alternatives to equation . Our 