$\mathcal{O}(Nm2^{Nm})$\                                         MPS QFT                               QI     $\mathcal{O}(Nm \times \text{Simp}_{Nm})$             
  ”                                                                                                                                                                                 

  Interpolation                                                                  Linear $(k=1)$                        C      $\mathcal{O}(2^{Nm})$                                 

  “ & MPS Linear $(k=1)$ & QI & $\sim\text{Simp}_{Nm}$\                          FFT                                   C      $\mathcal{O}(N(m+k)2^{N(m+k)})$                       
  ”                                                                                                                                                                                 

  “ & MPS QFT & QI & $\sim 3\times \text{QFT}_{N(m+k)}$\                         MPS differences                       QI     $\mathcal{O}(T_\text{cgs}\times\text{Simp}_{Nm})$     
  PDE Evolution & Finite differences & C & $\mathcal{O}(T_\text{cgs}2^{2Nm})$\                                                                                                      
  ”                                                                                                                                                                                 

  “ & FFT method & C & $\mathcal{O}((Nm+1)2^{Nm})$\                              MPS QFT                               QI     $\sim 2\times \text{QFT}_{N(m+k)}$                    
  ”                                                                                                                                                                                 

  State construct                                                                GR-like (Sect. \[sec:bounds\])        Q      $\mathcal{O}(Nm\chi^2)$                               

  “ & Explicit wavefunction & C & $\mathcal{O}(2^{Nm})$\                         MPS                                   QI     $\mathcal{O}(T_\text{steps}\times\text{Simp}_{Nm})$   
  ”                                                                                                                                                                                 

  MPS algorithms                                                                 Simplification $(\text{Simp}_{Nm})$   C      $\mathcal{O}(T_\text{sweeps} Nm 4d^3\chi^3)$          

  “ & Expected values & C & $\mathcal{O}(Nm\times 2d\chi^3)$\                    $\hat{O}_f\ket{p}$, MPO-MPS product   C      $\mathcal{O}(Nm(d\chi\chi_f)^2)$                      
  ”                                                                                                                                                                                 
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  : Algorithms and their costs. We compare the costs of different tasks when working with multivariate functions, from the construction of the state, to the simulation of their evolution. We use the following heuristic values: $\varepsilon,$ desired error bound; $N,$ number of variables; $m\sim \mathcal{O}(\log_2(\varepsilon)),$ number of qubits per variable for $2^m$ points in discretization; $d=2,$ physical dimension of qubits; $\chi, \chi_f,$ effective MPS and MPO bond dimensions; $T_\text{sweeps}$ number of iterations in the simplification algorithm (cf. Sect. \[sec:mps-approximation\]); $T_\text{cgs},$ number of iterations in conjugate gradient method (cf. Sect. \[sec:pde-finite-differences\]). We distinguish 