for the MPS algorithm using a $Nm=36$ qubits.[]{data-label="fig:entropies2d"}](fig-entropies2d.pdf){width="\linewidth"}

Let us begin the discussion using a general two-dimensional covariance matrix $$\Sigma = O(\theta)\left(
    \begin{array}{cc}
      \sigma_{max} & 0 \\ 0 & \sigma_{min}
    \end{array}
    \right)O^T,\;
  \mbox{with}\; O =\left(
    \begin{array}{cc}
      \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta)
    \end{array}
    \right).\label{eq:covariance2d}$$ One quadrature is squeezed by a factor $\sigma_{min}/\sigma_{max},$ while rotating the frame of reference an angle $\theta$, as shown in Fig. \[fig:entropies2d\]a. We sample this probability, reconstructing the exact wavefunction  with $Nm=28$ qubits. As shown in Table \[tab:state-sizes\], this size approaches the limits of a decent computer, using 2 gigabytes of data in real double precision form. In contrast, the same distributions using MPS, the techniques from Sect. \[sec:exponential\] and up to $Nm=36$ qubits consume less than 1 megabyte.

         Discretization         Nm    $\sigma_{min}/\sigma_{max}$   $\theta$          exact size          order       MPS size      
  ---------------------------- ----- ----------------------------- ---------- -------------------------- ------- ------------------ --
           $16,384^2$           28                 1                   0       $268\times 10^6$ (2 Gb)     (A)      1,040 (8 kb)    
   “ & 28 & 0.1 & $\pi/4$ & ”   (A)         126,628 (1 Mb)                                                                          
   “ & 28 & 0.1 & $\pi/4$ & ”   (B)          9,388 (71 kb)                                                                          
          $262,144^2$           36                0.1               $\pi/4$    $69\times 10^9$ (524 Tb)    (A)     183,220 (1 Mb)   
   “ & 36 & 0.1 & $\pi/4$ & ”   