the (B) order. This importance-based structure brings down the entanglement to about 3 e-bits and a reduction of 54,000 in the information required. We conjecture that for these Gaussian states—and other smooth functions—the (B) order consumes at most $N$ e-bits, giving a scaling of resources $\mathcal{O}(2^{2N}\times N)$ in both the time and memory costs of reproducing the probability distribution.

MPS quantum registers {#sec:quantum-inspired-algorithms}
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In the previous section we have seen that it is possible to encode single and multimode probability distributions in quantum registers, that these states are typically weakly entangled and admit an efficient MPS representation. We now review the representation of wavefunctions and operators, and the basic ingredients in the MPS toolbox—tensor contraction and reordering, tensor renormalization, time evolution, etc—that will be used in Sect. \[sec:analysis\] to implement actual algorithms. More precisely, Sect. \[sec:bilinear\] introduces two representations of multivariate functions: one following the precepts from Sect. \[sec:other-discretizations\], and another one that improves the computation of expected values (Sect. \[sec:expected\]) and equation solving. We will also discuss the algorithm of MPS simplification (Sect. \[sec:mps-approximation\]), which is an essential tool to implement all numerical analysis approximation schemes.

Quadratic and linear representation {#sec:bilinear}
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![Quadratic MPS representation, where functions are 