qubits. The second part of this work introduces the idea of MPS quantum registers, whereby we encode multivariate functions in arrangements of qubits that are represented, manipulated and interrogated using the MPS representation. Section \[sec:quantum-inspired-algorithms\] further develops this idea, recalling well known algorithms from the literature and how they specialize for our purposes. With these tools at hand, Section \[sec:analysis\] develops new quantum-inspired algorithms for the numerical analysis of multivariate functions. These include the mapping of functions to MPS format (Sect. \[sec:exponential\]), Fourier analysis (Sect. \[sec:qft\]), interpolation methods (Sect. \[sec:interpolation\]), and techniques for approximating derivatives of discretized functions (Sect. \[sec:derivatives\]), both through finite-difference (Sect. \[sec:finite-differences\]) and the Quantum Fourier Transform (QFT, Sect. \[sec:QFT-derivative\]). Sect. \[sec:time-evolution\] combines these techniques into higher level algorithms for solving partial derivative equations, demonstrating their performance in the Fokker-Planck model. This work is closed with a discussion of the results, including connections to recent advances in tensor-based numerical analysis, and an outlook of applications.

Summary {#sec:summary}
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  Problem                                                                        Algorithm                             Type   Cost                                                  
  ------------------------------------------------------------------------------ ------------------------------------- ------ ----------------------------------------------------- --
  Expected value                                                                 Monte Carlo                           C      $\mathcal{O}(1/\varepsilon^2)$                        

  “ & Amplitude estimation & Q & $\mathcal{O}(1/\varepsilon)$\                   MPS                                   QI     $\mathcal{O}(-N\chi^3\log(\varepsilon))$              
  ”                                                                                                                                                                                 

  Fourier transform                                                              QFT                                   Q      $\mathcal{O}(N^2m^2)$                                 

  “ & FFT & C & 