This approach suggests new algorithms for integrating probability distributions and computing expected values (Sect. \[sec:expected\]), for implementing discrete differentiation (Sect. \[sec:derivatives\]), Fourier transform (Sect. \[sec:qft\]), interpolation (Sect. \[sec:interpolation\]) and for the solution of partial differential equations (Sect. \[sec:time-evolution\]).

Table \[tab:algorithms\] summarizes the algorithms discussed in this work, paired with alternatives that already exist for quantum computers or in the field of numerical analysis. The table summarizes the costs of those algorithms, expressed in terms of well known quantities—discretization error, number of qubits, estimated entanglement and bond dimension size, etc—. In the case of quantum-inspired numerical analysis, we must emphasize that the performance metrics are heuristic. However, if entanglement remains bounded throughout the simulations, the quantum register method demands a small bond dimension $\chi,$ and the algorithms provide an exponential speedup over other classical techniques—from finite differences to the highly performant FFT techniques.

Storing multivariate functions in quantum registers {#eq:quantum-register}
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GR discretization {#sec:gr-construct}
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One of the earliest works suggesting the encoding of functions in quantum registers is the unpublished manuscript by Grover and Rudolph [@grover2002]. This designed a unitary operator $U_p$ that encodes a probability distribution $p(x)$ in an empty quantum register with $m$ qubits $$\begin{aligned}
  \ket{p_\text{GR}^{(m)}} &:= U_p\ket{0,0,\ldots,0} 