classical methods (C), from algorithms for a quantum computer (Q), or classical MPS techniques that rely on a quantum register approach (QI).[]{data-label="tab:algorithms"}

Quantum computing has introduced the clever idea of encoding probability distributions in quantum registers. In this encoding, a small number of qubits can store an exponentially large number of function samples. To fix ideas and notation, let us take a function $p(x_1,\ldots,x_N)$ of $N$ variables in a bounded interval. Let us use $m$ qubits to uniformly discretize the domain $$\begin{aligned}
  x_s^{i} &= a_i + \frac{b_i-a_i}{2^m}s_i = a_i + \delta^{i}_m \sum_{k=1}^m\frac{s_i^k}{2^k},\;i=1,2\ldots N.\label{eq:coordinates}\end{aligned}$$ The non-negative integer $s_i$ takes all possible values obtained by grouping $m$ bits $s_i^1s_i^2\cdots s_i^m,$ ordered in decreasing significance. The $N$ integers or $Nm$ bits can be associated to different states of a quantum register, enabling two representations of the function $p.$ The first one assumes a non-negative function, $$\ket{p} \propto \sum_{s_1,\ldots,s_N} \sqrt{p(s_1,\ldots,s_N)}\ket{s_1}\otimes\cdots\ket{s_N},\;s_i\in\{0,1,\ldots 2^{m}-1\},\label{eq:representation-1}$$ and associates $p(s)$ to the probability of the state $\ket{s}.$ The second encoding does not make this assumption and maps the distribution directly to the wavefunction $$\ket{p} \propto \sum_{s_1,\ldots,s_N} p(s_1,\ldots,s_N)\ket{s_1}\otimes\cdots\ket{s_N}.
  \label{eq:representation-2}$$ Both representations can be extended to situations where the functions are not normalized, just by keeping track of global prefactors.

In this 