non-empty set of mappings from $F$ to $K$. We seek to approximate $S(f)$ for $f\in F$ by means of quantum computations.

Usually $F$ is a set in a function space, $S$ is the solution operator, which maps the input $f\in F$ to the exact solution $S(f)$, and we want to approximate $S(f)$. $\Lambda$ is usually a set of linear functionals, supplying information $\lambda(f)$ about $f$ through which the algorithm can access the input $f$. $K$ is mostly ${{\mathbb R}}$ or $\C$, $G$ is a space containing both the solutions and the approximations, and the error is measured in the norm of $G$.

In the sequel it will be convenient to consider $f\in F$ also as a function on $\Lambda$ with values in $K$ by setting $f(\lambda):=\lambda(f)$. Let $\mathcal{F}(\Lambda,K)$ denote the set of all functions from $\Lambda$ to $K$.

Let $H_1$ be the two-dimensional complex Hilbert space $\C^2$, with its unit vector basis $\{e_0,e_1\}$, let $$H_m=\underbrace{H_1\otimes\dots\otimes H_1}_{m},$$ equipped with the tensor Hilbert space structure. Denote $${{\mathbb Z}}[0,N) := \{0,\dots,N-1\}$$ for $N\in{{\mathbb N}}$ (we write ${{\mathbb N}}= \{1,2,\dots \}$ and ${{\mathbb N}}_0={{\mathbb N}}\cup\{0\}$). Let $\mathcal{C}_m = \{\lg i\rs:\, i\in{{\mathbb Z}}[0,2^m)\}$ be the canonical basis of $H_m$, where $\lg i \rs$ stands for 