Quantum linear-systems state-output task

Let $A:{\mathbb{C}}^n\to {\mathbb{C}}^m$ be a finite-dimensional linear map, $m,n\ge 1$, and let $b\in \mathrm{ran}(A)\setminus \{0\}$, where $\mathrm{ran}(A)$ is the image defined in Kernel and image of a linear map. The quantum linear-systems state-output task (QLSP) is to prepare the normalized domain-register state $\mathrm{\mid }x⟩:=\frac{A^{+}b}{\mathrm{\parallel }{A}^{+}b\mathrm{\parallel }}\in {\mathbb{C}}^n,$ where $A^{+}$ is the Moore–Penrose pseudoinverse. By Every linear map between finite-dimensional real or complex inner product spaces admits a singular value decomposition, write $b={\sum }_{j:{\sigma }_j>0}{\beta }_j{u}_j$ in left singular vectors; the promised support gap is a supplied rational or exactly represented $\gamma >0$ such that every $j$ with ${\beta }_j\mathrm{\ne }0$ has singular value ${\sigma }_j\ge \gamma$. The input includes a costed unitary $B$ mapping its designated all-zero input to $\mathrm{\mid }b\mathrm{/}\mathrm{\parallel }b\mathrm{\parallel }⟩$ in the codomain register, tensored with a fixed work state. Its inverse, controlled access, and resource coordinates are specified in the separate HHL access model. If $b$ is multiplied by $c\mathrm{\ne }0$, the normalized representative is multiplied by $c\mathrm{/}\mathrm{\mid }c\mathrm{\mid }$, so its pure-state ray is unchanged. The task returns a quantum state (and may return a success flag); it does not promise a classical list of coordinates.