Exploiting all ancilla outcomes in linear combinations of unitaries: low-rank recovery and quantum trapdoor functions

Abstract: The linear combination of unitaries (LCU) is a fundamental quantum algorithm primitive that embeds non-unitary operators via post-selection on an ancilla register. In standard LCU, only the $|0\dots0\rangle$ ancilla outcome is retained; the remaining "junk" outcomes are discarded. We study these discarded parts by introducing an alternative LCU circuit which simplifies the coefficient preparation unitary with Hadamard gates and a single rotation qubit. Every computational basis measurement of the ancilla projects the system onto a different linear combination of the target unitaries. Collecting these outcome states and reshaping them into a $2K\times N$ matrix reveals a factorization $Φ= C X$, where $C$ encodes the coefficients and $X$ contains the action of each unitary on the input; this immediately shows $\operatorname{rank}(Φ)\le K$. This structure enables two complementary applications: (i) classical low-rank matrix completion can reconstruct the full output (including the target) from a fraction of its entries, turning every shot into useful information; (ii) treating $C$ as a secret key hides the input state, leading to a candidate quantum trapdoor function and symmetric encryption. The scheme thus turns the "junk" ancilla outcomes into a structured resource, possibly opening paths for further applications.