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

TL;DR

This paper leverages all ancilla measurement outcomes in a modified linear combination of unitaries (LCU) scheme to enable low-rank matrix recovery and propose a quantum trapdoor function, transforming discarded outcomes into valuable resources.

Contribution

It introduces an alternative LCU circuit that utilizes all ancilla outcomes, enabling low-rank recovery and quantum encryption applications.

Findings

Collected ancilla outcomes form a matrix with rank at most K.

Classical low-rank matrix completion can reconstruct the full output from partial data.

The scheme proposes a quantum trapdoor function for encryption based on ancilla outcomes.

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 $\Phi = C X$, where $C$ encodes the coefficients and $X$ contains the action of each unitary on the input; this immediately shows $\operatorname{rank}(\Phi)\le K$. 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.