Exploiting all ancilla outcomes in linear combinations of unitaries: low-rank recovery and quantum trapdoor functions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913121822572544 |
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| author | Daskin, Ammar |
| author_facet | Daskin, Ammar |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_02986 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Exploiting all ancilla outcomes in linear combinations of unitaries: low-rank recovery and quantum trapdoor functions Daskin, Ammar Quantum Physics 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. |
| title | Exploiting all ancilla outcomes in linear combinations of unitaries: low-rank recovery and quantum trapdoor functions |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2605.02986 |