Transducer-based linear combination of unitaries: theory and applications

Abstract

Linear combination of unitaries (LCU) is a fundamental primitive in quantum algorithms, whose cost is typically governed by the most expensive unitary appearing in the combination. We develop a transducer-based LCU framework that reduces this worst-case dependence to a weighted average query complexity, when the constituent unitaries share access to a common set of primitive oracles. Consider $A=\sum_j c_j U_j$, where $c_j>0$ and each unitary $U_j$ can be implemented using $C_j$ primitive queries. Given an upper bound $a\geq \|A\|$, our algorithm implements a block-encoding of $A/α$ with rescaling factor $α=\mathcal{O}(a)$, using $\widetilde{\mathcal{O}} (C_{\max}+\overline{C} λ/{a} )$ primitive queries, where $λ=\sum_j c_j$, $C_{\max}=\max_j C_j$, and $\overline{C}= {\sum_j c_j C_j}/λ$. By comparison, the standard LCU construction requires $\widetilde{\mathcal{O}} (C_{\max} λ/{a} )$ primitive queries. The improvement can therefore be substantial when costly unitaries have small weights and $λ/a$ is large. As applications, we obtain improved block-encodings of sparse matrices, leading to quantum algorithms for sparse Hamiltonian simulation and quantum linear systems with near-optimal query complexity up to polylogarithmic factors in all relevant parameters. Our main technique is the transducer framework developed by Belovs, Jeffery, and Yolcu [Quantum, 8:1444 (2024)]. Here we develop a complementary operator-level theory tailored to block-encodings. We identify the resolvent norm $K$ as a key complexity measure for transducer implementation besides the existing catalyst complexity. We show that the transducer action can be converted into an $ε$-approximate block-encoding using only $\mathcal{O}\left(K \log(1/ε)\right)$ queries to the transducer, improving the precision dependence from polynomial to logarithmic.