Query-Optimal and Gate-Efficient Lindbladian Simulation

We give a quantum algorithm for Lindbladian simulation given a block encoding of the Hamiltonian $H$ and a projected unitary encoding of the stacked jump operator $B={\sum }_{k=1}^m\mid k⟩\otimes L_k$, with normalization factors ${\alpha }_H$ and ${\alpha }_B$, respectively. For evolution time $t$, set $\tau =({\alpha }_H+{\alpha }_B^2)t$. The algorithm approximates the evolution channel to diamond-norm error $\epsilon$ using $O\text{\hspace{0.17em}⁣}(\tau +\frac{\log (1\mathrm{/}\epsilon )}{\log \text{\hspace{0.17em}⁣}(e+\log (1\mathrm{/}\epsilon )\mathrm{/}\tau )})$ oracle queries, matching the query lower bound for Hamiltonian simulation. The number of additional one- and two-qubit gates is linear in the query complexity up to polylogarithmic factors. The query- and gate-complexity bounds extend to Lipschitz-continuous time-dependent Lindbladians under coherent time-indexed oracle access. Our construction uses a one-query transducer that implements a product of rational approximations to short-time evolution when supplied with a catalyst. We bound the error from omitting the catalyst by exploiting orthogonality between different sequences of Kraus labels. The gate implementation combines a compressed Kraus-label representation, which stores only the positions and values of the nonzero labels, with the rotation factorization of Chen et al.