Query-Optimal and Gate-Efficient Lindbladian Simulation
TL;DR
A one-query transducer that implements a product of rational approximations to short-time evolution when supplied with a catalyst, and bound the error from omitting the catalyst by exploiting orthogonality between different sequences of Kraus labels.
Abstract
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 $B=\sum_{k=1}^m \lvert k\rangle\otimes L_k$, 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 $\varepsilon$ using $O\!\left(\tau+\frac{\log(1/\varepsilon)}{\log\!\left(e+\log(1/\varepsilon)/\tau\right)}\right)$ 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.