Shallow Quantum Circuits for Geometrically Local Lindblad Dynamics

Abstract

Open-system dynamics govern decoherence, transport, and engineered dissipation in quantum many-body systems. We propose a quantum algorithm for simulating time-independent, finite-range Lindblad dynamics with bounded local strength on a constant-dimensional lattice of $n$ sites, allowing noncommuting jump operators. For time $T\ge1$ and diamond-norm error at most $\epsilon$, it has depth $O(T\operatorname{polylog}(nT/\epsilon))$ and gate count $O(nT\operatorname{polylog}(nT/\epsilon))$. These results extend the HHKL spatial decomposition to open systems. The main obstacle is that the reversible overlap corrections used for unitary dynamics have no direct analogue for dissipative channels. We overcome this by retaining the environment in a shared unitary dilation, keeping information lost from the system coherently available and making the spatial corrections reversible. We prove exponential occupation bounds for the environmental states generated by the decomposition, allowing compact representation and coherent simulation of local factors on the inputs that arise. The construction reduces each short-time evolution to a constant number of parallel layers on regions of logarithmic diameter, using nearest-neighbour gates and polylogarithmic auxiliary density. We also prove unconditional worst-case lower bounds using time-independent, purely dissipative processes that transport quantum information. For circuits with local quantum and classical communication, the lower bounds are $\Omega(nT)$ for the expected gate count and $\Omega(T)$ for depth on time scales up to the lattice diameter. They match the upper bounds'leading time and volume dependence up to polylogarithmic factors. At fixed time and sufficiently small fixed error, we further prove a depth lower bound of $\Omega(\log n/\log\log n)$, showing that global accuracy can require increasing depth as the lattice grows.