Hamiltonian dynamics from pure dissipation

TL;DR

This paper demonstrates that pure dissipation can simulate Hamiltonian dynamics in open quantum systems, revealing fundamental limits and implications for quantum simulation and complexity.

Contribution

It shows that dissipative Lindbladians without Hamiltonian parts can approximate Hamiltonian evolution, establishing optimal scaling and implications for quantum complexity.

Findings

Dissipative generators can approximate Hamiltonian dynamics within epsilon error.

The scaling of $ ext{O}(t^2/ ext{epsilon})$ is necessary and optimal for time-independent dynamics.

Purely dissipative dynamics are BQP-complete even before reaching equilibrium.

Abstract

The fundamental difference between closed and open quantum dynamics lies in their environmental interaction: closed systems are perfectly isolated and evolve reversibly under unitary Hamiltonian dynamics, whereas open systems continuously couple to an external bath, resulting in irreversible dissipation and information loss. In this work, we show internal Hamiltonian dynamics can be "faked`` via external pure dissipation, i.e., Lindbladians without a coherent Hamiltonian part. More concretely, we show that, in a GKSL representation with zero explicit Hamiltonian term but nontraceless jump operators, bounded-norm dissipative generators can approximate Hamiltonian dynamics within $\epsilon$ error in diamond norm using $\mathcal{O}(t^2/\epsilon)$ evolution time. We further prove that for time-independent dynamics this $\mathcal{O}(t^2/\epsilon)$ scaling is in the worst case, necessary and optimal from a geometric perspective, which captures the fundamental decoherence cost for catching up with the speed of Hamiltonian dynamics. Our construction leads to various implications, including the BQP-completeness of purely dissipative dynamics even before reaching approximate equilibrium, a Zeno-adjacent state-independent freezing effect, the no super-quadratic fast-forwarding theorem of a class of purely dissipative dynamics, and reducing Lindbladian simulation cost via gauge changing.