High-order splitting of non-unitary operators on quantum computers

Dissipation and irreversibility are central to most physical processes, yet they lead to non-unitary dynamics that are challenging to realise on quantum processors. High-order operator splitting is an attractive approach for simulating unitary dynamics, yet conventional product formulas introduce negative time steps at high orders that are numerically unstable for dissipative dynamics. We show how complex-coefficient product formulas can decompose dissipative dynamics into a sequence of simple Hamiltonian evolutions in real and imaginary time with high-order accuracy. The unitary substages use positive real coefficients, while the dissipative substages use complex coefficients with positive real parts, where the real parts preserve the contractive evolution and the imaginary parts are additional unitary evolutions. We demonstrate the approach by simulating the classical problem of lossy mechanical wave propagation on a trapped-ion quantum processor. A step of order 4 achieves greater accuracy than the steps with low orders 1 and 2, despite the increased circuit depth on noisy hardware. The results suggest that high-order operator splitting is an accurate and practical approach for simulating dissipative dynamics on near-term quantum processors.