Quantum Simulation of Dynamical Transition Rates in Open Quantum Systems

TL;DR

The paper addresses the challenge of estimating transition rates in open quantum systems, which is computationally intensive on classical resources due to Lindblad-type dynamics. It proposes a derivative-to-expectation framework that recasts the time derivative of the equilibrium correlation function, $\\dot C(t)$, into a finite set of single-time expectation values obtained from parameter-tunable CPTP processes, enabling shallow-depth quantum circuits. The authors validate the approach on a spin-\\tfrac{1}{2}$ decoherence model implemented on IBMQ and apply it to the Caldeira–Leggett quantum Brownian motion model, demonstrating consistent results with theory and practical implementability on current hardware. The findings suggest substantial quantum advantages for studying open-system kinetics in quantum chemistry on intermediate-scale quantum computers, with clear pathways toward improved hardware, modular Lindbladian blocks, and non-Markovian extensions.

Abstract

Estimating transition rates in open quantum systems is hampered by computing-resource demands that grow rapidly with system size. We present a quantum-simulation framework that enables efficient estimation by recasting the transition rate, given as the time derivative of an equilibrium correlation function, into a set of independently measurable contributions. Each contribution term is evaluated as the expectation value of a parameter-tuned quantum process, thereby circumventing explicit Lindbladian numerics. We validate our method on a spin-1/2 decoherence model using an IBM quantum processor. Further, we apply the method to the Caldeira-Leggett model of quantum Brownian motion as a realistic and practically relevant setting and reaffirm the theoretical soundness and practical implementability. These results provide evidence that quantum simulation can deliver substantial computational advantages in studying open-system kinetics for quantum chemistry on an intermediate-scale quantum computer.

Figures (12)

• Figure 1: Metastable-transition dynamics: $C(t)$ (green) grows approximately linearly after an intrabasin transient, while the transition rate $\dot C(t)$ (violet) exhibits a plateau $\dot C(t)\!\approx\!k_{AB}$ for $\tau_{\rm intra}\!\ll\!t\!\ll\!\tau_{\rm eq}$ before decaying to zero at equilibrium ($\tau_{\rm intra}$: intrabasin relaxation time and $\tau_{\rm eq}$: global equilibrium time).
• Figure 2: Parameter-tunable quantum process for estimating $C(t)$ and $\dot{C}(t)$. $\hat{c}_{+}$ is a pure state while $\hat{\theta}_B$ and $\hat{\rho}_{\rm eq}$ are given as mixed states. The control qubit is phase-shifted by $\hat{R}(\chi)$ and read out after $\hat{H}_d$. $\hat{\theta}_{A,\epsilon}$ is the small-angle surrogate in Eq. \ref{['eq:thetaA_approx']}. $e^{{\mathcal{L}} t}$ is the CW-Lindblad propagator, and $\hat{N}$ and $\hat{M}$ are Hamiltonian/jump/anticommutator channels. The states $\hat{\Phi}$ in each step are explained in the Appendix.
• Figure 3: Simplified circuit of Fig. \ref{['fig:QCirc']} for evaluating $\mathcal{E}_C$.
• Figure 4: Spin-$\frac{1}{2}$ testbed. (a) $C(t)$ and (b) $\dot C(t)$ versus $t$ for $\mu=0.1$, $\gamma_0=1$, $\hbar=1$: analytics (blue, solid) vs CW-Lindblad QuTiP emulation at $N= 3$, 10, and 25 time steps.
• Figure 5: Six-qubit IBMQ circuit for $\mathcal{E}_{C}(t)$ in $C(t)$. Ancillary qubits $A_1$–$A_3$ implement the CW-Lindblad time-evolution gate. All qubits start in $\ket{0}$ and we choose $\rho_{\rm eq} = \ket{0}\bra{0}$ based on the combination in Table \ref{['tab:CTermTable']}. The gates inside the dashed red box correspond to $\hat{\theta}_A=\hat{\sigma}_z$ and are omitted for $\hat{\theta}_A=\hat{\mathds{1}}$ due to $\ket{0}\bra{0} = (\hat{\mathds{1}} + \hat{\sigma}_z)/2$ in Table \ref{['tab:CTermTable']}. Single-qubit rotations in the $\hat{\rho}_{\mathrm{eq}}$ channel are parametrised as $\hat{R}_z(\theta_z)$ in blue boxes and $\hat{R}_y(\theta_y)$ in purple boxes.