Towards Efficient Quantum Thermal State Preparation via Local Driving: Lindbladian Simulation with Provable Guarantees

TL;DR

This work tackles the challenge of preparing thermal states $\rho_β \propto e^{-βH}$ on quantum devices by proposing a practical local-driving Lindbladian scheme that uses only analog Hamiltonian evolution, time-dependent local couplings to ancillas, and ancilla resets. By encoding an approximate Gibbs sampler through a local drive and a Gaussian filter, the authors derive rigorous bounds on the fixed-point error in terms of protocol parameters and the mixing time $τ_{\rm mix}$, and they provide a resource estimate showing polynomial scaling with system size for local Hamiltonians. The approach reproduces the dissipative structure of exact Gibbs samplers under KMS detailed balance, while replacing the exact coherent term with a physically implementable Lamb shift, and yields a tractable pathway toward near-term thermal-state preparation. Numerical case studies on small spin models illustrate the method’s promise, with energy convergence to the thermal value and a decaying trace distance to $\rho_β$, and reveal practical considerations such as the potential benefits of omitting the rewinding step. Overall, the paper lays a rigorous foundation for efficiently approximating thermal states on quantum simulators and highlights key open questions about mixing-time behavior and optimal parameter regimes.

Abstract

Preparing the thermal density matrix $ρ_β \propto e^{-βH}$ corresponding to a given Hamiltonian $H$ is a task of central interest across quantum many-body physics, and is particularly salient when attempting to study it with quantum computers. Although solved in principle by recent constructions of efficiently simulable Lindblad master equations -- that provably have $ρ_β$ as a steady state [C.-F.~Chen \emph{et al.}, Nature \textbf{646}, pp.~561--566 (2025)] -- the implementation of these ``exact Gibbs samplers'' requires large-scale quantum computational resources and is hence challenging \emph{in practice} on current or even near-term quantum devices. Here, we propose a scheme for approximately simulating an exact Gibbs sampler that only requires the repeated implementation of three readily available ingredients: (a) analog simulation of $H$; (b) strictly local but time-dependent couplings to ancilla qubits; and (c) reset of the ancillas. We give rigorous guarantees on the difference between the fixed point reached by our protocol and the exact thermal state, which only depend on parameters of the protocol and its \emph{mixing time}. The procedure is efficiently implementable on near-term devices if $H$ is local, and the mixing time scales mildly with both system size and protocol parameters. While guaranteeing the latter for Hamiltonians of interest remains an important problem for future work, here we lay the groundwork for developing fully efficient thermal state preparation protocols on quantum simulators.

Figures (4)

• Figure 1: Schematic of one time step of the local driving sampler. In each step, the "bath" qubits are initialized in the all-zero state, followed by an entangling step consisting of time evolution with the system Hamiltonian $H$ together with a dynamically driven system-bath coupling $V(t) = \sum_i V_i(t)$ [cf. \ref{['eq:HSB_schroedinger']}]. Finally, the bath qubits are measured and the outcomes discarded. The final unitary "rewinding" $e^{iHT}$ can be dropped with only a modest increase to the error bound, and can be ignored with no penalty if a quasiparticle picture applies.
• Figure 2: Summary of different approximations made during the derivation of the local driving sampler protocol. Overall, the setup in \ref{['fig:setup']}, with the parameter choices indicated in the rightmost column above, implements a channel $\mathcal{K}$ that approximates an exact Gibbs sampler and hence has an approximately thermal fixed point. As discussed in \ref{['sec:optimal_accuracy']}, the parameter choices indicated in the last column may be non-trivial.
• Figure 3: Comparison of the time evolution of thermal state preparation for the mixed-field Ising model \ref{['eq:MFI']} under the exact Gibbs sampler $\mathcal{L}$ [cf. \ref{['eq:Gibbsexact']}] (black), the Lindbladian $\mathcal{L}_T$ [cf. \ref{['eq:LDS']}] (red), the local driving protocol $K[\rho]$ [cf. \ref{['eq:channelev']}] (blue), and the protocol $K^\prime[\rho]$ without rewinding (dashed blue) starting from the maximally mixed state $\rho_0 = 2^{-n_{\rm S}}\mathds{1}$. Parameters: $\beta = 1.0$, $\sigma = 0.5$, $J=0.5$ , $n_S = 8$, $T=6\sigma$. (a) Time evolution of the error in the average energy density $\Delta e(t)=\frac{\langle H-H_\beta\rangle}{\langle H_\beta \rangle}$. The energy density approaches the exact result within less than two percent. (b) Evolution of the trace distance to the Gibbs state $\rho_\beta$. The trace distance decays, indicating convergence toward a state close to the Gibbs state.
• Figure 4: The mixing time of the maximally mixed state $t^*$ as a function of the parameter $\sigma$ in the filter function, $\beta=1.0$. (a) For the mixed-field Ising model defined in Eq. \ref{['eq:MFI']}, (b) the transverse field Ising model defined in Eq. \ref{['eq:TFI']}. For large $\sigma$, $t^*$ scales exponentially with $\sigma$.