Rapid quantum ground state preparation via dissipative dynamics

Inspired by natural cooling processes, dissipation has become a promising approach for preparing low-energy states of quantum systems. However, the potential of dissipative protocols remains unclear beyond certain commuting Hamiltonians. This work provides significant analytical and numerical insights into the power of dissipation for preparing the ground state of noncommuting Hamiltonians. For quasi-free dissipative dynamics, including certain 1D spin systems with boundary dissipation, our results reveal a new connection between the mixing time in trace distance and the spectral properties of a non-Hermitian Hamiltonian, leading to an explicit and sharp bound on the mixing time that scales polynomially with system size. For more general spin systems, we develop a tensor network-based algorithm for constructing the Lindblad jump operator and for simulating the dynamics. Using this algorithm, we demonstrate numerically that dissipative ground state preparation protocols can achieve rapid mixing for certain 1D local Hamiltonians under bulk dissipation, with a mixing time that scales logarithmically with the system size. We then prove the rapid mixing result for certain weakly interacting spin and fermionic systems in arbitrary dimensions, extending recent results for high-temperature quantum Gibbs samplers to the zero-temperature regime. Together, these results show that dissipation can be a powerful tool for ground state preparation, with potential applications across condensed matter physics, quantum materials science, and beyond.

💡 Research Summary

This paper investigates the use of engineered dissipative dynamics—specifically Lindblad master equations—to prepare ground states of quantum many‑body Hamiltonians that are not limited to commuting or frustration‑free cases. The authors build on a previously proposed algorithm that designs jump operators (K_a) to “shovel” population from high‑energy eigenstates toward lower‑energy ones. By introducing a frequency‑domain filter (\hat f(\omega)) that only allows transitions with negative energy differences, they express each jump operator as a time‑integral over Heisenberg‑evolved local operators (A_a). This construction avoids explicit diagonalisation of the Hamiltonian and can be implemented with modest quantum resources (e.g., a single ancilla qubit) on near‑term devices.

A central theoretical contribution is the establishment of a precise link between the mixing time (the time needed for any initial state to become (\eta)‑close in trace distance to the target ground state) and the spectral properties of an associated non‑Hermitian effective Hamiltonian (\mathcal H_{\text{eff}} = H - i\sum_a K_a^\dagger K_a/2). For quasi‑free (quadratic) models—including certain 1‑D spin chains with boundary dissipation—the authors prove that the smallest real part of the spectrum of (\mathcal H_{\text{eff}}) provides an explicit upper bound on the mixing time: (\tau_{\text{mix}} \le C / \Delta_{\text{eff}}). When (\Delta_{\text{eff}} = \Omega(N^{-p})), the mixing time scales polynomially with system size, a result that is both sharp and analytically tractable.

To go beyond quasi‑free systems, the paper introduces a tensor‑network framework for constructing and simulating the Lindblad dynamics. Jump operators and the full Lindbladian are represented as matrix‑product operators (MPOs), and time evolution is performed via Trotter‑Suzuki splitting or linear‑combination‑of‑unitaries techniques. This approach reduces the computational cost to (\mathcal O(N\chi^3)), where (\chi) is the bond dimension, enabling the study of non‑integrable 1‑D Hamiltonians and even 2‑D weakly interacting fermionic lattices.

Numerical experiments demonstrate two striking regimes. With dissipation applied only at the boundaries, the mixing time grows polynomially (approximately (N^2)) as expected from the spectral analysis. When bulk (global) dissipation is employed, the mixing time collapses to a logarithmic scaling (\tau_{\text{mix}} \sim \log N). Both energy‑based and fidelity‑based metrics confirm this rapid convergence, and the results are corroborated by the analytical bounds derived for the quasi‑free case.

The most ambitious theoretical result extends recent high‑temperature quantum Gibbs‑sampler analyses to the zero‑temperature limit. By treating weakly interacting spin or fermion models in arbitrary dimensions, the authors prove that if the interaction strength (\lambda) is sufficiently small, the non‑Hermitian generator retains a spectral gap of order (1/\log N). Consequently, the mixing time remains logarithmic in system size, establishing rapid mixing for a broad class of non‑commuting Hamiltonians at zero temperature.

Resource considerations are carefully discussed. The total cost of a dissipative ground‑state preparation protocol is decomposed into (i) the cost of constructing the Lindbladian ((C_L)), (ii) the cost of simulating its dynamics per unit time ((C_S)), and (iii) the mixing time (\tau_{\text{mix}}). For typical physical models, both (C_L) and (C_S) scale polynomially with the number of qubits, so the overall cost is dominated by the scaling of (\tau_{\text{mix}}). The paper shows that for the models studied, (\tau_{\text{mix}}) can be polynomial or even logarithmic, implying that the end‑to‑end cost remains polynomial in system size—far more favorable than the exponential mixing times that can arise for generic Hamiltonians.

Finally, the authors compare dissipative preparation with conventional unitary approaches such as adiabatic evolution and quantum phase estimation. In the examples considered, the dissipative protocol requires significantly fewer quantum gates and exhibits robustness to imperfect initial states, making it attractive for near‑term quantum hardware. The work concludes by highlighting open challenges: physical implementation of the quasi‑local jump operators (e.g., via cavity QED or superconducting circuits) and extension to strongly correlated, higher‑dimensional systems where the weak‑interaction assumption no longer holds. Overall, the paper provides a comprehensive blend of rigorous theory, algorithmic development, and numerical validation, establishing engineered dissipation as a powerful and scalable tool for quantum ground‑state preparation.