Rapid quantum ground state preparation via dissipative dynamics (2503.15827v2)

Abstract: 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 non-commuting 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. Our theoretical approaches are applicable to systems with singular stationary states, and are thus expected to have applications beyond the specific systems considered in this study.

Rapid Quantum Ground State Preparation via Dissipative Dynamics

The paper, "Rapid quantum ground state preparation via dissipative dynamics," presents a comprehensive analysis of utilizing dissipative approaches to prepare low-energy ground states of quantum systems. The paper, conducted by Zhan, Ding, Huhn, Gray, Preskill, Chan, and Lin, establishes both theoretical and numerical foundations for the effectiveness of dissipative dynamics, specifically through the Lindblad master equation framework. The team focuses on non-commuting Hamiltonians, extending the boundaries of what has previously been understood predominantly for commuting scenarios.

Key Contributions

1. Dissipative Dynamics for Quasi-Free Systems: The authors explore quasi-free dissipative dynamics, which include certain one-dimensional (1D) spin systems with boundary dissipation. A notable finding is the establishment of a novel connection between mixing time in trace distance and the spectral properties of a non-Hermitian Hamiltonian. This leads to explicit, polynomially scaling bounds on mixing time concerning system size.
2. Tensor Network Methods for Non-Quasi-Free Systems: For more complex spin systems, a tensor network-based algorithm is developed to construct Lindblad jump operators and simulate system dynamics. The numerical simulations demonstrate the algorithm's capacity for rapid ground state preparation, marked by a logarithmically scaling mixing time with system size for specific 1D local Hamiltonians under bulk dissipation.
3. Generalization to Weakly Interacting Systems: The paper extends rapid mixing results to weakly interacting spin and fermionic systems, applicable across arbitrary dimensions. This generalization is particularly significant as it transitions recent findings for quantum Gibbs samplers at high temperatures to the zero-temperature regime, thereby directly focusing on ground state preparations.

Theoretical Insights and Implications

• The rigorous mathematical treatment underscores that the spectral properties of a non-Hermitian Hamiltonian can effectively determine the convergence rate of the system's dynamics in trace distance. This insight alone fortifies the theoretical underpinnings of dissipative quantum state preparation methods.
• The tensor network-based simulation approach offers a scalable computational technique, suggesting practicality for near-term quantum technologies. Given the exponential complexity of traditional quantum simulations, such scalable solutions are indispensable for leveraging quantum computing in practical applications.

Numerical Validation

Through simulations of the Transverse Field Ising Model (TFIM) and the cluster state Hamiltonian, various characteristics of dissipative dynamics are validated. The TFIM studies with boundary dissipation confirm the theoretically predicted polynomial scaling of mixing time, aligning with an $\mathcal{O}(N^3)$ complexity. Importantly, the role of coherent terms in Lindblad dynamics emerges as crucial, establishing that their absence may preclude convergence.

Conclusions and Future Directions

This work pushes the frontier of quantum state preparation through dissipative dynamics, highlighting its potential advantages over unitary or adiabatic methods, particularly in robustness to noise and potential for simpler initialization procedures. The implications for early fault-tolerant quantum devices are palpable, as both theoretical insights and demonstrated numerical efficiency align well with the constraints and capabilities of current quantum hardware.

Future research directions could explore the spectral gap properties of more complex Hamiltonian systems, refining our understanding of dissipative dynamics' role. Extensions to broader classes of quantum systems and further optimization of tensor network algorithms could also be promising areas to explore to enhance the applicability and efficiency of quantum simulations.