Rapid quantum ground state preparation via dissipative dynamics

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 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.

Figures (21)

• Figure 1: (a) Schematic representation of the ground state preparation algorithm, in which high-energy components are systematically dissipated into lower-energy states until convergence to the ground state is achieved. (b) and (c) depict the associated filter function in the frequency and time domains, respectively.
• Figure 2: Tensor network representation of the jump operator $K_a$ in the MPO form associated from a local coupling operator $A_a$. First, the MPO representation of the operator $e^{iH s_j} A_a e^{-iH s_j}$ is constructed for a set of time steps $\{s_j\}$. Next, a weighted summation from discretizing the integral $\sum_j p_j f(s_j) e^{iH s_j} A_a e^{-iH s_j}$ is performed to combine these operators. Finally, the resulting MPO is compressed to reduce the bond dimension, yielding an efficient representation of $K_a$.
• Figure 3: Illustration of vectorizing an MPO into an MPS using the Choi isomorphism and then converting it back. Each pair of local site indices in the MPO is reshaped into a single combined index, allowing standard MPS techniques, such as TEBD, to be applied. The reverse process restores the original MPO from the MPS by reshaping the combined indices back into pairs.
• Figure 4: Illustration of the tensor network computation associated with a single jump operator $K_a$ in $\mathcal{L}(\rho)$.
• Figure 5: Numerical results of 1D TFIM \ref{['eqn:H_TFIM']} with $J=1,g=1.5$, using $\{A_a\}=\{X_1, Y_1, X_N, Y_N\}$ as coupling operators. (a) Convergence of energy starting from the maximally mixed state. The dashed lines are exponential fits of the asymptotic behaviour of energy decay, meaning $(E(t)-{E_0})/N=\Theta(\exp(-\kappa_N t))$ for constants $\kappa_N$ when $t$ is sufficiently large. (b) The scaling of the inverse of the Liouvillian gap $\Delta^{-1}_{\mathcal{L}}$ with respect to the system size $N$. Red points are the energy convergence rate $\kappa_N$ calculated from fitting the data in (a).

Theorems & Definitions (22)

• Theorem 1
• Theorem 2: Informal
• Theorem 3: Informal
• Theorem 4: Fuchs--van de Graaf FuchsVanDeGraaf2002NielsenChuang2000
• Proposition 5
• proof
• Definition 6: Gevrey function
• Theorem 8: Rigorous version of Theorem \ref{['thm:rapid_mixing_2D_TFIM']}
• Proposition 9
• proof : Proof of Theorem \ref{['thm:rapid_mixing_2D_TFIM_rigo']}