Fast mixing of all-to-all quantum systems at high temperatures

Abstract: It is shown that arbitrary quantum $k$-local Hamiltonians with bounded strength interactions admit a quantum Gibbs sampler [CKG23] with a system-size independent spectral gap, at sufficiently high temperatures. This generalizes the existing quantum fast-mixing results beyond the geometrically-local setting. As a consequence, such systems admit fully-polynomial time quantum approximation algorithms for partition functions and global expectation values.

• The paper establishes a system-size-independent spectral gap for quantum Gibbs samplers applied to all-to-all k-local Hamiltonians at high temperatures through non-commutative cluster expansion techniques.
• It employs a non-commutative cluster expansion and a Dobrushin-type criterion to control operator spreading and derive sharp mixing time bounds.
• The results imply fully polynomial-time quantum algorithms for thermal state preparation and expectation estimation in non-local quantum systems.

Fast Mixing Dynamics in All-to-All Quantum Systems at High Temperatures

Introduction and Motivation

The mixing time and thermalization dynamics of quantum systems are fundamental in both quantum many-body physics and quantum algorithms. Locality, most naturally present in lattice models, traditionally constrains these dynamics via Lieb-Robinson bounds, limiting operator spreading and informing efficient simulation and algorithmic strategies. However, a range of models with all-to-all interactions—such as quantum LDPC codes, certain mean-field models, and the Sachdev-Ye-Kitaev (SYK) model—lack such geometric intuition, complicating the analysis of their mixing times and the performance of quantum Markov processes for Gibbs sampling.

This work establishes a system-size-independent spectral gap for a class of quantum Gibbs samplers in all-to-all, $k$-local Hamiltonians with bounded coupling strengths, at sufficiently high temperatures. This generalizes quantum fast-mixing results in the high-temperature regime previously limited to spatially local systems.

All-to-All $k$-Local Hamiltonians and High Temperature Gibbs Sampling

The considered systems are generic $k$-local Hamiltonians

$\vH = \sum_{e \in \Gamma} \vh_e \otimes \vI_{[n] \setminus e}$

over $n$ qudits, constrained such that each term acts non-trivially on at most $k$ qudits, and the summed interaction strengths incident to a pair of qudits is at most $\sJ$. The degree $\sd$ bounds the number of distinct qudits any qudit interacts with via two-body terms.

At high temperatures (small inverse temperature $\beta$), the system approaches the trivial maximally mixed phase, and both classical and quantum cluster expansions converge. The main result is that for these Hamiltonians, quantum Gibbs samplers generated from the thermal Lindbladian construction of Chen, Kastoryano, Gharibian 2023 possess a spectral gap that is independent of system size, provided $k$0.

Technical Approach and Main Results

Cluster Expansion and Complex-Time Locality: The work leverages a non-commutative cluster expansion—generalizing classical Koteck\'{y}-Preiss methods to quantum Hamiltonians—to capture the quasi-locality of operator spreading under complex-time evolution. For $k$1 sufficiently small, evolved operators can be decomposed into sums over clusters, and the terms' norms decay rapidly in cluster size, with bounds depending only on $k$2, $k$3, $k$4, and $k$5.

Non-Commutative Dobrushin Condition: The spectral gap is established by adapting a Dobrushin-type criterion for quantum Markov generators, previously used almost exclusively in commuting or lattice models. The argument replaces the physical Lindbladian $k$6 with a “pseudo-Lindbladian” generator $k$7, for which a precise mean-field Dobrushin analysis is feasible due to improved locality properties in the high-temperature regime. The required bounds on the corresponding Dobrushin coefficients are shown to follow from cluster expansion convergence, with both $k$8 (maximum deviation of local operator evolution) and $k$9 (maximum pairwise evolved commutator norm sum) controlled by rapidly decaying cluster sums.

Dirichlet Form Comparison: To translate the spectral gap for $k$0 into one for the physical $k$1, the argument performs a Dirichlet form comparison. Through careful use of reverse triangle inequalities and Duhamel expansions, the Dirichlet forms associated with the filtered (frequency-resolved) Lindbladian are compared to those of $k$2, with an explicit quantitative loss that remains polynomial in $k$3.

Result: For any all-to-all $k$4-local Hamiltonian of bounded degree and pairwise interaction strength, the constructed quantum Gibbs sampler (CKG Lindbladian) exhibits a system-size-independent gap at sufficiently high temperatures. Consequently, polynomial-time quantum algorithms exist for approximating both partition functions and global expectation values in this regime.

Implications and Theoretical Significance

Algorithmic Consequences: The established gap yields fully polynomial-time quantum algorithms for thermal state preparation and observable estimation, extending high-temperature efficiency guarantees to all-to-all interacting models. This marks a qualitative improvement in algorithmic tractability for non-local models, as previous results for quantum mixing at high temperature strictly relied on geometric locality.

Classical vs Quantum Complexity: The convergence conditions for the cluster expansion mirror those for classical spin systems and suggest a similar boundary of computational hardness at the phase transition. The obtained fast-mixing threshold for quantum models aligns with the computational threshold for approximate counting in classical random constraint satisfaction problems, indicating optimality under standard complexity assumptions.

Towards Disordered and Mean-Field Models: While the all-to-all Dobrushin argument closes the prior gap between lattice and non-lattice systems for fast mixing, the methods remain too weak to address the mixing times of highly disordered models such as quantum spin glasses or the SYK Hamiltonian at moderate $k$5. The classical theory saw a delay of decades between Dobrushin-based mixing for random graphs and the subsequent advances via spectral independence; a similar phase is anticipated for the quantum setting.

Theoretical Insights: The pseudo-Lindbladian path circumvents some shortcomings of direct Lindbladian locality arguments in non-geometric systems, highlighting new directions in the non-commutative probability and operator growth theory. The use of a KMS oscillator norm, instead of standard sup-operator or trace norms, is pivotal and may yield further advances in non-local quantum Markov process theory.

Conclusion

This work generalizes the theory of quantum mixing at high temperatures to dense, non-geometric Hamiltonians, using cluster expansions and Dobrushin-type arguments adapted to the non-commutative setting. The explicit, system-size-independent spectral gap established for quantum Gibbs samplers ensures efficient quantum approximation of partition functions and expectation values in the high-temperature phase for all-to-all interacting quantum systems (2606.26090). The techniques and locality analysis developed herein lay new groundwork for tackling the mixing properties of more complex, disordered, or frustrated quantum models in future research.