Predicting observables in equilibrium states is a central yet notoriously hard question in quantum many-body systems. In the physically relevant thermodynamic limit, certain mathematical formulations of this task have even been shown to result in undecidable problems. Using a finite-size scaling of algorithms devised for finite systems often fails due to the lack of certified convergence bounds for this limit. In this work, we design certified algorithms for computing expectation values of observables in the equilibrium states of local quantum Hamiltonians, both at zero and positive temperature. Importantly, our algorithms output rigorous lower and upper bounds on these values. This allows us to show that expectation values of local observables can be approximated in finite time, contrasting related undecidability results. When the Hamiltonian is commuting on a 2-dimensional lattice, we prove fast convergence of the hierarchy at high temperature and as a result for a desired precision ε, local observables can be approximated by a convex optimization program of quasi-polynomial size in 1/ε.
Convex relaxation hierarchies provide lower bounds to the ground state energy of quantum many-body systems that can be computed in polynomial time on a classical computer, at any fixed hierarchy level. However, scaling these methods to large systems and accurate approximations remains challenging due to the computational cost of traditional solvers and the scarcity of efficiency guarantees. In this work, we develop local relaxation hierarchies and efficient, highly parallelisable message passing algorithms for estimating the relaxed ground state energies. We show that the first level of the hierarchy---based on local consistency of pairwise reduced density matrices---is exact for commuting Hamiltonians on trees. We further establish that another hierarchy, based on consistent intervals, converges exponentially fast in the interval size to the ground state energy for weak perturbations of separable Hamiltonians on a chain, thereby providing an efficient classical algorithm for these systems. Then, we introduce two variants of message passing algorithms that run in
O(n/ε2) and
O(n/ε) time for any fixed level of the local hierarchy on bounded-degree graphs, where
ε is the precision for the relaxed ground state energy per site. This assumes that the optimal messages have
O(1) norm---a condition we observe in practical settings in our experiments. These algorithms are based on the subgradient method and the Nesterov-type accelerated gradient descent method applied to an entropy-smoothed objective. Finally, we benchmark the message passing algorithms across different quantum Hamiltonians, lattice geometries, and relaxation levels, validating the theoretical predictions and their potential to surpass standard convex optimisation solvers for this problem. We release the resulting library at github.com/rick1924/gse-message-passing.
In this work, we study the problems of certifying and learning quantum
k-local Hamiltonians, for a constant
k. Our main contributions are as follows:
- Certification of Hamiltonians. We show that certifying a local Hamiltonian in normalized Frobenius norm via access to its time-evolution operator can be achieved with only
O(1/ε) evolution time. This is optimal, as it matches the Heisenberg-scaling lower bound of
Ω(1/ε). To our knowledge, this is the first optimal algorithm for testing a Hamiltonian property. A key ingredient in our analysis is the Bonami Hypercontractivity Lemma from Fourier analysis.
- Learning Gibbs states. We design an algorithm for learning Gibbs states of local Hamiltonians in trace norm that is sample-efficient in all relevant parameters. In contrast, previous approaches learned the underlying Hamiltonian (which implies learning the Gibbs state), and thus inevitably suffered from exponential sample complexity scaling in the inverse temperature.
- Certification of Gibbs states. We give an algorithm for certifying Gibbs states of local Hamiltonians in trace norm that is both sample and time-efficient in all relevant parameters, thereby solving a question posed by Anshu (Harvard Data Science Review, 2022).
A fundamental challenge in quantum physics is determining the ground-state properties of many-body systems. Whereas standard variational approaches posit a wave-function ansatz and minimize over the possible states expressible by that ansatz, the problem can alternatively be formulated as a noncommutative polynomial optimization problem and treated through a hierarchy of semidefinite programming relaxations. In contrast to variational calculations, these relaxations provide lower bounds on ground-state energies and both lower and upper bounds on observable expectation values. However, this approach typically suffers from severe scalability issues, limiting its applicability to small-to-medium-scale systems. In this article, we demonstrate that systematically leveraging the inherent structures of the system can substantially mitigate these scalability challenges and thus permits computing meaningful bounds for quantum spin systems on square lattices of size up to
16×16.
We establish general conditions for the rapid convergence of propagation algorithms for computing expectation values of local observables in quantum many-body systems, covering both spin and fermionic systems. Our results apply to evolutions that are sufficiently contractive, providing a general mechanism by which contractivity controls the growth generated by local interactions and enables efficient classical simulation. As a first application, we consider noisy time evolution generated by local Hamiltonians on arbitrary interaction graphs. We show that the dynamics can be efficiently simulated when the noise strength
λ is sufficiently large compared to the degree of the interaction graph. More generally, for systems with an interaction strength
u, our bounds determine a time horizon
tmax as a function of the interaction strength
u, the noise strength
λ, and the interaction structure below which the algorithm is efficient. In the noiseless limit,
λ=0, our analysis extends the efficiently simulable time scale from
log(1/u) proved in Facelli, Fawzi, and Fawzi (2026) to
tmax∼1/u matching the recent improvement by Zhao, Marvian and Tong (2026). As a second application, we apply the same framework to the computation of local observables in Gibbs states of local Hamiltonians. We consider a pseudo-Lindbladian whose steady state is the Gibbs state and show that, at sufficiently high temperature, its propagation algorithm converges rapidly. In particular, for weakly-interacting systems with interaction strength
u, our result applies up to inverse temperatures
β∼log(1/u).
We consider the problem of learning the Hamiltonian of a quantum system from estimates of Gibbs-state expectation values. Various methods for achieving this task were proposed recently, both from a practical and theoretical point of view. On the one hand, some practical algorithms have been implemented and used to analyze experimental data but these algorithms often lack correctness guarantees or fail to scale to large systems. On the other hand, theoretical algorithms with provable asymptotic efficiency guarantees have been proposed, but they seem challenging to implement. Recently, a semidefinite family of Hamiltonian learning algorithms was proposed which was numerically demonstrated to scale well into the 100-qubit regime, but provided no provable accuracy guarantees. We build on this work in two ways, by extending it to provide certified a posteriori lower and upper bounds on the parameters to be learned, and by proving a priori convergence in the special case where the Hamiltonian is commuting.