We prove that the Gibbs state of any spin chain can be exactly decomposed into a mixture of product states over contiguous blocks of qubits with bounded length, at any constant temperature. As a consequence, any such Gibbs state can be prepared by local unitary circuits of constant depth, with efficient classical preprocessing. We place strong limits on the spatial structure of thermal entanglement: both the entanglement depth and entanglement width (measures of multiparticle entanglement) are bounded by constants, while the localizable entanglement (the bipartite entanglement that can be generated with the aid of local measurements and classical communication) vanishes exactly beyond a constant threshold distance. Our result also has implications for quantum thermalization: for any translation-invariant spin-chain Hamiltonian with nondegenerate spectral gaps, typical low-complexity pure states drawn from a maximally entropic ensemble at any constant effective temperature become locally indistinguishable from the Gibbs state upon unitary evolution. Additionally, we obtain an analogous decomposition for fermionic systems: the Gibbs state of any 1D local fermionic Hamiltonian is exactly a mixture of constant-depth local fermionic circuits acting on Gaussian states.
Entanglement is the hallmark of quantum physics, yet its characterization in interacting many-body systems at thermal equilibrium remains one of the most important challenges in quantum statistical physics. We prove that the Gibbs state of any quantum spin chain can be exactly decomposed into a mixture of matrix product states with a bond dimension that is independent of the system size, at any finite temperature. As a consequence, the Schmidt number, arguably the most stringent measure of bipartite entanglement, is strictly finite for thermal states, even in the thermodynamic limit. Our decomposition is explicit and is accompanied by an efficient classical algorithm to sample the resulting matrix product states.
We prove that a quasi-local quantum Gibbs sampler rapidly mixes to the Gibbs state of any 1D finite-range Hamiltonian at all finite temperatures in
O(log(n/ε)) time. Each update traces out a contiguous block of constant length and reconstructs it using the Petz recovery map, following early proposals for quantum heat-bath Gibbs samplers by Kastoryano and Brandão~\cite{kastoryano2016quantum}. Our proof directly bounds the distance between the updated state and the target Gibbs state using quantum Wasserstein distance of order
1. We further provide
O(polylog(n/ε)) depth of quantum circuits to prepare these Gibbs states. We hope these techniques will facilitate the analysis and design of other Gibbs samplers.
We establish sufficient conditions for preparing quantum thermal states of noncommuting local Hamiltonians with polylogarithmic circuit depth in arbitrary fixed spatial dimension. Our conditions combine locality and stability bounds on the effective interactions of reduced density matrices of a Gibbs state with a quantitative high-temperature condition and access to coarse classical reference Hamiltonians. When these references can be generated locally, the classical preprocessing cost is
N1+o(1) for
N sites at inverse-polynomial global accuracy. We construct the preparation circuit from spatially localized Petz recovery maps. Quantum corrections derived from the microscopic Hamiltonian enable accurate recovery while the classical references remain coarse. After preprocessing, the resulting circuit prepares the canonical purification using
Npolylog(N/ε) gates and qubits, where
ε is the preparation error. These results link two fundamental questions: how correlations are organized in thermal equilibrium, and how efficiently the corresponding states can be realized through operations allowed by quantum mechanics. By translating static equilibrium structure into explicit preparation circuits, they give equilibrium locality a constructive computational interpretation and a physically grounded role in quantum algorithm design.
Efficient simulation of a quantum system generally relies on structural properties of the quantum state. Motivated by the recent results by Bakshi et al. on the sudden death of entanglement in high-temperature Gibbs states of quantum spin systems, we study the high-temperature Gibbs states of bounded-degree local fermionic Hamiltonians, which include the special case of geometrically local fermionic systems. We prove that, at a sufficiently high temperature that is independent of the system size, the Gibbs state is a probabilistic mixture of fermionic Gaussian states. This forms the basis of an efficient classical algorithm to prepare the Gibbs state by sampling from a distribution of fermionic Gaussian states.
We show that thermal states of local Hamiltonians are separable above a constant temperature. Specifically, for a local Hamiltonian
Hon a graph with degree
g, its Gibbs state at inverse temperature
β, denoted by
ρ=e−βH/tr(e−βH), is a classical distribution over product states for all
β<1/(cg), where
cis a constant. This sudden death of thermal entanglement upends conventional wisdom about the presence of short-range quantum correlations in Gibbs states. Moreover, we show that we can efficiently sample from the distribution over product states. In particular, for any
β<1/(cg3), we can prepare a state
ε-close to
ρin trace distance with a depth-one quantum circuit and
poly(n)log(1/ε)classical overhead.11In independent and concurrent work, Rouzé, França, and Alhambra [37] obtain an efficient quantum algorithm for preparing high-temperature Gibbs states via a dissipative evolution.
A central question in quantum information theory is the circuit complexity of states arising from standard many-body models. We study this question for quantum
p-spin glasses, random Hamiltonians whose interactions act on
p-tuples of qubits through Pauli strings. Anschuetz, Gamarnik, and Kiani (arXiv:2404.07231) showed that the optimum energy is separated from the best energy achievable by product states. This leaves open whether shallow circuits can close the gap, since even depth-one circuits can generate entanglement.
We show that the entanglement needed to close the product-state gap cannot be generated at shallow depth. When the average interaction degree grows with
n, we prove that, for all sufficiently large fixed
p, any circuit preparing an
n-qubit state whose normalized energy is within a fixed positive constant of the optimum must have depth
Ωp(logn). In the bounded-average-degree regime, we prove a fixed-depth obstruction: for every fixed
D, a sufficiently large degree prefactor rules out depth-
D preparation of near-ground states. Both results hold uniformly over circuits with an arbitrary number of ancilla qubits.
Our results give an obstruction in the spirit of the No Low-Energy Trivial States problem of Freedman and Hastings (arXiv:1301.1363), but for random quantum spin glasses rather than code-based Hamiltonians such as those of Anshu, Breuckmann, and Nirkhe (arXiv:2206.13228), whose ground states admit polynomial-size preparation circuits. This setting opens a probabilistic route to NLTS-like questions: we recast state-preparation lower bounds for random quantum Hamiltonians as uniform control of Gaussian processes indexed by shallow circuits.