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arXiv:2403.16850 · Annual Symposium on Foundations of Computer Science

High-Temperature Gibbs States are Unentangled and Efficiently Preparable

Ainesh Bakshi, Allen Liu, Ankur Moitra, Ewin Tang

Published Oct 27, 2024• 18 citations
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Abstract

We show that thermal states of local Hamiltonians are separable above a constant temperature. Specifically, for a local Hamiltonian$H$on a graph with degree$\mathfrak{g}$, its Gibbs state at inverse temperature$\beta$, denoted by$\rho=e^{-\beta H}/\text{tr}(e^{-\beta H})$, is a classical distribution over product states for all$\beta < 1/ (c \mathfrak{{g}})$, where$c$is 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$\beta < 1/(c\mathfrak{g}^{3})$, we can prepare a state$\varepsilon$-close to$\rho$in trace distance with a depth-one quantum circuit and$\text{poly}(n)\log(1/\varepsilon)$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.

DOI
10.1109/focs61266.2024.00068

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We show that thermal states of local Hamiltonians are separable above a constant temperature. Specifically, for a local HamiltonianHHHon a graph with degreeg\mathfrak{g}g, its Gibbs state at inverse temperatureβ\betaβ, denoted byρ=e−βH/tr(e−βH)\rho=e^{-\beta H}/\text{tr}(e^{-\beta H})ρ=e−βH/tr(e−βH), is a classical distribution over product states for allβ<1/(cg)\beta < 1/ (c \mathfrak{{g}})β<1/(cg), wherecccis 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 anyβ<1/(cg3)\beta < 1/(c\mathfrak{g}^{3})β<1/(cg3), we can prepare a stateε\varepsilonε-close toρ\rhoρin trace distance with a depth-one quantum circuit andpoly(n)log⁡(1/ε)\text{poly}(n)\log(1/\varepsilon)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.
At high temperature, quantum Gibbs states retain several classical features of the maximally mixed state: the absence of entanglement, the absence of magic, analyticity of the partition function, correlation decay, and algorithmic tractability. We prove new and sharp bounds showing that these features persist down to finite temperatures independent of system size, but fail at distinct inverse-temperature scales, forming a hierarchy of classical-to-quantum transitions. Our results hold for long-range Pauli interactions with bounded strength at every site. Despite such all-to-all interactions, we show that the death of entanglement occurs at constant temperature, resolving an open question of Rouze, Franca and Alhambra (STOC'25). We give a polynomial-time classical algorithm that prepares Gibbs states up to the death of entanglement transition. Notably, this is asymptotically colder than temperatures at which quantum Gibbs samplers are known to mix quickly, as well as the original separability temperature of Bakshi et al. (FOCS'24), which we improve to be tight up to constants. At asymptotically even colder temperatures, we show that the Gibbs state remains in the thermodynamic infinite-temperature phase. We give polynomial-time classical algorithms for estimating thermal expectations up to the phase transition (despite both entanglement and magic), and as a corollary resolve a correlation decay conjecture of Harrow, Mehraban and Soleimanifar (STOC'20).
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.
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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.
Learning the Hamiltonian underlying a quantum many-body system in thermal equilibrium is a fundamental task in quantum learning theory and experimental sciences. To learn the Gibbs state of local Hamiltonians at any constant inverse temperature, the state-of-the-art provable algorithms fall short of the optimal sample and computational complexity, in sharp contrast with the locality and simplicity in the classical cases. In this work, we present a learning algorithm that learns each local term of a n-qubit Hamiltonian on any bounded-degree graph to a constant additive error with the optimal sample complexity O(log⁡n)\mathcal{O}(\log n)O(logn). The protocol uses parallelizable local quantum measurements that act within bounded neighborhoods of the graph and near-linear-time classical post-processing. We also give a learning algorithm for lattice Hamiltonians with near-optimal scaling on the learning precision and the inverse temperature. At the heart of our algorithm is the interplay between locality, the Kubo-MartinSchwinger condition, and the operator Fourier transform at arbitrary temperatures.