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Finite correlation length implies efficient preparation of quantum thermal states
Fernando G. S. L. Brandao, Michael J. Kastoryano
TL;DR
Preparing quantum thermal states lacks a general efficient quantum procedure. The paper uses uniform correlation decay and approximate Markov structure to construct local-channel preparation circuits, obtaining logarithmic depth and efficient local-observable estimation under these assumptions.
Problem
General-purpose efficient procedures for preparing or sampling quantum thermal states are not known, and existing approaches have important restrictions or difficult runtime analysis.
Source: §1 · p. 1Method
The paper combines uniform clustering, the uniform Markov property, local recovery maps, and quantum belief propagation to construct Gibbs-state preparation circuits.
Source: §1 · pp. 2–3Results
Under the two uniform assumptions, a constant-depth quasi-local circuit and an iterated logarithmic-depth circuit of strictly local quantum channels prepare the Gibbs state, while uniform clustering enables local estimation of observables.
Source: §1 · pp. 2–3Takeaways & Limitations
All one-dimensional Gibbs states of local bounded Hamiltonians can be prepared efficiently, and commuting Hamiltonians reduce the approximation error to that from uniform clustering.
Source: §1 · p. 3, §2.5 · p. 7The assumptions can fail for topologically ordered systems and have not been proved for Gibbs states of non-commuting Hamiltonians in dimensions greater than one.
Source: §2.2 · p. 4, §4 · pp. 14–15
Abstract
from arXiv · showhide
Preparing quantum thermal states on a quantum computer is in general a difficult task. We provide a procedure to prepare a thermal state on a quantum computer with a logarithmic depth circuit of local quantum channels assuming that the thermal state correlations satisfy the following two properties: (i) the correlations between two regions are exponentially decaying in the distance between the regions, and (ii) the thermal state is an approximate Markov state for shielded regions. We require both properties to hold for the thermal state of the Hamiltonian on any induced subgraph of the original lattice. Assumption (ii) is satisfied for all commuting Gibbs states, while assumption (i) is satisfied for every model above a critical temperature. Both assumptions are satisfied in one spatial dimension. Moreover, both assumptions are expected to hold above the thermal phase transition for models without any topological order at finite temperature. As a building block, we show that exponential decay of correlation (for thermal states of Hamiltonians on all induced subgraph) is sufficient to efficiently estimate the expectation value of a local observable. Our proof uses quantum belief propagation, a recent strengthening of strong sub-additivity, and naturally breaks down for states with topological order.
1 Introduction
The paper introduces a Gibbs-state preparation procedure using local quantum channels, with logarithmic depth under uniform clustering and uniform Markov assumptions. It also establishes local estimation of observables from uniform clustering and identifies extensions and topological limitations.
Preparation procedure
- The procedure patches thermal-state regions and locally recovers erased patches, relying on exponentially decaying correlations and approximate Markov structure.The proof combines quantum belief propagation with recovery maps from strengthened strong subadditivity.Source: §1 · p. 2
Limitations
- The approach does not apply to topologically ordered states with nonzero topological entanglement entropy, because such states cannot be prepared by finite-depth local-channel circuits.Uniform clustering can also fail for non-contractible regions in topologically ordered systems.Source: §1 · p. 2
Assumptions
- Uniform clustering requires exponentially decaying correlations for Gibbs states on every subset, while the uniform Markov property requires shielded-region conditional mutual information to decay with shield width.Both assumptions are formulated uniformly over induced subgraphs or regions of the lattice.Source: §1 · p. 2
Local estimation
- Uniform clustering alone allows local estimation of expectation values, connecting the result to local indistinguishability and local topological quantum order.This provides the building block for the full preparation theorem.Source: §1 · pp. 2–3
Main preparation result
- A constant-depth circuit of quasi-local quantum channels prepares the Gibbs state under uniform clustering and uniform Markovianity, and iteration yields logarithmic depth with strictly local channels.The quasi-locality can be chosen as O(log(N)) when assumption errors decay exponentially.Source: §1 · p. 3
Extensions and special cases
- In one dimension, both assumptions hold, so all one-dimensional Gibbs states of local bounded Hamiltonians can be prepared efficiently.For commuting Hamiltonians, the Markov-related approximation errors vanish beyond the interaction range, leaving uniform clustering as the remaining source of error.Source: §1 · p. 3
2 Preliminaries
The preliminaries define the lattice Gibbs-state framework and introduce uniform clustering, quantum belief propagation, recovery maps, and approximation errors used by the paper’s theorems.
2.1 Notation
- The paper studies finite-dimensional quantum spins on a finite D-dimensional square lattice with local, bounded Hamiltonian terms and Gibbs states defined on full or restricted regions.Hamiltonian terms have bounded support range and norm; restricted Gibbs states are defined using Hamiltonians confined to induced regions.Source: §2.1 · pp. 3–4
2.2 Clustering
- Uniform clustering requires exponentially or otherwise decreasing covariance between observables in separated regions for Gibbs states on every lattice subset, not only the full lattice.This uniform formulation differs from conventional bulk clustering and must also cover non-contractible regions for the main preparation theorem.Source: §2.2 · p. 4
2.3 Quantum Belief Propagation
- Quantum belief propagation represents a local Hamiltonian perturbation through a bounded operator that is supported quasi-locally near the perturbation, with errors decaying exponentially with radius.The relevant constants depend on inverse temperature and Lieb-Robinson velocity, but not on the lattice volume.Source: §2.3 · pp. 5–6
2.4 The Markov property and recovery maps
- Small conditional mutual information implies a local recovery map, and separated regions with individually local recovery maps can be recovered jointly through the union property.The recovery-map framework formalizes the approximate Markov condition for shielded regions.Source: §2.4 · pp. 6–7
2.5 Approximation Errors
- The proofs combine errors from uniform Markovness, belief propagation, and clustering; commuting local Hamiltonians eliminate the first two beyond a fixed interaction-dependent distance.For general non-critical thermal systems, the clustering and Markov errors are typically exponential, although decay faster than 1/poly(ℓ) suffices.Source: §2.5 · p. 7
3 Results
The results establish local estimation of observables under uniform clustering, then use clustering and Markov assumptions to construct efficient Gibbs-state preparation channels. The implications include dimension-dependent circuit structure, unconditional one-dimensional preparation, and limits imposed by topological order.
3.1 Estimating local expectation values
- Uniform clustering lets local expectation values be estimated quasi-locally, with errors controlled by boundary size and exponentially decaying correlation functions.The result is formulated as local indistinguishability and applies without restrictions on the region shape or observable structure.Source: §3.1 · pp. 8–9
3.3 Implications and strengthening of Theorem 6
- For commuting Hamiltonians, the Markov-related errors vanish beyond the interaction range, so preparation efficiency depends only on uniform clustering.For topologically ordered commuting-projector states, nonzero topological entanglement entropy prevents the required non-contractible-region conditions and therefore blocks Theorem 6.Source: §3.1 · p. 9, §3.3 · pp. 13–14
3.2 The main theorem
- For a D-dimensional Gibbs state satisfying uniform clustering and uniform Markov properties, Theorem 6 gives a depth D + 1 mixed quantum circuit with local channels of range O(ℓ).The construction uses engineered dissipation, parallel channels on disjoint regions, and a lattice decomposition into holes, connecting structures, and higher-dimensional regions.Source: §3.2 · pp. 10–12, Figure 3 · p. 10
- The preparation error has an O(|Λ|) prefactor multiplying exponentially decaying clustering and Markov error terms, while the construction uses only constant-range channels independent of system size.In two dimensions, the proof uses concentric tile regions and removes disconnected inner regions before composing the channel layers.Source: §3.2 · pp. 11–12
3.3 Implications and strengthening of Theorem 6
- In one dimension, all finite-temperature Gibbs states of local lattice systems can be efficiently prepared because one-dimensional Gibbs states always satisfy clustering and the Markov condition.The one-dimensional construction uses two overlapping local quantum channels and does not require the full uniform assumptions on every induced subgraph.Source: §3.3 · pp. 12–13
4 Outlook
The outlook identifies unresolved conditions for extending the preparation guarantees and proposes replacing finite-support channels with constant-support sequences. It also places the work alongside related patching approaches while emphasizing broader applicability of its proof strategy.
Broader scope
- The proof strategy may extend beyond Gibbs states to other classes of pure and mixed states.The authors state that the argument does not depend decisively on Gibbs-state structure.Source: §4 · p. 14
Open problems
- The main unresolved question is when the Markov property holds for Gibbs states of non-commuting Hamiltonians in dimensions greater than one.The authors conjecture that clustering implies the uniform Markov condition but have not proved it.Source: §4 · pp. 14–15
- A second open problem is whether clustering or the Markov property can differ between contractible and non-contractible regions at finite temperature.The paper connects this question to the role of topology in the ground-state setting.Source: §4 · p. 15
Future directions
- The authors propose replacing log(N)-depth channels with finite support by log(N)-depth channels with constant support.A non-commuting Gibbs sampler and semigroup mixing-time analysis are suggested as possible routes.Source: §4 · p. 15
- Such a replacement could permit asynchronous circuit application and provide a route to proving semigroup gaps and gaps of frustration-free Hamiltonians.These are presented as potential advantages of the semigroup approach.Source: §4 · p. 15
Related work
- Related patching work by Swingle and McGreevy also studies efficient quantum-state preparation and connects vanishing conditional mutual information to the task [41].This note added after the work was completed identifies related approaches rather than a new result of this paper.Source: §4 · p. 15