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Quantum Gibbs Samplers: the commuting case

Michael J. Kastoryano, Fernando G. S. L. Brandao

arXiv:1409.3435v3quant-phcond-mat.stat-mechmath-ph

TL;DR

The paper asks how to characterize and efficiently prepare thermal states of quantum lattice systems. It develops a non-commutative Lp framework for Gibbs correlations and two commuting-Hamiltonian Gibbs samplers, proving that system-size-independent gaps are equivalent to strong clustering and yielding efficient preparation results in one dimension and at high temperature.

  • Problem

    The paper addresses how Gibbs-state correlations determine thermalization and whether physically relevant quantum Gibbs states can be prepared efficiently.

  • Method

    The paper develops a non-commutative Lp framework with Davies and Heat-Bath Gibbs samplers, using conditional expectations to analyze correlations and convergence.

  • Results

    For commuting local Hamiltonians, both samplers have a system-size-independent gap if and only if the Gibbs state satisfies strong clustering.

  • Takeaways & Limitations

    Gibbs states satisfying strong clustering can be prepared in polynomial time, including all one-dimensional commuting systems at constant temperature and commuting systems above a size-independent critical temperature.

Abstract

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We analyze the problem of preparing quantum Gibbs states of lattice spin Hamiltonians with local and commuting terms on a quantum computer and in nature. Our central result is an equivalence between the behavior of correlations in the Gibbs state and the mixing time of the semigroup which drives the system to thermal equilibrium (the Gibbs sampler). We introduce a framework for analyzing the correlation and mixing characteristics of quantum Gibbs states and quantum Gibbs samplers, which is rooted in the theory of non-commutative Lp spaces. We consider two distinct classes of Gibbs samplers, one of which being the well-studied Davies generators modelling the dynamics on the system due to weak-coupling with a large Markovian environment. We show that their gap is independent of system size if, and only if, a certain strong form of clustering of correlations holds in the Gibbs state. As concrete applications of our formalism, we show that for every one-dimensional lattice system, or for systems in lattices of any dimension at high enough temperatures, the Gibbs samplers of commuting Hamiltonians are always gapped, giving an efficient way of preparing these states on a quantum computer.

I. INTRODUCTION

This work develops a framework for quantum Gibbs samplers of commuting Hamiltonians and establishes an equivalence between strong correlation clustering and system-size-independent spectral gaps. It applies this framework to obtain polynomial-time preparation results for one-dimensional systems and sufficiently hot systems.

  • I. INTRODUCTION: The paper introduces a quantum framework, rooted in non-commutative Lp spaces, for relating Gibbs-state correlations to Gibbs-sampler mixing.It defines conditional expectations and two sampler classes: Davies and Heat-Bath generators.
  • I. INTRODUCTION: The framework is restricted to commuting local terms, although the resulting preparation approach may offer resilience to noise through heat-bath coupling.One-dimensional commuting Gibbs states also admit matrix-product-operator preparation methods.
  • I. INTRODUCTION: Strong clustering implies the standard weak clustering of correlations and connects to local indistinguishability under distant perturbations.This extends the correlation framework beyond a direct quantum analogue of classical DLR theory.
  • I. INTRODUCTION: For commuting local Hamiltonians, Davies and Heat-Bath samplers have a system-size-independent gap if and only if the Gibbs state satisfies strong clustering.The gap controls the convergence rate of the sampler to the Gibbs state.
  • I. INTRODUCTION: Both samplers provide polynomial-time quantum algorithms for preparing Gibbs states of commuting Hamiltonians satisfying strong clustering.The samplers are local and efficiently implementable on quantum computers or quantum simulators.
  • A. Summary of results: Both samplers prepare Gibbs states of every one-dimensional commuting Hamiltonian at constant temperature and of commuting Hamiltonians above a system-size-independent critical temperature.In one dimension, this follows from clustering properties of Gibbs states; at high temperature, the samplers are also shown to be gapped.

III. CONDITIONAL EXPECTATIONS

The paper introduces conditional expectations as local quasi-projectors onto a Gibbs state, requiring complete positivity, unitality, consistency, reversibility, and monotonicity. Two constructions are developed: a dynamical local Liouvillian projector and a static map determined by the reference state.

  • Conditional expectations are maps that serve as local quasi-projectors onto the Gibbs state.
  • A conditional expectation is completely positive and unital.
  • The framework also imposes consistency, reversibility, and monotonicity conditions related to classical conditional expectation and detailed balance.
  • The local Liouvillian projector associated with a local generator acts only on a subsystem and a finite neighborhood around it.
  • For a frustration-free, locally reversible Liouvillian, the local Liouvillian projector is a conditional expectation with respect to the stationary state.
  • When the Liouvillian is primitive, the local projector on the whole system is equivalent to expectation with respect to the full state.

B. Minimal conditional expectations

The minimal conditional expectation is a static map defined directly from a full-rank state and designed to minimally affect observables outside a chosen subsystem. For commuting Gibbs states, it becomes local, although locality is not guaranteed for arbitrary full-rank states.

  • The minimal conditional expectation is designed to minimally affect observables outside A while satisfying the conditional-expectation requirements.
  • For a full-rank state, the map is defined from the reduced state on A and acts as the identity on A while acting nontrivially on its complement.
  • In the eigenbasis of the reference state, the minimal conditional expectation reduces to the classical conditional expectation.
  • The map is completely positive, unital, reversible, and monotone with respect to the reference state.
  • Unlike the local Liouvillian projector, the minimal conditional expectation is uniquely defined for every full-rank state without using a dynamical description.
  • For Gibbs states of commuting Hamiltonians, the minimal conditional expectation acts on A plus a neighborhood around A, whereas locality can fail for general full-rank states.

A. Davies generators

The paper studies Davies generators and Heat-Bath generators as Gibbs samplers for local commuting Hamiltonians. Both are local, reversible, frustration-free semigroups, while the Davies construction additionally models weak coupling to a thermal bath.

  • Davies generators arise from weak coupling between a finite quantum system and a large thermal bath.
  • The system-bath construction uses local Hermitian operators spanning each site's observable algebra, with generalized Pauli matrices as one possible choice.
  • For local commuting potentials, the Davies generator generates a completely positive unital semigroup.
  • The Davies generator is local, locally primitive, locally reversible with respect to the Gibbs state, and frustration free.
  • The Heat-Bath generator is a second, simpler Gibbs sampler constructed from local conditional expectations rather than a physical bath model.
  • The Heat-Bath generator shares complete positivity, locality, local primitivity, reversibility, and frustration freeness under the commuting-potential assumptions.
  • Replacing the Gibbs state by another faithful state preserves the Heat-Bath framework but generally makes locality bounds difficult to obtain.

V. DECAY OF CORRELATIONS

The paper formulates weak and strong clustering through conditional covariances and connects these correlation properties to locality and Gibbs-state structure. For commuting Gibbs states, the framework establishes equivalences between covariance formulations and restricts relevant observables to local boundary regions.

  • The paper studies exponential decay of correlations between local observables as their supports become separated.
  • Weak clustering describes decay of covariance with the distance between observable supports, while strong clustering incorporates conditional expectations and separated regions.
  • The relevant distance is the width of A ∪ B in the disjoint-region setting and the separation between the boundaries of B and A when B ⊆ A.
  • For the conditional expectations considered, strong clustering can be tested using observables supported on A ∪ B and its boundary.
  • The clustering definitions use covariance weighted by L2 norms, making them stronger than analogous definitions based on operator norms.
  • The covariance inner product is unconventional and does not reduce to the usual pure-state covariance when the reference state is pure.
  • For commuting Hamiltonian Gibbs states, the paper proves that weak clustering in the modified covariance is equivalent to weak clustering in the other covariance formulation.
  • The proof uses the fact that conjugating a local observable by powers of the Gibbs state enlarges its support only to the local boundary.

A. Local indistinguishably

The paper introduces local indistinguishability as a strong correlation condition and shows its equivalence to a strengthened form of weak clustering. This condition connects system-size-independent log-Sobolev behavior with correlation decay, while its relation to strong clustering remains unresolved.

  • The paper introduces extended clustering notions to characterize rapid correlation decay in quantum Gibbs states.
  • Local indistinguishability is defined through fixed points of conditional expectations associated with local regions.
  • A strengthened weak-clustering bound using products of 1-norm and infinity-norm is equivalent to local indistinguishability.
  • This clustering form follows from a system-size-independent log-Sobolev constant, whereas weak clustering follows from merely a constant spectral gap.
  • The relation between strong clustering and local indistinguishability is not known, although local indistinguishability implies weak clustering.

VI. MAIN RESULTS

This section develops conditional variances and their overlap inequalities as technical tools for relating local regions to global Gibbs-sampler gaps. The argument uses general properties of conditional expectations rather than Gibbs-state structure alone.

  • Conditional variance is introduced as a central quantity for analyzing local regions and reduces to ordinary variance on the full lattice.
  • The section studies how conditional variances on two overlapping subsets relate to the variance on their union.
  • The proof derives an identity involving conditional expectations and uses positivity and contractivity to obtain the desired variance inequality.
  • The proposition requires only general properties of conditional expectations, without assuming a Gibbs state or local structure.

A. Strong clustering implies gapped Gibbs sampler

The paper proves that strong clustering yields a system-size-independent spectral gap for Davies and Heat-Bath Gibbs samplers of commuting local Hamiltonians. An iterative overlapping-region construction establishes the bound, with applications extending to polynomially decaying clustering and highlighting limits of weaker assumptions.

  • Main theorem: Strong clustering implies a spectral gap independent of lattice size for the associated Heat-Bath or Davies Gibbs sampler.
  • Main theorem: The argument applies to Davies and Heat-Bath generators because their associated conditional expectations and generators share locality, reversibility, and kernel properties.
  • Proof strategy: The proof compares conditional gaps on overlapping regions and iteratively grows rectangles so that doubling system size does not substantially reduce the gap.
  • Proof strategy: The decomposition uses overlaps of order L whose distinct overlap regions do not intersect, enabling the conditions required by Lemma 24.
  • Proof strategy: The threshold L0 depends on clustering constants rather than system size, yielding a constant lower bound on the conditional gap.
  • Scope and limitations: Strong clustering need not decay exponentially; sufficiently high-degree polynomial decay is enough, and every rectangular region has a gapped generator.
  • Scope and limitations: Extending the proof to weak clustering would require additional boundary-condition control, and evidence from the 4D toric code suggests this may fail in general dimensions.

B. Gapped Gibbs sampler implies strong clustering

The paper proves the converse direction: a system-size-independent gap for a local Gibbs sampler implies strong clustering of the Gibbs state. The proof maps the sampler to a local frustration-free Hamiltonian and uses the detectability lemma to control correlations.

  • Detectability-lemma method: The detectability lemma organizes overlapping local projectors into non-overlapping layers and yields an approximate local projection property.The number of layers and pyramid sets depends on locality parameters rather than system size.
  • Hamiltonian correspondence: The sampler’s operator representation is a local frustration-free Hamiltonian with ground-state energy zero.This correspondence allows techniques for frustration-free Hamiltonians to be applied to Gibbs samplers.
  • Main theorem: A gapped Gibbs sampler for a local commuting Hamiltonian implies that the Gibbs state satisfies strong clustering.The result is stated for projective conditional expectations and applies to the local Gibbs sampler under the theorem’s assumptions.
  • Correlation bound: Applying these approximate projections to separated regions bounds conditional covariances and establishes strong clustering.The proof constructs inner and outer projectors for complementary regions and uses reversibility together with Hölder’s inequality.
  • Consequences: A gapped Gibbs sampler also implies weak clustering, while strong clustering implies weak clustering under the paper’s stated correspondence.The converse result is weaker in general than the theorem for strong clustering but extends to the standard covariance-decay notion.
  • Scope of the proof: The equivalence proof for one conditional expectation requires a projective conditional expectation, while the non-projective case is handled only through a limiting construction.For Eρ, the argument relates the original generator to a limiting iterated conditional expectation.

VII. ONE-DIMENSIONAL MODELS

For one-dimensional commuting Hamiltonians, weak and strong clustering are equivalent, and the associated Davies and Heat-Bath Gibbs samplers are always gapped. The argument exploits the zero-dimensional boundary of one-dimensional regions and matrix-product-state methods.

  • One-dimensional geometry: In one dimension, the boundary of separated regions is zero-dimensional, so boundary effects contribute only a constant factor to clustering bounds.This makes strong clustering, which accounts for worst-case boundary conditions, equivalent to weak clustering for commuting potentials.
  • Clustering equivalence: Weak clustering is equivalent to strong clustering for one-dimensional commuting potentials and both conditional expectations considered in the paper.The equivalence is stated for Eρ and EL.
  • Proof strategy: The proof reduces covariance estimates to observables supported near the boundaries of the regions.The relevant boundary support is represented using modified Gibbs states and decompositions across ∂A and ∂B.
  • Matrix-product-state argument: A one-dimensional commuting Gibbs state can be represented as a matrix product state with bond dimension bounded by 2^r after blocking sites.The finite interaction range permits a two-local blocked description, and only crossing terms increase Schmidt rank.
  • Sampler consequence: All one-dimensional Gibbs samplers of commuting Hamiltonians are gapped, including Davies and Heat-Bath generators.This follows by combining one-dimensional clustering with the general equivalence between strong clustering and a system-size-independent gap.
  • From MPS to samplers: The resulting transfer-operator and parent-Hamiltonian gap properties provide the clustering estimates needed for the Gibbs-sampler theorem.These estimates are transferred to the relevant covariance and L2 norms before invoking the one-dimensional clustering equivalence.

VIII. THE HIGH TEMPERATURE PHASE

At sufficiently high temperatures, commuting Gibbs samplers on finite-dimensional lattices have system-size-independent spectral gaps. The proof maps the generators to frustration-free Hamiltonians and applies local spectral-gap bounds.

  • Main result: There is a critical temperature Tc(r, d), independent of lattice size, above which both Heat-Bath and Davies generators have constant spectral gaps.The result applies to r-local bounded commuting Hamiltonians on d-dimensional lattices.
  • Main result: The high-temperature result implies that every such Gibbs state can be prepared efficiently on a quantum computer.The constant gap supplies the sampler-mixing guarantee used for efficient state preparation.
  • Heat-Bath generator: The Heat-Bath proof maps the generator to a local frustration-free Hamiltonian whose terms become sufficiently close to commuting at high temperature.The limiting infinite-temperature Hamiltonian is non-interacting and has gap one.
  • Spectral-gap method: Knabe’s bound reduces the global gap estimate to local connected sublattices of a fixed size independent of the volume.The bound uses constants N(k, d) and λ(k, d) determined by locality and dimension.
  • Heat-Bath generator: At temperatures T ≥ Tc, the local spectral estimates imply a constant gap for the Heat-Bath generator.The threshold is obtained by comparing the high-temperature local Hamiltonian with the Knabe bound.
  • Davies generator: The Davies generator is likewise gapped at high temperature, first at infinite temperature and then above a volume-independent critical temperature.The proof uses locality, Parseval’s theorem, and the same perturbative argument used for the Heat-Bath case.

IX. OUTLOOK

The outlook emphasizes the framework’s unified treatment of commuting quantum Gibbs samplers, its connection to correlation clustering, and several open extensions and applications.

  • Main contributions: The framework gives two prescriptions for local quantum Gibbs samplers and relates rapid convergence to strong exponential clustering.It also connects this clustering notion with more conventional correlation-decay properties.
  • Physical relevance: Davies generators model thermal dynamics arising from weak coupling to a Markovian thermal reservoir.This makes the analysis relevant to thermalization times in optical-lattice and related experiments.
  • Limitations and open problems: The framework is poorly suited to Hamiltonians with non-commuting local terms because key operators generally become non-local.Extending the results beyond commuting Hamiltonians is identified as an important direction.
  • Limitations and open problems: An open question is whether the spectral gap of commuting Gibbs samplers is equivalent to a Log-Sobolev inequality.Such an equivalence could rule out intermediate mixing regimes by yielding either exponential or logarithmic relaxation times.

1. The equivalence of weak and strong clustering in higher dimensions

The paper discusses conjectured extensions of clustering equivalences to higher-dimensional commuting systems and their possible implications for thermal stability and quantum memories.

  • Higher-dimensional clustering: The equivalence between weak and strong clustering proved for one-dimensional systems is conjectured to extend to two-dimensional commuting Hamiltonians.The existing proof relies on a boundary Schmidt decomposition and does not directly generalize to higher dimensions.
  • Low-temperature questions: The framework raises whether correlation clustering of a commuting Hamiltonian’s ground state persists in its Gibbs state at sufficiently low non-zero temperature.A related question concerns whether LTQO implies local indistinguishability for the Gibbs sampler at low temperatures.
  • Technical obstacles: Degenerate ground spaces and the mismatch between Gibbs-sampler Lp norms and ground-state operator norms remain obstacles to a full answer.This difficulty is especially relevant for commuting models with topological order.
  • Quantum-memory implications: A related conditional argument would make Davies generators gapped for LTQO Hamiltonians, ruling out memory lifetimes longer than polynomial in system size.This would provide a no-go result for self-correcting quantum memories under the stated assumptions.
  • Potential implications: If the proposed low-temperature and higher-dimensional clustering connections hold, topological order in two-dimensional commuting Hamiltonians would imply a gapped Gibbs sampler at finite temperatures.The text presents this as a conditional implication rather than an established result.

Appendix

The appendix develops technical properties of localized operators and generalized eigenvalue expressions used in analyzing the Gibbs samplers and their associated projectors.

  • Geometric arguments: The appendix analyzes expressions for λ_Λ(A) using geometric decompositions of regions in R^d.The argument tracks coordinate intervals and nested subsets to establish the required containment relations.
  • Spectral analysis: The operators Q_A and related transformed operators are treated as Hermitian, allowing the relevant relation to be recast as a generalized eigenvalue equation.This supplies an algebraic route for studying the sampler’s spectral properties.
  • Boundary localization: The transformed conditional and generator-related operators act non-trivially only on boundary regions such as A∂.They can consequently be represented using Hermitian operators supported on the boundary tensor identity on its complement.
  • Tensor-product algebra: The appendix uses the determinant identity det(A ⊗ 1) = det(A)^n when reducing tensor-product operator expressions.Here n is the dimension of the matrix A.
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