Rapid mixing for high-temperature Gibbs states with arbitrary external fields

Abstract: Gibbs states are a natural model of quantum matter at thermal equilibrium. We investigate the role of external fields in shaping the entanglement structure and computational complexity of high-temperature Gibbs states. External fields can induce entanglement in states that are otherwise provably separable, and the crossover scale is $h\asymp β^{-1} \log(1/β)$, where $h$ is an upper bound on any on-site potential and $β$ is the inverse temperature. We introduce a quasi-local Lindbladian that satisfies detailed balance and rapidly mixes to the Gibbs state in $\mathcal{O}(\log(n/ε))$ time, even in the presence of an arbitrary on-site external field. Additionally, we prove that for any $β<1$, there exist local Hamiltonians for which sampling from the computational-basis distribution of the corresponding Gibbs state with a sufficiently large external field is classically hard, under standard complexity-theoretic assumptions. Therefore, high-temperature Gibbs states with external fields are natural physical models that can exhibit entanglement and classical hardness while also admitting efficient quantum Gibbs samplers, making them suitable candidates for quantum advantage via state preparation.

• The paper constructs a field-resonant Lindbladian that mixes to high-temperature Gibbs states in O(log(n/ε)) time, with rates independent of external-field strength h.
• The paper proves Gibbs states remain separable mixtures of product states up to h = Θ(β⁻¹ log(1/β)), matching the known entanglement crossover up to constants.
• The paper uses field refrigeration to show that strong fields can make computational-basis Gibbs sampling classically hard at any β < 1 unless the polynomial hierarchy collapses to its third level.

Overview

This paper studies how on-site external fields affect both the structure and the computational complexity of high-temperature Gibbs states of local Hamiltonians. The Hamiltonians considered have the form $H = W + V$, where $W = \sum_a W_a$ is a bounded $L$-local, degree-$D$ interaction term and $V = \sum_i V_i$ is an external field with $\|V_i\| \le h$. The paper delivers three results. First, it constructs a "field-resonant" Lindbladian that satisfies KMS detailed balance and mixes to the Gibbs state in $O(\log(n/\epsilon))$ time, with a mixing bound that is entirely independent of the field strength $h$, for any inverse temperature $\beta \le (DL)^{-3}/28800$. Second, it proves that the Gibbs state remains a convex combination of product states whenever $h \lesssim \beta^{-1}\log(1/\beta)$, matching up to constants the entanglement threshold of Kuwahara and Hatano and thereby pinning down the correct crossover scale for field-induced entanglement. Third, it shows that for any $W = \sum_a W_a$0, sufficiently large external fields make sampling from the computational-basis distribution of the Gibbs state classically hard unless the polynomial hierarchy collapses to the third level.

Rapid mixing via a field-resonant Lindbladian

The central algorithmic contribution addresses the question of whether external fields obstruct efficient quantum Gibbs-state preparation. Prior approaches fail in this regime for distinct reasons. The Lindbladian of Bakshi, Liu, Moitra, and Tang uses jump operators given by imaginary-time evolutions of single-site Paulis, whose quasi-locality is controlled by Araki expansionals and degrades precisely at $W = \sum_a W_a$1. The Chen–Kastoryano–Gilyén (CKG) construction with canonical parameters $W = \sum_a W_a$2 retains quasi-locality—thanks to a field-independent Lieb–Robinson bound—but its dissipative contraction decays as $W = \sum_a W_a$3, becoming exponentially small once $W = \sum_a W_a$4. The authors exhibit an explicit two-qubit example ($W = \sum_a W_a$5) showing that no Dobrushin-type proof can recover the needed suppression for the canonical parameters, since the dependence on $W = \sum_a W_a$6 cancels exactly when the on-site and interaction terms commute.

The field-resonant Lindbladian resolves this by making the CKG filter and transition-weight parameters site-dependent. For each site $W = \sum_a W_a$7, the authors set

$W = \sum_a W_a$8

The Gaussian transition weight $W = \sum_a W_a$9 is then centered at the locally relevant Bohr frequency scale, restoring an order-one dissipative contraction regardless of field strength. Two structural facts underpin the analysis:

Field-independent Lieb–Robinson bound. By passing to the interaction picture and factoring out the on-site evolution (following Nachtergaele, Raz, Schlein, and Sims), combined with a generalization of the Haah–Hastings–Kothari–Low bound to time-dependent Hamiltonians, the authors obtain a shell decomposition of Heisenberg evolution whose tails decay as $L$0 with $L$1, uniformly in $L$2. This yields quasi-local expansions for both the jump operators and the coherent terms; the latter requires new moment bounds on the kernel functions $L$3 and $L$4, which grow only like $L$5 and thus cancel the factorial from the Lieb–Robinson estimate.

Dissipative contraction. When the disagreement is already at the updated site, rotating $L$6 into a Pauli-$L$7 basis reduces the contraction analysis to a Gaussian overlap integral $L$8. For the field-resonant choice this integral equals $L$9 when $D$0 and at least $D$1 otherwise—a universal constant $D$2 independent of $D$3.

Combining these ingredients within the transport-plan formalism and the quantum Dobrushin condition of Bakshi et al., the update matrix takes the form "negative diagonal plus geometrically decaying quasi-local tail," giving column sums strictly below one and hence

$D$4

with no dependence on $D$5 anywhere in the rate. This is the strongest quantitative claim of the paper: mixing remains logarithmic even for exponentially large external fields. The authors also argue (without a full compilation theorem) that the evolution should be implementable with $D$6 gates on constant-dimensional lattices when $D$7, using a truncation radius $D$8 and the large-field Hamiltonian simulation result of Low and Wiebe; the circuit complexity acquires only a logarithmic dependence on the field strength.

Separability and the entanglement threshold

The second result shows that if $D$9 and $V = \sum_i V_i$0, then the Gibbs state is a convex combination of product states—and can therefore be prepared by a depth-2 circuit. Combined with the Kuwahara–Hatano construction showing entanglement at $V = \sum_i V_i$1, this establishes that $V = \sum_i V_i$2 is, up to constant factors, the correct threshold at which external fields first induce entanglement.

The proof follows the blueprint of Bakshi, Liu, Moitra, and Tang but requires a refined combinatorial treatment of the Araki expansional, since the nested-commutator expansion now contains arbitrarily many on-site insertions interleaved with interaction terms. These insertions do not enlarge support but proliferate the number and magnitude of admissible clusters; the authors control them with a cluster-counting identity yielding per-cluster coefficients bounded by $V = \sum_i V_i$3, which stays small up to the stated threshold. A notable caveat: the required temperature scales exponentially worse in the locality parameter than the field-free analysis, which the authors attribute to their techniques rather than to a fundamental obstruction.

Classical hardness from field refrigeration

The third result shows that external fields do not merely preserve classical hardness at high temperature—they can create it. The key device is a field-refrigeration gadget: given a commuting projector Hamiltonian $V = \sum_i V_i$4, attach $V = \sum_i V_i$5 ancilla qubits per projector term and define $V = \sum_i V_i$6. Tracing out the ancillas recovers exactly the Gibbs state of $V = \sum_i V_i$7 at effective inverse temperature

$V = \sum_i V_i$8

Since $V = \sum_i V_i$9, choosing $\|V_i\| \le h$0 with $\|V_i\| \le h$1 and $\|V_i\| \le h$2 achieves $\|V_i\| \le h$3 at physical temperature $\|V_i\| \le h$4. Applying this to the commuting-projector family of Rajakumar and Watson—whose computational-basis distributions are hard to sample at any $\|V_i\| \le h$5—yields a family of 6-local, degree-$\|V_i\| \le h$6 Hamiltonians with $\|V_i\| \le h$7 ancillas such that classical sampling to total variation distance $\|V_i\| \le h$8 would collapse PH to the third level. Notably, the required field strength $\|V_i\| \le h$9 coincides with the entanglement threshold, though the authors do not claim a causal connection.

Limitations and open questions

Several gaps are acknowledged explicitly. The rapid-mixing theorem requires $O(\log(n/\epsilon))$0, while the hardness reduction operates at $O(\log(n/\epsilon))$1; these regimes do not overlap, so the paper does not exhibit a single Hamiltonian family that is simultaneously provably fast-mixing and classically hard. The separability threshold's exponential dependence on $O(\log(n/\epsilon))$2 is likely suboptimal. The implementation claims are presented as expected costs rather than a proven compilation theorem, with the normalization factor $O(\log(n/\epsilon))$3 carrying a possibly avoidable logarithmic dependence on $O(\log(n/\epsilon))$4. Finally, the nature and correlation-length scaling of field-induced entanglement above the threshold remain unquantified, and whether natural Hamiltonian families realize an overlap between the quantum-easy and classically-hard regimes—an overlap that would yield an unconditional quantum-advantage statement—is left open.

Conclusion

This paper establishes that arbitrary on-site external fields are compatible with logarithmic-time quantum Gibbs sampling, identifies $O(\log(n/\epsilon))$5 as the sharp scale separating separable from entangled high-temperature Gibbs states, and demonstrates via field refrigeration that strong fields can import low-temperature classical hardness into the high-temperature regime. Together these results position high-temperature Gibbs states with external fields as concrete candidates for quantum advantage through state preparation, while leaving the precise alignment of the mixing, separability, and hardness thresholds as the principal open problem.