An efficient and exact noncommutative quantum Gibbs sampler

TL;DR

The authors construct an exactly detailed-balanced Lindbladian whose stationary state is the quantum Gibbs state for arbitrary noncommuting Hamiltonians, providing a robust quantum analogue of Metropolis-Hastings. They achieve this with Gaussian or Metropolis-like transition weights and a carefully engineered coherent term, ensuring exact stationarity without energy estimation. The framework supports efficient implementation via operator Fourier transforms, time-domain LCU methods, and block-encodings, and it naturally purifies to a family of parent Hamiltonians whose ground state corresponds to the purified Gibbs state, enabling a quasi-adiabatic preparation path. For lattice systems, the Lindbladian is quasi-local, yielding favorable scaling with beta and locality, and purification yields a practical route to preparing purified Gibbs states with controlled complexity. Overall, the work positions a quantum Gibbs sampler as a principled quantum counterpart to classical MCMC, with concrete algorithmic constructs and complexity guarantees.

Abstract

Preparing thermal and ground states is an essential quantum algorithmic task for quantum simulation. In this work, we construct the first efficiently implementable and exactly detailed-balanced Lindbladian for Gibbs states of arbitrary noncommutative Hamiltonians. Our construction can also be regarded as a continuous-time quantum analog of the Metropolis-Hastings algorithm. To prepare the quantum Gibbs state, our algorithm invokes Hamiltonian simulation for a time proportional to the mixing time and the inverse temperature $β$, up to polylogarithmic factors. Moreover, the gate complexity reduces significantly for lattice Hamiltonians as the corresponding Lindblad operators are (quasi-) local (with radius $\simβ$) and only depend on local Hamiltonian patches. Meanwhile, purifying our Lindbladians yields a temperature-dependent family of frustration-free "parent Hamiltonians", prescribing an adiabatic path for the canonical purified Gibbs state (i.e., the Thermal Field Double state). These favorable features suggest that our construction serves as a quantum algorithmic counterpart to classical Markov chain Monte Carlo sampling.

Figures (4)

• Figure 1: (Left) For the classical Gibbs distribution, the detailed balance condition is a pairwise relation between heating (red) and cooling (blue) transition rates, depending on the energy difference $\nu$ of states. (Right) For the quantum Gibbs state, the detailed balance condition refers to pairs of matrix elements of the density operator (expanded in the energy basis), where each matrix element is described by a pair of energies (of the basis elements in the ket and bra respectively) therefore the relation depends on both of the respective energy differences $\nu_1$ and $\nu_2$.
• Figure 2: (Left) For lattice Hamiltonians, our Lindbladian is a sum of quasi-local terms $\mathcal{L}_{\beta}^{a}$ localized around each jump $\bm{A}^a$ with radius $\tilde{\mathcal{O}}(\beta)$. Indeed, detailed balance is really about the energy difference, which can be diagnosed by Fourier Transforming the Heisenberg evolution $\bm{A}^a(t) = \mathrm{e}^{\mathrm{i} \bm{H} t}\bm{A}^a \mathrm{e}^{-\mathrm{i} \bm{H} t}$. Due to the Lieb-Robinson bounds, the localized Lindbladian terms effectively only depend on the local Hamiltonian patch nearby (up to exponentially decaying tail). (Right) This locality persists after purification, where two copies of the system are glued together.
• Figure 3: A plot of the filter functions $\gamma(\omega)$ for Metropolis, Glauber and our filters arising from Gaussian linear combination \ref{['eq:convolutionFilter']}-\ref{['eq:convolutionFilterAPX']} (with $\sigma_{E}=\frac{1}{\beta}$).
• Figure 4: Circuit for block-encoding \ref{['eq:fAA']}. The gate $\bm{Prep}$ is a shorthand for $\bm{prep}_{\sqrt{\left\vert {f_+} \right\vert}}$ and $\bm{Prep}'$ for $\bm{prep}_{f_+/\sqrt{\left\vert {f_+} \right\vert}}$.

Theorems & Definitions (91)

• Theorem 1.1: Gibbs state is stationary
• Theorem 1.2: Efficient implementation
• Proposition 1.1: Purifying Lindbladians
• Theorem 1.3: Block encodings for the discriminants
• Definition 2.1: Kubo‚ÄìMartin‚ÄìSchwinger detailed balance condition
• Proposition 2.1: Fixed point
• Lemma 2.1: Prescribing the coherent term
• Corollary 2.1: $\bm{ \rho}$-DB Lindbladians
• proof : Proof of \ref{['lem:FindingCoherenceTerm']}
• Proposition 2.2: Detailed balance in the Energy domain