Quantum Thermal State Preparation

TL;DR

This work introduces a physically motivated, quantum Gibbs-sampling framework grounded in open-system thermodynamics. By crafting continuous-time Lindbladians inspired by Davies dynamics and harnessing a Gaussian-weighted operator Fourier Transform, the authors obtain provable approximate Gibbs fixed points even under finite-energy-resolution constraints. They present two algorithmic streams: incoherent Lindbladian simulation (with both weak-measurement and LCUs) and a coherent, Szegedy-type Gibbs sampler that yields a quadratic speedup for preparing purified Gibbs states via simulated annealing, all supported by explicit block-encodings for both the Lindbladian and its discriminant proxy. A rigorous analytic scaffold—approximate detailed balance, secular approximation, and perturbation bounds—underpins finite-time thermalization results and the robustness of fixed points. The framework promises a versatile, scalable approach to thermal-state preparation on quantum devices and offers a principled path toward practical quantum simulations of thermodynamic regimes.

Abstract

Preparing ground states and thermal states is essential for simulating quantum systems on quantum computers. Despite the hope for practical quantum advantage in quantum simulation, popular state preparation approaches have been challenged. Monte Carlo-style quantum Gibbs samplers have emerged as an alternative, but prior proposals have been unsatisfactory due to technical obstacles rooted in energy-time uncertainty. We introduce simple continuous-time quantum Gibbs samplers that overcome these obstacles by efficiently simulating Nature-inspired quantum master equations (Lindbladians). In addition, we construct the first provably accurate and efficient algorithm for preparing certain purified Gibbs states (called thermal field double states in high-energy physics) of rapidly thermalizing systems; this algorithm also benefits from a quantum walk speedup. Our algorithms' costs have a provable dependence on temperature, accuracy, and the mixing time (or spectral gap) of the relevant Lindbladian. We complete the first rigorous proof of finite-time thermalization for physically derived Lindbladians by developing a general analytic framework for nonasymptotic secular approximation and approximate detailed balance. Given the success of classical Markov chain Monte Carlo (MCMC) algorithms and the ubiquity of thermodynamics, we anticipate that quantum Gibbs sampling will become indispensable in quantum computing.

Figures (12)

• Figure 1: The Metropolis-Hastings algorithm iterates a Markov chain to sample from the Gibbs distribution. Each step begins with a (random) jump: if the energy decreases, accept; if the energy increases, accept with a carefully chosen probability. Otherwise, reject the move. Remarkably, detailed balance can be enforced in a lazy manner via rejection sampling without storing the whole matrix.
• Figure 2: (Up) Davies' generator gives a continuous-time Markov generator on the energy spectrum (assuming the Hamiltonian is nondegenerate and that the input state is diagonal in the energy basis.). The transitions are weighted by $\gamma(\omega)$: the heating transitions (red) are suppressed by a Boltzmann factor relative to the cooling transitions (blue), entailing detailed balance. The operator $\bm{A}_{\nu}^a$ contains the transitions with energy difference $\nu$, which requires an infinite-time Fourier Transform. (Down) Our Lindbladian Gibbs sampler can be considered a "semi-classical" random walk where nearby Bohr-frequencies $\omega\pm \mathcal{O}(\sigma_t^{-1})$ cannot be distinguished. The operator Fourier Transform $\hat{\bm{A}}^a(\omega)$ contains a band of transitions. This breaks the detailed balance condition; the fixed point deviates from the Gibbs state.
• Figure 3: Quantum circuit implementation of an approximate $\delta$-time step via a weak measurement scheme.
• Figure 4: Circuit $\bm{U}$ for block-encoding the Lindbladian. Practically, if we use the simpler weak-measurement-based simulation (\ref{['thm:weakMeasSim']}), then by \ref{['cor:rndWeakMeasSim']}, we can use a single randomly chosen Lindblad operator $\bm{A}^a$ at a time. Moreover, if $\bm{A}^a$ is unitary, we can simply replace $\bm{V}_{jump}$ with $\bm{A}^a$, implying $b=c=0$, i.e., the third and the forth registers can be omitted, thus $n+\lceil\log(N)\rceil+2$ qubits suffice to simulate the Lindbladian $\mathrm{e}^{\mathcal{L} t }$
• Figure 5: Circuit for operator Fourier Transform $\mathcal{F}$ for an operator $\bm{O}$ acting on the system $\bm{ \rho}$. Of course, in our use, the operator may also act nontrivially on other ancillas.

Theorems & Definitions (143)

• Theorem 1.1: Gibbs state is thermodynamic
• Definition 1.1: Lindbladian mixing time
• Definition 1.2: Block-encoding of a Lindbladian
• Theorem 1.2: Linear-time Lindbladian simulation, simplified
• Lemma 1.1: Efficient block-encoding
• Theorem 1.3: Approximate Gibbs fixed point
• Lemma 1.2: Efficient block-encoding
• Theorem 1.4: Approximate purified Gibbs state
• Lemma 1.3: Efficient block-encoding
• Definition 2.1: Detailed balance condition