Chi-Fang Chen

The Markov property entails the conditional independence structure inherent in Gibbs distributions for general classical Hamiltonians, a feature that plays a crucial role in inference, mixing time analysis, and algorithm design. However, much less is known about quantum Gibbs states. In this work, we show that for any Hamiltonian with a bounded interaction degree (e.g., D-dimensional lattices), the quantum Gibbs state is locally Markov at arbitrary temperature, meaning there exists a quasi-local recovery map for every local region. Notably, this recovery map is obtained by applying a detailed-balanced Lindbladian with jumps acting on the region. Consequently, we prove that (i) the conditional mutual information (CMI) for a shielded small region decays exponentially with the shielding distance, and (ii) under the assumption of uniform clustering of correlations, Gibbs states of general non-commuting Hamiltonians on $D$-dimensional lattices can be prepared by a quantum circuit of depth $\e^{\mathcal{O}(\log^D(n/\epsilon))}$. Our proofs introduce a regularization scheme for imaginary-time-evolved operators at arbitrarily low temperatures and reveal a connection between the Dirichlet form, a dynamic quantity, and the commutator in the KMS inner product, a static quantity. We believe these tools pave the way for tackling further challenges in quantum thermodynamics and mixing times, particularly in low-temperature regimes.

Learning the Hamiltonian underlying a quantum many-body system in thermal equilibrium is a fundamental task in quantum learning theory and experimental sciences. To learn the Gibbs state of local Hamiltonians at any inverse temperature $\beta$, the state-of-the-art provable algorithms fall short of the optimal sample and computational complexity, in sharp contrast with the locality and simplicity in the classical cases. In this work, we present a learning algorithm that learns each local term of an $n$-qubit $D$-dimensional Hamiltonian to an additive error $\epsilon$ with sample complexity $\tilde{O}( \frac{e^{\poly\beta}}{\beta^2\epsilon^2}) \log(n)$. The protocol uses parallelizable local quantum measurements that act within bounded regions of the lattice and near-linear-time classical post-processing. Thus, our complexity is near optimal with respect to $n,\epsilon$ and is polynomially tight with respect to $\beta$. We also give a learning algorithm for Hamiltonians with bounded interaction degree with sample and time complexities of similar scaling on $n$ but worse on $\beta, \epsilon$. At the heart of our algorithm is the interplay between locality, the Kubo-Martin-Schwinger condition, and the operator Fourier transform at arbitrary temperatures.

It is shown that every one-dimensional Hamiltonian with short-range interacting spins admits a quantum Gibbs sampler [CKG23] with a system-size independent spectral gap at all finite temperatures. Consequently, their Gibbs states can be prepared in polylogarithmic depth, and satisfy exponential clustering of correlations, generalizing [Ara69].

Statistical mechanics assumes that a quantum many-body system at low temperature can be described by its Gibbs state. However, many complex quantum systems only exist as metastable states of dissipative open system dynamics, which substantially deviate from true thermal equilibrium. Why, then, should the predictions of thermal equilibrium--such as the area law--be so unreasonably effective in explaining low-temperature phenomena? In this work, we model metastable states as approximate stationary states of a quasi-local, (KMS)-detailed-balanced master equation representing Markovian system-bath interaction. We show that all metastable states exhibit universal structures that parallel true quantum Gibbs states: an area law of mutual information and a local Markov property. The more metastable the states are, the larger the regions to which these structural results apply. Behind our structural results lies a systematic framework encompassing sharp equivalences between local minima of free energy, a non-commutative Fisher information, as well as approximate detailed-balance and Kubo-Martin-Schwinger conditions, ultimately building towards a quantitative theory of thermal metastability.

Markov Chain Monte Carlo (MCMC) methods are an essential tool in classical algorithms design. Especially, the Metropolis sampling algorithm and Glauber dynamics have drastically advanced our understanding of material properties, reaction dynamics, phase transitions, and thermodynamics. Recently, there has been a new wave of quantum MCMC algorithms that draws inspiration from the cooling process in Nature to design continuous-time Quantum Markov chains (i.e., Lindbladians) satisfying (approximate) detailed balance. Nevertheless, the quantum analog of detailed balance, which has been central to classical Markov chain design and analysis, has posed a challenge to quantum algorithms design and has only recently been achieved exactly and (quasi)-locally for an efficiently implementable Lindbladian by [CKG23]. The construction of [CKG23] provably leads to an efficient Gibbs state preparation method in the high-temperature regime. However, proving fast mixing for low temperatures remains an open problem, apart from some (almost) integrable systems. Here we introduce (i) a new continuous-time Lindbladian construction that also leads to quasi-local and detailed-balanced dynamics, and (ii) show that it is fast mixing for high-temperature lattice Hamiltonians. The new construction's major advantage is that it does not increase the number of Kraus operators, which is particularly helpful for numerical studies. We exploit the resulting low Kraus rank through a (iii) novel custom variant of density matrix renormalization group (DMRG) for superoperators to provide numerical evidence for various 1D models (Transverse-field Ising, Heisenberg XXZ) that the Gibbs sampler is mixing fast. We also introduce (iv) new detailed-balanced discrete-time quantum channel variants of all existing continuous-time detailed-balanced Lindbladian construction and (v) show that they are also mixing fast at high-temperatures, and provide some preliminary (vi) resource estimates for their implementation confirming their algorithmic efficiency.