Algorithms 2

Abstract: We introduce the notion of polynomial-depth duality transformations, which relates two sets of operator algebras through a conjugation by a poly-depth quantum circuit, and make use of this to construct efficient Gibbs samplers for a variety of interesting quantum Hamiltonians as they are poly-depth dual to classical Hamiltonians. This is for example the case for the 2D toric code, which is demonstrated to be poly-depth dual to two decoupled classical Ising spin chains for any system size, and we give evidence that such dualities hold for a wide class of stabilizer Hamiltonians. Additionally, we extend the above notion of duality to Lindbladians in order to show that mixing times and other quantities such as the spectral gap or the modified logarithmic Sobolev inequality are preserved under duality.

Abstract: Despite extensive study, our understanding of quantum Markov chains remains far less complete than that of their classical counterparts. [Temme'13] observed that the Davies Lindbladian, a well-studied model of quantum Markov dynamics, contains an embedded classical Markov generator, raising the natural question of how the convergence properties of the quantum and classical dynamics compare. While [Temme'13] showed that the spectral gap of the Davies Lindbladian can be exponentially smaller than that of the embedded classical generator for certain highly structured Hamiltonians, we show that if the spectrum of the Hamiltonian does not contain long arithmetic progressions, then the two spectral gaps must be comparable. As a consequence, we prove that for a large class of Hamiltonians, including those obtained by perturbing a fixed Hamiltonian with a generic external field, the quantum spectral gap remains within a constant factor of the classical spectral gap. Our result aligns with physical intuition and enables the application of classical Markov chain techniques to the quantum setting. The proof is based on showing that any ``off-diagonal'' eigenvector of the Davies generator can be used to construct an observable which commutes with the Hamiltonian and has a Lindbladian Rayleigh quotient which is comparably small. Thus, a spectral gap for such observables implies a spectral gap for the full Davies generator.

Abstract: We present a novel approach to quantizing Markov chains. The approach is based on the Markov chain coupling method, which is frequently used to prove fast mixing. Given a particular coupling, e.g., a grand coupling, we construct a completely positive and trace preserving map. This quantum map has a unique fixed point, which corresponds to the quantum sample (qsample) of the classical Markov chain's stationary distribution. We show that the convergence time of the quantum map is directly related to the coupling time of the Markov chain coupling.

Abstract: 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.