Quantum Gibbs Samplers: The Commuting Case

We analyze the problem of preparing quantum Gibbs states of lattice spin Hamiltonians with local and commuting terms on a quantum computer and in nature. Our central result is an equivalence between the behavior of correlations in the Gibbs state and the mixing time of the semigroup which drives the system to thermal equilibrium (the Gibbs sampler). We introduce a framework for analyzing the correlation and mixing properties of quantum Gibbs states and quantum Gibbs samplers, which is rooted in the theory of non-commutative \({\mathbb{L}_p}\) spaces. We consider two distinct classes of Gibbs samplers, one of them being the well-studied Davies generator modelling the dynamics of a system due to weak-coupling with a large Markovian environment. We show that their spectral gap is independent of system size if, and only if, a certain strong form of clustering of correlations holds in the Gibbs state. Therefore every Gibbs state of a commuting Hamiltonian that satisfies clustering of correlations in this strong sense can be prepared efficiently on a quantum computer. As concrete applications of our formalism, we show that for every one-dimensional lattice system, or for systems in lattices of any dimension at temperatures above a certain threshold, the Gibbs samplers of commuting Hamiltonians are always gapped, giving an efficient way of preparing the associated Gibbs states on a quantum computer.     

Hydrogenation of

The course of the hydrogenation of [5]- and [6]metacyclophane (1b and 1c) and their thermochemistry is described. Both compounds are hydrogenated rapidly (within 10 s) to furnish the bridgehead olefins 13b and 12c. The accompanying hydrogenation enthalpies are -220 and -141 kJmol(-1), respectively. Strain energies (SE) and olefinic strains (OS) of a number of bridgehead olefins have been evaluated by DFT calculations; it was concluded that 13b belongs to the class of hyperstable olefins which correlates nicely with its reluctance to undergo hydrogenation. By combining experimental hydrogenation enthalpies and DFT calculations, SE of 187 and 121 kJmol(-1) were derived for 1b and 1c.     

Dissipative quantum Church-Turing theorem

We show that the time evolution of an open quantum system, described by a possibly time dependent Liouvillian, can be simulated by a unitary quantum circuit of a size scaling polynomially in the simulation time and the size of the system. An immediate consequence is that dissipative quantum computing is no more powerful than the unitary circuit model. Our result can be seen as a dissipative Church-Turing theorem, since it implies that under natural assumptions, such as weak coupling to an environment, the dynamics of an open quantum system can be simulated efficiently on a quantum computer. Formally, we introduce a Trotter decomposition for Liouvillian dynamics and give explicit error bounds. This constitutes a practical tool for numerical simulations, e.g., using matrix-product operators. We also demonstrate that most quantum states cannot be prepared efficiently.