Purification and uniqueness of quantum Gibbs states

Yong‐Moon Park, Hyun Jae Yoo

Igor D. Novikov

Stephen Dias Barreto

In this paper we study a quantum spin glass [3,17] as a quantum spin system on an infinite graph. We construct the global dynamics for the quantum spin system on an infinite graph with deterministic nearest neighbour type of interactions. As expected, at a fixed temperature T , the equilibrium state [6,7,12] satisfies the kms [14,18,12] condition and is invariant under the action of the time evolution group. Furthermore, it is shown that the equilibrium state is unique if additional assumptions are made regarding the structure of the graph. Such results have been proved in the past mainly for translation invariant interactions on lattices, see [1,2,8,13]. More recently Matsui has established the uniqueness of the the Gibbs States for quantum spin models on lattices based on the ergodicity of Markov semigroups through the purification of quantum Gibbs states [10]. In our case the nearest neighbour type of interaction we have considered is more general than similar type of interactions considered in some earlier works referred to above.

Cambyse Rouzé, Daniel Stilck França, Alvaro Martin Alhambra

The preparation of quantum Gibbs states is a crucial task in quantum computing. In this work, we prove that a recently introduced, efficiently implementable dissipative evolution thermalizes to the Gibbs state in time scaling polynomially or even logarithmically with system size at high enough temperatures for any Hamiltonian that satisfies a Lieb-Robinson bound, such as local Hamiltonians on a lattice. Furthermore, we show the efficient adiabatic preparation of the associated purifications or "thermofield double"states. To the best of our knowledge, these are the first results rigorously establishing the efficient preparation of high-temperature Gibbs states and their purifications. In the low-temperature regime, we show that implementing this family of dissipative evolutions for inverse temperatures polynomial in the system's size is computationally equivalent to standard quantum computations. On a technical level, for high temperatures, our proof makes use of the mapping of the generator of the evolution into a Hamiltonian, and then connecting its convergence to that of the infinite temperature limit. We further present an alternative proof that is based on showing the exponential decay of the so-called oscillator norm, yielding convergence in logarithmic times. For low temperature, we instead perform a perturbation at zero temperature and resort to circuit-to-Hamiltonian mappings akin to the proof of universality of quantum adiabatic computing. Taken together, our results show that a family of quasi-local dissipative evolutions efficiently prepares a large class of quantum many-body states of interest, and has the potential to mirror the success of classical Monte Carlo methods for quantum many-body systems.

Vittorio Gorini, Alberto Frigerio, Maurizio Verri

Ingemar Bengtsson, Karol Życzkowski