A state on a tripartite quantum system A⊗B⊗C forms a Markov chain if it can be reconstructed from its marginal on A⊗B by a quantum operation from B to B⊗C . We show that the quantum conditional mutual information I(A : C|B) of an arbitrary state is an upper bound on its distance to the closest reconstructed state. It thus quantifies how well the Markov chain property is approximated.
In this work we investigate a model of thermalization wherein a single ancillary qubit randomly interacts with the system to be thermalized. This not only sheds light on the emergence of Gibbs states in nature, but also provides a routine for preparing arbitrary thermal states on a digital quantum computer. For desired β and random interaction G the routine boils down to time independent Hamiltonian simulation and is represented by the channel Φ:ρ↦EGTrEnv[e−i(H+αG)t(ρ⊗Ze−βHE)ei(H+αG)t]. We rigorously prove that these dynamics reduce to a Markov chain process in the weak-coupling regime with the thermal state as the approximate fixed point. We upper bound the total simulation time required in terms of the Markov chain spectral gap λ⋆, which we compute exactly in the ground state limit. These results are independent of any eigenvalue knowledge of the system, but we are further able to show that with knowledge of eigenvalue differences λS(i)−λS(j), then the total simulation time is dramatically reduced. The ratio of the complete ignorance simulation cost to the perfect knowledge simulation cost scales as O(δmin7ε3.5λ⋆(β)3.5∥HS∥7), where δmin is related to the eigenvalue differences of the system. Additionally, we provide more specific results for single qubit and harmonic oscillator systems as well as numeric experiments with hydrogen chains. In addition to the algorithmic merits, these results can be viewed as broad extensions of the Repeated Interactions model to generic Hamiltonians with unknown interactions, giving a complete picture of the thermalization process for quantum systems.
We construct a simple quantum version of the classical Metropolis algorithm to prepare and observe quantum thermal states. It induces both a quantum Markov chain that mixes the quantum thermal state and a classical Markov chain that mixes its observable measurements and enables a low-depth quantum circuit implementation. A Gaussian-filtered variant of quantum phase estimation enables thermalization times proportional to the reciprocal of temperature and the logarithm of biasing error. This matches the thermalization time of imaginary-time evolution, against which our algorithm performs favorably.