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We prove an upper bound on the diffusivity of a general local and translation invariant quantum Markovian spin system: $D leq D_0 + left(alpha , v_text{LR} tau + beta , xi right) v_text{C}$. Here $v_text{LR}$ is the Lieb-Robinson velocity, $v_text{C}$ is a velocity defined by the current operator, $tau$ is the decoherence time, $xi$ is the range of interactions, $D_0$ is a microscopically determined diffusivity and $alpha$ and $beta$ are precisely defined dimensionless coefficients. The bound constrains quantum transport by quantities that can either be obtained from the microscopic interactions ($D_0, v_text{LR}, v_text{C},xi$) or else determined from independent local non-transport measurements ($tau,alpha,beta$). We illustrate the general result with the case of a spin half XXZ chain with on-site dephasing. Our result generalizes the Lieb-Robinson bound to constrain the sub-ballistic diffusion of conserved densities in a dissipative setting.
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We investigate the time evolution of an open quantum system described by a Lindblad master equation with dissipation acting only on a part of the degrees of freedom ${cal H}_0$ of the system, and targeting a unique dark state in ${cal H}_0$. We show that, in the Zeno limit of large dissipation, the density matrix of the system traced over the dissipative subspace ${cal H}_0$, evolves according to another Lindblad dynamics, with renormalized effective Hamiltonian and weak effective dissipation. This behavior is explicitly checked in the case of Heisenberg spin chains with one or both boundary spins strongly coupled to a magnetic reservoir. Moreover, the populations of the eigenstates of the renormalized effective Hamiltonian evolve in time according to a classical Markov dynamics. As a direct application of this result, we propose a computationally-efficient exact method to evaluate the nonequilibrium steady state of a general system in the limit of strong dissipation.
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.
Microscopic speed limits that constrain the motion of matter, energy, and information abound in physics, from the ultimate speed limit set by light to Lieb-Robinson speed limits in quantum spin systems. In addition to these state-independent speed limits, systems can also be governed by emergent state-dependent speed limits indicating slow dynamics arising, for example, from slow low-energy quasiparticles. Here we describe a different kind of speed limit: a situation where complex information/entanglement spreads rapidly, in a fashion inconsistent with any speed limit, but where simple signals continue to obey an approximate speed limit. If we take the point of view that the motion of simple signals defines the local spacetime geometry of the universe, then the effects we describe show that spacetime locality can be compatible with a high degree of non-local interactions provided these are sufficiently chaotic. With this perspective, we sharpen a puzzle about black holes recently raised by Shor and propose a schematic resolution.
Modeling the dynamics of a quantum system connected to the environment is critical for advancing our understanding of complex quantum processes, as most quantum processes in nature are affected by an environment. Modeling a macroscopic environment on a quantum simulator may be achieved by coupling independent ancilla qubits that facilitate energy exchange in an appropriate manner with the system and mimic an environment. This approach requires a large, and possibly exponential number of ancillary degrees of freedom which is impractical. In contrast, we develop a digital quantum algorithm that simulates interaction with an environment using a small number of ancilla qubits. By combining periodic modulation of the ancilla energies, or spectral combing, with periodic reset operations, we are able to mimic interaction with a large environment and generate thermal states of interacting many-body systems. We evaluate the algorithm by simulating preparation of thermal states of the transverse Ising model. Our algorithm can also be viewed as a quantum Markov chain Monte Carlo (QMCMC) process that allows sampling of the Gibbs distribution of a multivariate model. To demonstrate this we evaluate the accuracy of sampling Gibbs distributions of simple probabilistic graphical models using the algorithm.
Open quantum systems can exhibit complex states, which classification and quantification is still not well resolved. The Kerr-nonlinear cavity, periodically modulated in time by coherent pumping of the intra-cavity photonic mode, is one of the examples. Unraveling the corresponding Markovian master equation into an ensemble of quantum trajectories and employing the recently proposed calculation of quantum Lyapunov exponents [I.I. Yusipov {it et al.}, Chaos {bf 29}, 063130 (2019)], we identify `chaotic and `regular regimes there. In particular, we show that chaotic regimes manifest an intermediate power-law asymptotics in the distribution of photon waiting times. This distribution can be retrieved by monitoring photon emission with a single-photon detector, so that chaotic and regular states can be discriminated without disturbing the intra-cavity dynamics.
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