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Register a new userImproved thermal area law and quasi-linear time algorithm for quantum Gibbs states
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One of the most fundamental problems in quantum many-body physics is the characterization of correlations among thermal states. Of particular relevance is the thermal area law, which justifies the tensor network approximations to thermal states with a bond dimension growing polynomially with the system size. In the regime of sufficiently low temperatures, which is particularly important for practical applications, the existing techniques do not yield optimal bounds. Here, we propose a new thermal area law that holds for generic many-body systems on lattices. We improve the temperature dependence from the original $mathcal{O}(beta)$ to $tilde{mathcal{O}}(beta^{2/3})$, thereby suggesting diffusive propagation of entanglement by imaginary time evolution. This qualitatively differs from the real-time evolution which usually induces linear growth of entanglement. We also prove analogous bounds for the Renyi entanglement of purification and the entanglement of formation. Our analysis is based on a polynomial approximation to the exponential function which provides a relationship between the imaginary-time evolution and random walks. Moreover, for one-dimensional (1D) systems with $n$ spins, we prove that the Gibbs state is well-approximated by a matrix product operator with a sublinear bond dimension of $e^{sqrt{tilde{mathcal{O}}(beta log(n))}}$. This proof allows us to rigorously establish, for the first time, a quasi-linear time classical algorithm for constructing an MPS representation of 1D quantum Gibbs states at arbitrary temperatures of $beta = o(log(n))$. Our new technical ingredient is a block decomposition of the Gibbs state, that bears resemblance to the decomposition of real-time evolution given by Haah et al., FOCS18.
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The extension of many-body quantum dynamics to the non-unitary domain has led to a series of exciting developments, including new out-of-equilibrium entanglement phases and phase transitions. We show how a duality transformation between space and time on one hand, and unitarity and non-unitarity on the other, can be used to realize steady state phases of non-unitary dynamics that exhibit a rich variety of behavior in their entanglement scaling with subsystem size -- from logarithmic to extensive to emph{fractal}. We show how these outcomes in non-unitary circuits (that are spacetime-dual to unitary circuits) relate to the growth of entanglement in time in the corresponding unitary circuits, and how they differ, through an exact mapping to a problem of unitary evolution with boundary decoherence, in which information gets radiated away from one edge of the system. In spacetime-duals of chaotic unitary circuits, this mapping allows us to uncover a non-thermal volume-law entangled phase with a logarithmic correction to the entropy distinct from other known examples. Most notably, we also find novel steady state phases with emph{fractal} entanglement scaling, $S(ell) sim ell^{alpha}$ with tunable $0 < alpha < 1$ for subsystems of size $ell$ in one dimension. These fractally entangled states add a qualitatively new entry to the families of many-body quantum states that have been studied as energy eigenstates or dynamical steady states, whose entropy almost always displays either area-law, volume-law or logarithmic scaling. We also present an experimental protocol for preparing these novel steady states with only a very limited amount of postselection via a type of teleportation between spacelike and timelike slices of quantum circuits.
We develop two cutting-edge approaches to construct deep neural networks representing the purified finite-temperature states of quantum many-body systems. Both methods commonly aim to represent the Gibbs state by a highly expressive neural-network wave function, exemplifying the idea of purification. The first method is an entirely deterministic approach to generate deep Boltzmann machines representing the purified Gibbs state exactly. This strongly assures the remarkable flexibility of the ansatz which can fully exploit the quantum-to-classical mapping. The second method employs stochastic sampling to optimize the network parameters such that the imaginary time evolution is well approximated within the expressibility of neural networks. Numerical demonstrations for transverse-field Ising models and Heisenberg models show that our methods are powerful enough to investigate the finite-temperature properties of strongly correlated quantum many-body systems, even when the problematic effect of frustration is present.
Classical chimera states are paradigmatic examples of partial synchronization patterns emerging in nonlinear dynamics. These states are characterized by the spatial coexistence of two dramatically different dynamical behaviors, i.e., synchronized and desynchronized dynamics. Our aim in this contribution is to discuss signatures of chimera states in quantum mechanics. We study a network with a ring topology consisting of N coupled quantum Van der Pol oscillators. We describe the emergence of chimera-like quantum correlations in the covariance matrix. Further, we establish the connection of chimera states to quantum information theory by describing the quantum mutual information for a bipartite state of the network.
We study measures of decoherence and thermalization of a quantum system $S$ in the presence of a quantum environment (bath) $E$. The entirety $S$$+$$E$ is prepared in a canonical thermal state at a finite temperature, that is the entirety is in a steady state. Both our numerical results and theoretical predictions show that measures of the decoherence and the thermalization of $S$ are generally finite, even in the thermodynamic limit, when the entirety $S$$+$$E$ is at finite temperature. Notably, applying perturbation theory with respect to the system-environment coupling strength, we find that under common Hamiltonian symmetries, up to first order in the coupling strength it is sufficient to consider $S$ uncoupled from $E$, but entangled with $E$, to predict decoherence and thermalization measures of $S$. This decoupling allows closed form expressions for perturbative expansions for the measures of decoherence and thermalization in terms of the free energies of $S$ and of $E$. Large-scale numerical results for both coupled and uncoupled entireties with up to 40 quantum spins support these findings.
Preparation of Gibbs distributions is an important task for quantum computation. It is a necessary first step in some types of quantum simulations and further is essential for quantum algorithms such as quantum Boltzmann training. Despite this, most methods for preparing thermal states are impractical to implement on near-term quantum computers because of the memory overheads required. Here we present a variational approach to preparing Gibbs states that is based on minimizing the free energy of a quantum system. The key insight that makes this practical is the use of Fourier series approximations to the logarithm that allows the entropy component of the free-energy to be estimated through a sequence of simpler measurements that can be combined together using classical post processing. We further show that this approach is efficient for generating high-temperature Gibbs states, within constant error, if the initial guess for the variational parameters for the programmable quantum circuit are sufficiently close to a global optima. Finally, we examine the procedure numerically and show the viability of our approach for five-qubit Hamiltonians using Trotterized adiabatic state preparation as an ansatz.
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