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Physical review. B./Physical review. B

Dissipation-assisted operator evolution method for capturing hydrodynamic transport

Tibor Rakovszky, Curt W. von Keyserlingk, Frank Pollmann

Published Feb 16, 2022• 105 citations
Publisher

Abstract

Transport properties are among the most practically important features of materials, yet their precise calculation in strongly interacting systems poses a formidable challenge. One might even expect that it requires a quantum computer. The authors introduce an algorithm that tackles this challenge with only moderate resources on a classical computer. It relies on introducing an artificial dissipation process that only affects nonlocal observables, thus reducing the computational cost without strongly influencing physically relevant quantities. The algorithm is tested on various paradigmatic models with good convergence results.

DOI
10.1103/physrevb.105.075131

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Transport properties are among the most practically important features of materials, yet their precise calculation in strongly interacting systems poses a formidable challenge. One might even expect that it requires a quantum computer. The authors introduce an algorithm that tackles this challenge with only moderate resources on a classical computer. It relies on introducing an artificial dissipation process that only affects nonlocal observables, thus reducing the computational cost without strongly influencing physically relevant quantities. The algorithm is tested on various paradigmatic models with good convergence results.
Using artificial dissipation to tame entanglement growth, we chart the emergence of diffusion in a generic interacting lattice model for varying U(1) charge densities. We follow the crossover from ballistic to diffusive transport above a scale set by the scattering length, finding the intuitive result that the diffusion constant scales as D ∝ 1 / ρ at low densities ρ . Our numerical approach generalizes the Dissipation-Assisted Operator Evolution algorithm: in the spirit of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy, we effectively approximate nonlocal operators by their ensemble averages, rather than discarding them entirely. This greatly reduces the operator entanglement entropy, while still giving accurate predictions for diffusion constants across all density scales. We further construct a minimal model for the transport crossover, yielding charge correlation functions which agree well with our numerical data. Our results clarify the dominant contributions to hydrodynamic correlation functions of conserved densities, and serve as a guide for generalizations to low-temperature transport.
Tensor product states have proved extremely powerful for simulating the area-law entangled states of many-body systems, such as the ground states of gapped Hamiltonians in one dimension. The applicability of such methods to the dynamics of many-body systems is less clear: The memory required grows exponentially in time in most cases, quickly becoming unmanageable. New methods reduce the memory required by selectively discarding/dissipating parts of the many-body wave function which are expected to have little effect on the hydrodynamic observables typically of interest: For example, some methods discard fine-grained correlations associated with nnn-point functions, with nnn exceeding some cutoff ${\ensuremath{\ell}}_{*}$. In this paper, we present a theory for the sizes of backflow corrections, i.e., systematic errors due to discarding this fine-grained information. In particular, we focus on their effect on transport coefficients. Our results suggest that backflow corrections are exponentially suppressed in the size of the cutoff ${\ensuremath{\ell}}_{*}$. Moreover, the backflow errors themselves have a hydrodynamical expansion, which we elucidate. We test our predictions against numerical simulations run on random unitary circuits and ergodic spin chains. These results lead to the conjecture that transport coefficients in ergodic diffusive systems can be captured to a given precision $\ensuremath{\epsilon}$ with an amount of memory scaling as $exp[\mathcal{O}(log{(\ensuremath{\epsilon})}^{2})]$, significantly better than the naive estimate of memory $exp[\mathcal{O}(\mathrm{poly}({\ensuremath{\epsilon}}^{\ensuremath{-}1}))]$ required by more brute-force methods.
We introduce the dissipation-assisted operator evolution (DAOE) method for calculating transport properties of strongly interacting lattice systems in the high temperature regime. DAOE is based on evolving observables in the Heisenberg picture, and applying an artificial dissipation that reduces the weight on non-local operators. We represent the observable as a matrix product operator, and show that the dissipation leads to a decay of operator entanglement, allowing us to capture the dynamics to long times. We test this scheme by calculating spin and energy diffusion constants in a variety of physical models. By gradually weakening the dissipation, we are able to consistently extrapolate our results to the case of zero dissipation, thus estimating the physical diffusion constant with high precision.
The real world obeys quantum physics and quantum computing presents an alternative way to map physical problems to systems that follow the same laws. Such computation fundamentally constitutes a better way to understand the most challenging quantum problems. One such problem is the accurate simulation of highly correlated quantum systems. Due to the high dimensionality of the problem classical computers require considerable computer power to accurately predict material properties, especially when strong electron interactions are present. Still, modern day quantum hardware has many limitations and only allows for modeling of very simple systems. Here we present for the first time a quantum computer model simulation of a complex hemocyanin molecule, which is an important respiratory protein involved in various physiological processes such as oxygen transport and immune defence, and is also used as a key component in therapeutic vaccines for cancer. To better characterise the mechanism by which hemocyanin transports oxygen, variational quantum eigensolver (VQE) based on fermionic excitations and quantum embedding methods is used in the context of dynamic mean field theory to solve Anderson impurity model (AIM). Finally, it is concluded that the magnetic structure of hemocyanin is largely influenced by the many-body correction and that the computational effort for solving correlated electron systems could be substantially reduced with the introduction of quantum computing algorithms. We encourage the use of the Hamiltonian systems presented in this paper as a benchmark for testing quantum computing algorithms efficiency for chemistry applications.