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arXiv:2307.13576 · Physical Review A

Simulating quantum transport via collisional models on a digital quantum computer

Rebecca Erbanni, Xiansong Xu, Tommaso F. Demarie, Dario Poletti

Published Sep 28, 2023• 9 citations
arXivHTMLPDF

Abstract

Digital quantum computers have the potential to study the dynamics of many-body quantum systems. Nonequilibrium open quantum systems are, however, less straightforward to be implemented. Here we explore the feasibility of studying steady-state transport in strongly interacting many-body quantum systems on a digital quantum computer. To do so, we consider a collisional model representation of the nonequilibrium open dynamics for a boundary-driven $XXZ$ spin chain. More specifically, we investigate how the depth of the quantum circuit is affected by how close we want the steady state to be to the one expected from the underlying master equation. We study the simulation of a boundary-driven spin chain in regimes of weak and strong interactions, which would lead in large systems to diffusive and ballistic dynamics, considering also possible errors in the implementation of the protocol. Last, we analyze the effectiveness of digital simulation via the collisional model of current rectification when the $XXZ$ spin chains are subject to nonuniform magnetic fields and show that, although the circuit depths required to reach steady states are still prohibitive for today's hardware, few collisions are enough to suggest a strong rectifying power.

DOI
10.1103/physreva.108.032619

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Digital quantum computers have the potential to study the dynamics of many-body quantum systems. Nonequilibrium open quantum systems are, however, less straightforward to be implemented. Here we explore the feasibility of studying steady-state transport in strongly interacting many-body quantum systems on a digital quantum computer. To do so, we consider a collisional model representation of the nonequilibrium open dynamics for a boundary-driven XXZXXZXXZ spin chain. More specifically, we investigate how the depth of the quantum circuit is affected by how close we want the steady state to be to the one expected from the underlying master equation. We study the simulation of a boundary-driven spin chain in regimes of weak and strong interactions, which would lead in large systems to diffusive and ballistic dynamics, considering also possible errors in the implementation of the protocol. Last, we analyze the effectiveness of digital simulation via the collisional model of current rectification when the XXZXXZXXZ spin chains are subject to nonuniform magnetic fields and show that, although the circuit depths required to reach steady states are still prohibitive for today's hardware, few collisions are enough to suggest a strong rectifying power.
State-of-the-art approaches to extract transport coefficients of many-body quantum systems broadly fall into two categories: (i) they target the linear-response regime in terms of equilibrium correlation functions of the closed system; or (ii) they consider an open-system situation typically modeled by a Lindblad equation, where a nonequilibrium steady state emerges from driving the system at its boundaries. While quantitative agreement between (i) and (ii) has been found for selected model and parameter choices, also disagreement has been pointed out in the literature. Studying magnetization transport in the spin-1/21/21/2 XXZ chain, we here demonstrate that at weak driving, the nonequilibrium steady state in an open system, including its buildup in time, can remarkably be constructed just on the basis of correlation functions in the closed system. We numerically illustrate this direct correspondence of closed-system and open-system dynamics, and show that it allows the treatment of comparatively large open systems, usually only accessible to matrix product state simulations. We also point out potential pitfalls when extracting transport coefficients from nonequilibrium steady states in finite systems.
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Understanding the transport properties of quantum many-body systems is a central challenge in condensed matter and statistical physics. Theoretical studies usually rely on two main approaches: Dynamics of linear-response functions in closed systems and boundary-driven dynamics governed by Markovian master equations for open systems. While the equivalence of their dynamical behavior has been explored in recent studies, a systematic comparison of the transport coefficients obtained from these two classes of approaches remains a largely open question. Here, we address this gap by comparing and contrasting the dc diffusion constant Ddc\mathcal{D}_{\text{dc}}Ddc​ according to the two approaches, focusing on the specific example of magnetization transport in the spin-1/21/21/2 XXZ chain. Using exact numerical simulations for finite system sizes, we find (i) a clear mismatch between the two Ddc\mathcal{D}_{\text{dc}}Ddc​ and (ii) a strong dependence of Ddc\mathcal{D}_{\text{dc}}Ddc​ on the system-bath coupling strength for the open system, where neither (i) nor (ii) tend to vanish in the thermodynamic limit. These findings suggest limitations of the open-system approach to transport coefficients. To gain insight into the origin of (i) and (ii), we go beyond Ddc\mathcal{D}_{\text{dc}}Ddc​ and analyze the full time dependence of the diffusion coefficient D(t)D(t)D(t) in the open system. In this way, we find that both (i) and (ii) vanish up to a finite time scale. While this time scale gradually increases with system size and tends to be macroscopic in the thermodynamic limit, this increase is still slow compared to the increase of the time to reach the steady state, where (i) and (ii) do not vanish. This observation can be seen as a wrong, yet unavoidable order of limits of long times first and large system sizes afterwards.
Digital quantum computers offer a promising route for studying complex many-body systems that are otherwise inaccessible by their classical counterparts. Capabilities including mid-circuit measurements and feedback allow for simulating the dynamics of interacting open quantum systems. Using the Quantinuum System Model H1 trapped-ion quantum computer, we experimentally realise quantum trajectories for a two-dimensional system of (interacting) particles-hard-core bosons or fermions-undergoing stochastic driving at a source and drain at opposite corners of a square lattice. We study the non-equilibrium steady state with persistent current resulting from the this in/out flow of particles. The particle statistics, presence of interactions, and introduction of a magnetic field produce measurable effects on the steady state. Our findings highlight the rich physics in this corner driven two-dimensional setup and showcases both the power and current limitations of quantum computers as a platform to study it.
Abstract Simulating charge and energy transfer in extended molecular networks requires an effective model to include the environment because it significantly affects the quantum dynamics. A prototypical effect known as environment-assisted quantum transport (ENAQT) consists in the enhancement of the transfer efficiency by the interaction with an environment. A simple description of this phenomenon is obtained by a quantum master equation describing a quantum walk over the molecular network in the presence of inter-site decoherence. We consider the problem of simulating the dynamics underlying ENAQT in a digital quantum computer. Two different quantum algorithms are introduced, the first one based on stochastic Hamiltonians and the second one based on a collision scheme. We test both algorithms by simulating ENAQT in a small molecular network on a quantum computer emulator and provide a comparative analysis of the two approaches. Both algorithms can be implemented in a memory efficient encoding with the number of required qubits scaling logarithmically with the size of the simulated system while the number of gates increases quadratically. We discuss the algorithmic quantum trajectories generated by the two simulation strategies showing that they realize distinct unravellings of the site-dephasing master equation. In our approach, the non-unitary dynamics of the open system is obtained through effective representations of the environment, paving the way to digital quantum simulations of quantum transport influenced by structured environments.
In the study of quantum transport, much has been known about dynamics near thermal equilibrium. However, quantum transport far away from equilibrium is much less well understood---the linear response approximation does not hold for physics far out of equilibrium in general. In this work, motivated by recent cold atom experiments on probing quantum many-body dynamics of a one-dimensional XXZ spin chain, where a transition from ballistic to diffusive dynamics has been established by increasing the interaction strengths, we study the strong interaction limit of the one-dimensional spinless fermion model, which is dual to the XXZ spin chain. We develop a highly efficient computation algorithm for simulating the nonequilibrium dynamics of this system exactly, and examine the nonequilibrium dynamics starting from a density modulation quantum state. We find ballistic transport in this strongly correlated setting and show that a plane-wave description emerges at long-time evolution. We also observe a sharp distinction between transport velocities in short and long times as induced by interaction effects and provide a quantitative interpretation for the long-time transport velocity. We expect our results to shed light on the understanding of the dynamics of the XXZ spin chain in the strong interaction regime.