Session X37: Lieb-Robinson Bounds in Systems with Power-Law Decaying Interactions

Finite speed of quantum scrambling with long range interactions

In a locally interacting many-body system, two isolated qubits, separated by a large distance r, become correlated and entangled with each other at a time t≥r/v. This finite speed v of quantum information scrambling limits quantum information processing, thermalization and even equilibrium correlations. Yet most experimental systems contain long range power law interactions -- qubits separated by r have potential energy V(r)∝r^(−α). Examples include the long range Coulomb interactions in plasma (α=1) and dipolar interactions between spins (α=3). In one spatial dimension, we prove that the speed of quantum scrambling remains finite for sufficiently large α. This result parametrically improves previous bounds, compares favorably with recent numerical simulations, and can be realized in quantum simulators with dipolar interactions. Our new mathematical methods lead to improved algorithms for classically simulating quantum systems, and improve bounds on environmental decoherence in experimental quantum information processors.   

Emergent locality in systems with power-law interactions

In generic systems with local interactions transport is diffusive and information propagates inside a quasicausal "light-cone" parametrized by a constant velocity known as the Lieb-Robinson velocity. Introducing long-range interactions, should intuitively enhance transport by long-range hops, and deform the linear "light-cones," into a superballistic form. Using two numerically exact techniques I will show that this is not the case for a number of generic one-dimensional systems. All studied systems, for sufficiently short-range interactions, show universal behaviour of asymptotically emergent locality and a unique composite transport comprised of diffusive and superdiffusive features.   

Locality and Digital Quantum Simulation of Power-Law Interactions

The propagation of information in non-relativistic quantum systems obeys a speed limit known as a Lieb-Robinson bound. We show that Lieb-Robinson bounds provide tighter estimates for the gate count in quantum algorithms for simulating power-law interacting Hamiltonians. In addition, we prove new Lieb-Robinson bounds using techniques from quantum simulation algorithms. This brings the analysis full circle, revealing a deep connection between Lieb-Robinson bounds and digital quantum simulation.   

Some Aspects of Quantum Dynamics in Many-Body Systems with Power-Law Interactions

In this talk, I will discuss two recent works on dynamics of quantum systems with power-law interactions.

In the first half, I describe a proof of a new family of Lieb-Robinson bounds for lattice spin systems with long-range interactions. Our results apply for arbitrary k-body interactions, so long as they decay as a sufficiently rapid power-law with distance. I will discuss what is appropriate notion of light cone in power-law interacting systems, and compare our results with previous works in this light.

In the second half, I describe a proof of the existence of non-equilibrium phases of matter in the prethermal regime of periodically-driven, power-law interacting systems. In this context, we predict the existence of a disorder-free, prethermal discrete time crystal in one dimension -- a phase strictly forbidden for short-range interactions.   

Signaling and scrambling for strongly long-range interacting quantum systems

Strongly long-range interacting quantum systems—those with interactions decaying as a power-law 1/r^α in the distance r on a D-dimensional lattice for α ≤ D—have received significant interest in recent years. They are present in leading experimental platforms for quantum computation and simulation, as well as in theoretical models of quantum information scrambling and fast entanglement creation. Since no notion of locality is expected in such systems, a general understanding of their dynamics is lacking. As a first step towards rectifying this problem, we prove two new Lieb-Robinson-type bounds that constrain the time for signaling and scrambling in strongly long-range interacting systems, for which no tight bounds were previously known. Our first bound applies to systems mappable to free-particle Hamiltonians with long-range hopping, and is saturable for α ≤ D/2. Our second bound pertains to generic long-range interacting spin Hamiltonians, and leads to a tight lower bound for the signaling time to extensive subsets of the system for all α < D. This result also lower-bounds the scrambling time, and suggests a path towards achieving a tight scrambling bound that can prove the long-standing fast scrambling conjecture.