The propagation of information in nonrelativistic quantum systems obeys a speed limit known as a Lieb-Robinson bound.We derive a new Lieb-Robinson bound for systems with interactions that decay with distance r as a power law, 1=r α .The bound implies an effective light cone tighter than all previous bounds.Our approach is based on a technique for approximating the time evolution of a system, which was first introduced as part of a quantum simulation algorithm by Haah et al., FOCS'18.To bound the error of the approximation, we use a known Lieb-Robinson bound that is weaker than the bound we establish.This result brings the analysis full circle, suggesting a deep connection between Lieb-Robinson bounds and digital quantum simulation.In addition to the new Lieb-Robinson bound, our analysis also gives an error bound for the Haah et al. quantum simulation algorithm when used to simulate power-law decaying interactions.In particular, we show that the gate count of the algorithm scales with the system size better than existing algorithms when α > 3D (where D is the number of dimensions).
The propagation of information in nonrelativistic quantum systems obeys a speed limit known as a Lieb-Robinson bound.We derive a new Lieb-Robinson bound for systems with interactions that decay with distance r as a power law, 1=r α .The bound implies an effective light cone tighter than all previous bounds.Our approach is based on a technique for approximating the time evolution of a system, which was first introduced as part of a quantum simulation algorithm by Haah et al., FOCS'18.To bound the error of the approximation, we use a known Lieb-Robinson bound that is weaker than the bound we establish.This result brings the analysis full circle, suggesting a deep connection between Lieb-Robinson bounds and digital quantum simulation.In addition to the new Lieb-Robinson bound, our analysis also gives an error bound for the Haah et al. quantum simulation algorithm when used to simulate power-law decaying interactions.In particular, we show that the gate count of the algorithm scales with the system size better than existing algorithms when α > 3D (where D is the number of dimensions).
The Lieb-Robinson theorem states that information propagates with a finite velocity in quantum systems on a lattice with nearest-neighbor interactions. What are the speed limits on information propagation in quantum systems with power-law interactions, which decay as 1/r^{α} at distance r? Here, we present a definitive answer to this question for all exponents α>2d and all spatial dimensions d. Schematically, information takes time at least r^{min{1,α-2d}} to propagate a distance r. As recent state transfer protocols saturate this bound, our work closes a decades-long hunt for optimal Lieb-Robinson bounds on quantum information dynamics with power-law interactions.
Recent works have revealed the intricate effect of long-range interactions on information transport in quantum many-body systems: In D spatial dimensions, interactions decaying as a power-law r−α with α > 2D + 1 exhibit a Lieb-Robinson bound (LRB) with a linear light cone and the threshold 2D + 1 is sharp in general. Here, we observe that mutually commuting, long-range interactions satisfy an enhanced LRB of the form t r−α for any α > 0, and this scaling is sharp. In particular, the linear light cone occurs at α = 1 in any dimension. Part of our motivation stems from quantum error-correcting codes. As applications, we derive enhanced bounds on ground state correlations and an enhanced local perturbations perturb locally principle for which we adapt a recent subharmonicity argument of Wang-Hazzard. Similar enhancements hold for commuting interactions with stretched exponential decay.
The speed limit of information propagation is one of the most fundamental features in non-equilibrium physics. The region of information propagation by finite-time dynamics is approximately restricted inside the effective light cone that is formulated by the Lieb-Robinson bound. To date, extensive studies have been conducted to identify the shape of effective light cones in most experimentally relevant many-body systems. However, the Lieb-Robinson bound in the interacting boson systems, one of the most ubiquitous quantum systems in nature, has remained a critical open problem for a long time. This study reveals a tight effective light cone to limit the information propagation in interacting bosons, where the shape of the effective light cone depends on the spatial dimension. To achieve it, we prove that the speed for bosons to clump together is finite, which in turn leads to the error guarantee of the boson number truncation at each site. Furthermore, we applied the method to provide a provably efficient algorithm for simulating the interacting boson systems. The results of this study settle the notoriously challenging problem and provide the foundation for elucidating the complexity of many-body boson systems.
In this paper, we prove a family of Lieb-Robinson bounds for discrete spin systems with long-range interactions. Our results apply for arbitrary k-body interactions, so long as they decay with a power law greater than kd, where d is the dimension of the system. More precisely, we require that the sum of the norm of terms with diameter greater than or equal to R, acting on any one site, decays as a power law $1/{R}^{\ensuremath{\alpha}}$, with $\ensuremath{\alpha}>d$. These bounds allow us to prove that, at any fixed time, the spatial decay of a time evolved operator follows arbitrarily closely to $1/{r}^{\ensuremath{\alpha}}$. Moreover, we introduce an alternative light cone definition for power-law interacting quantum systems which captures the region of the system where changing the Hamiltonian can affect the evolution of a local operator. In short-range interacting systems, this light cone agrees with the conventional definition. However, in long-range interacting systems, our definition yields a stricter light cone, which is more relevant in most physical contexts.
The Lieb-Robinson (LR) bound rigorously shows that in quantum systems with short-range interactions, the maximum amount of information that travels beyond an effective "light cone" decays exponentially with distance from the light-cone front, which expands at finite velocity.Despite being a fundamental result, existing bounds are often extremely loose, limiting their applications.We introduce a method that dramatically and qualitatively improves LR bounds in models with finite-range interactions.Most prominently, in systems with a large local Hilbert space dimension D, our method gives a LR velocity that grows much slower than previous bounds with D as D → ∞.For example, in the Heisenberg model with spin S, we find v ≤ const.compared to the previous v ∝ S, which diverges at large S, and in multiorbital Hubbard models with N orbitals, we find v ∝ √ N instead of previous v ∝ N , and similarly in the Nstate truncated Bose-Hubbard model and Wen's quantum rotor model.Our bounds also scale qualitatively better in some systems when the spatial dimension or certain model parameters become large, for example in the d-dimensional quantum Ising model and perturbed toric code models.Even in spin-1/2 Ising and Fermi-Hubbard models, our method improves the LR velocity by an order of magnitude with typical model parameters, and significantly improves the LR bound at large distance and early time.
Recent works have revealed the intricate effect of long-range interactions on information transport in quantum many-body systems: In D spatial dimensions, interactions decaying as a power-law r−α with α>2D+1 exhibit a Lieb-Robinson bound (LRB) with a linear light cone and the threshold 2D+1 is sharp in general. Here, we observe that mutually commuting, long-range interactions satisfy an enhanced LRB of the form tr−α for any α>0, and this scaling is sharp. In particular, the linear light cone occurs at α=1 in any dimension. Part of our motivation stems from quantum error-correcting codes. As applications, we derive enhanced bounds on ground state correlations and an enhanced local perturbations perturb locally (LPPL) principle for which we adapt a recent subharmonicity argument of Wang-Hazzard. Similar enhancements hold for commuting interactions with stretched exponential decay.