Locality and Digital Quantum Simulation of Power-Law Interactions

Abstract

The propagation of information in non-relativistic 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).

Finite Speed of Quantum Scrambling with Long Range Interactions

“…This result parametrically improves previous bounds [4][5][6][7], compares favorably with recent numerical simulations [8,9], and can be realized in quantum simulators with dipolar interactions [10,11]. Our new mathematical methods lead to improved algorithms for classically simulating quantum systems [6,12], and improve bounds on environmental decoherence in experimental quantum information processors.Almost five decades ago, Lieb and Robinson proved that spatial locality implies the ballistic propagation of quantum information [1]. Intuitively defining a "scrambling time" t s (r) by the time at which an initially isolated qubit can significantly entangle with another a distance r away, locality implies that t s (r)r.…”

“…Our dramatic improvement over existing results is made possible by new mathematics [16]: identities for unitary time evolution expanded as a sum over flexibly chosen equivalence classes of sequences of couplings.Our work has clear physical consequences. Scrambling in dipolar spin chains is hardly faster than in a spin chain with nearest neighbor interactions; hence, it should be far more efficient to simulate numerically [6,12]. Nor does decoherence seriously limit the quantum information processing capabilities of a nuclear spin chain, no matter how large the environment.…”

“…Such objects generically interact with long range interactions, and until now, whether locality of quantum scrambling necessarily persists in the presence of long range interactions has remained unclear. In 2005, Hastings and Koma used the canonical Lieb-Robinson theorem to prove that when α > d, t s (r) log r [4]; more recently, this bound has been improved for α > 2d to t s (r) r (α−2d)/(α−d) [6]. If such bounds were tight, then insulating a quantum processor from its environment would be absolutely crucial.…”

Nonlocal emergent hydrodynamics in a long-range quantum spin system

“…In particular, it conserves the total S z magnetization, with product states in the S z basis evolving radically differently depending on the complexity of the corresponding magnetization sector. For just a few spin flips on top of the completely polarized state, the dynamics can be exactly solved and are described in terms of ballistically propagating spin waves, with a diverging group velocity at α = 1 [24,26,27] related to the algebraic leakage of the Lieb-Robinson bound [28][29][30][31].…”

Quantum kinetic perturbation theory for near-integrable spin chains with weak long-range interactions

“…m-independent) initial correlations, followed by a 'lightcone'-like spreading of correlations in space and time [16,[32][33][34][35]. In the presence of long-range interactions, a variety of analytical, numerical, as well as experimental results indicate that, at least for sufficiently small values of α, the linear shape of the cone gets replaced by a curved shape [36][37][38][39][40][41]. For α=4 as used in figure 7 a curved shape is not visible, and it has in fact been conjectured that correlations spread strictly linearly for α larger than some critical value [42].…”