On entropy growth and the hardness of simulating time evolution

Abstract

The simulation of quantum systems is a task for which quantum computers are believed to give an exponential speedup as compared to classical ones. While ground states of one-dimensional systems can be efficiently approximated using Matrix Product States (MPS), their time evolution can encode quantum computations, so that simulating the latter should be hard classically. However, one might believe that for systems with high enough symmetry, and thus insufficient parameters to encode a quantum computation, efficient classical simulation is possible. We discuss supporting evidence to the contrary: We provide a rigorous proof of the observation that a time independent local Hamiltonian can yield a linear increase of the entropy when acting on a product state in a translational invariant framework. This criterion has to be met by any classical simulation method, which in particular implies that every global approximation of the evolution requires exponential resources for any MPS based method.

Lieb-Robinson Bounds and the Simulation of Time-Evolution of Local Observables in Lattice Systems

“…Indeed, t-DMRG simulates time evolution for short times to essentially machine precision. For long times, the entropy will in general grow too much, as then sites are in the space time cone of too many sites of the lattice, and an efficient simulation in terms of matrix-product states is hence [60] no longer possible [12,61]. That is, the power of the t-DMRG approach can be rigorously grasped in terms of Lieb-Robinson bounds.…”

Optical implementation and entanglement distribution in Gaussian valence bond states

“…The global Gaussian state of the chain is described by a CM Γ = N i=1 γ [i] . As the interest in GVBS lies mainly in their connections with ground states of Hamiltonians invariant under translation [3], we can focus on pure (Det…”

“…The projection corresponds mathematically to taking a Schur complement (see Refs. [4,3,22] for details), yielding an output pure GVBS of N modes on a ring with a CM…”

“…In this context, entanglement is an essential resource. Recently, the valence bond formalism, originally developed for spin systems [2], has been generalized to the CV scenario [3,4] for the special class of Gaussian states, which play a central role in theoretical and practical CV quantum information and communication [5].…”

“…After recalling the necessary notation (Sec. 2) and the construction of Gaussian valence bond states [3] (Sec. 3), we discuss the characterization of entanglement and its distribution in such states as regulated by the entanglement properties of simpler states involved in the valence bond construction [4] (Sec.…”