On entropy growth and the hardness of simulating time evolution
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The simulation of quantum systems is a task for which quantum computers are believed to give an exponential speed up as compared with 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.
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According to the Lieb-Robinson theorem, after a sufficiently large time, states can become strongly entangled.
“Indeed, this fact is related to correlation and entanglement spreading according to the Lieb-Robinson theorem that states that, after a sufficiently large time, states can become strongly entangled ([91-96], see also [97]).”
From Can one trust quantum simulators? (2012) · cites this paper as [94]
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- Authors
- Schuch, N., Wolf, M. M., Vollbrecht, K. G. H., & Cirac, J. I.
- Affiliations
- Max Planck Society (DE)
- Venue
- New Journal of Physics
- Publisher
- IOP Publishing
- Published
- 2008-03-20
- Volume
- 10
- Issue
- 3
- Pages
- 033032
- ISSN
- 1367-2630 (ISSN-L)
- OpenAlex
- W1970054335
- Cited by
- 113
- References
- 36
- Citations by year
- 2026: 5 · 2025: 11 · 2024: 6 · 2023: 8 · 2022: 4 · 2021: 11 · 2020: 13 · 2019: 10 · 2018: 9 · 2017: 2 · 2016: 2 · 2015: 4 · 2014: 4 · 2013: 2 · 2012: 2
- FWCI
- 3.86
- Type
- journal article
- Language
- en
- Open access
- gold
- doi.org
- Licence
- cc-by
- Locations
- New Journal of Physics (publishedVersion, cc-by)
- arXiv (Cornell University) (publishedVersion)
- Max Planck Institute for Plasma Physics (submittedVersion)
- MPG.PuRe (Max Planck Society) (submittedVersion, cc-by)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
- Keywords
- entropy increase, classical simulation of quantum dynamics, quantum advantage, matrix product states, local Hamiltonians, translation invariance, product states, ground state, one-dimensional systems
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