TY - GEN

T1 - On preparing ground states of gapped hamiltonians

T2 - 58th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2017

AU - Gilyen, Andras Pal

AU - Sattath, Or

N1 - Publisher Copyright:
© 2017 IEEE.

PY - 2017/11/10

Y1 - 2017/11/10

N2 - A frustration-free local Hamiltonian has the property that its ground state minimises the energy of all local terms simultaneously. In general, even deciding whether a Hamiltonian is frustration-free is a hard task, as it is closely related to the QMA1-complete quantum satisfiability problem (QSAT) - the quantum analogue of SAT, which is the archetypal NP-complete problem in classical computer science. This connection shows that the frustration-free property is not only relevant to physics but also to computer science.The Quantum Lovsz Local Lemma (QLLL) provides a sufficient condition for frustration-freeness. Is there an efficient way to prepare a frustration-free state under the conditions of the QLLL? Previous results showed that the answer is positive if all local terms commute. These works were based on Mosers compression argument which was the original analysis technique of the celebrated resampling algorithm. We generalise and simplify the compression argument, so that it provides a simplified version of the previous quantum results, and improves on some classical results as well.More importantly, we improve on the previous constructive results by designing an algorithm that works efficiently for non-commuting terms as well, assuming that the system is uniformly gapped, by which we mean that the system and all its subsystems have an inverse polynomial energy gap. Similarly to the previous results, our algorithm has the charming feature that it uses only local measurement operations corresponding to the local Hamiltonian terms.

AB - A frustration-free local Hamiltonian has the property that its ground state minimises the energy of all local terms simultaneously. In general, even deciding whether a Hamiltonian is frustration-free is a hard task, as it is closely related to the QMA1-complete quantum satisfiability problem (QSAT) - the quantum analogue of SAT, which is the archetypal NP-complete problem in classical computer science. This connection shows that the frustration-free property is not only relevant to physics but also to computer science.The Quantum Lovsz Local Lemma (QLLL) provides a sufficient condition for frustration-freeness. Is there an efficient way to prepare a frustration-free state under the conditions of the QLLL? Previous results showed that the answer is positive if all local terms commute. These works were based on Mosers compression argument which was the original analysis technique of the celebrated resampling algorithm. We generalise and simplify the compression argument, so that it provides a simplified version of the previous quantum results, and improves on some classical results as well.More importantly, we improve on the previous constructive results by designing an algorithm that works efficiently for non-commuting terms as well, assuming that the system is uniformly gapped, by which we mean that the system and all its subsystems have an inverse polynomial energy gap. Similarly to the previous results, our algorithm has the charming feature that it uses only local measurement operations corresponding to the local Hamiltonian terms.

KW - Local Hamiltonians

KW - Moser-Tardos Algorithm

KW - QMA

KW - QSAT

KW - Quantum Algorithm

KW - Quantum Lovász Local Lemma

UR - http://www.scopus.com/inward/record.url?scp=85041094183&partnerID=8YFLogxK

U2 - 10.1109/FOCS.2017.47

DO - 10.1109/FOCS.2017.47

M3 - Conference contribution

AN - SCOPUS:85041094183

T3 - Annual Symposium on Foundations of Computer Science - Proceedings

SP - 439

EP - 450

BT - Proceedings - 58th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2017

PB - Institute of Electrical and Electronics Engineers

Y2 - 15 October 2017 through 17 October 2017

ER -