TY - JOUR

T1 - When a local Hamiltonian must be frustration-free

AU - Sattath, Or

AU - Morampudi, Siddhardh C.

AU - Laumann, Chris R.

AU - Moessner, Roderich

N1 - Funding Information:
We thank Dorit Aharonov, David Gosset, Antonello Scardicchio, Shivaji Sondhi, Mario Szegedy, and Umesh Vazirani for valuable discussions. This work was supported by the ARO Grant W922NF-09-1-0440, NSF Grants CCF-0905626 and PHY-1520535, the DFG Grant SFB 1143, ERC Grant 030-8301, and a Sloan Research Fellowship.

PY - 2016/6/7

Y1 - 2016/6/7

N2 - A broad range of quantum optimization problems can be phrased as the question of whether a specific system has a ground state at zero energy, i.e., whether its Hamiltonian is frustration-free. Frustrationfree Hamiltonians, in turn, play a central role for constructing and understanding new phases of matter in quantum many-body physics. Unfortunately, determining whether this is the case is known to be a complexity-theoretically intractable problem. This makes it highly desirable to search for efficient heuristics and algorithms to, at least, partially answer this question. Here we prove a general criterion-a sufficient condition-under which a local Hamiltonian is guaranteed to be frustration-free by lifting Shearer's theorem from classical probability theory to the quantum world. Remarkably, evaluating this condition proceeds via a fully classical analysis of a hardcore lattice gas at negative fugacity on the Hamiltonian's interaction graph, which, as a statistical mechanics problem, is of interest in its own right. We concretely apply this criterion to local Hamiltonians on various regular lattices, while bringing to bear the tools of spin glass physics that permit us to obtain new bounds on the satisfiable to unsatisfiable transition in random quantum satisfiability.We are then led to natural conjectures for when such bounds will be tight, as well as to a novel notion of universality for these computer science problems. Besides providing concrete algorithms leading to detailed and quantitative insights, this work underscores the power of marrying classical statistical mechanics with quantum computation and complexity theory.

AB - A broad range of quantum optimization problems can be phrased as the question of whether a specific system has a ground state at zero energy, i.e., whether its Hamiltonian is frustration-free. Frustrationfree Hamiltonians, in turn, play a central role for constructing and understanding new phases of matter in quantum many-body physics. Unfortunately, determining whether this is the case is known to be a complexity-theoretically intractable problem. This makes it highly desirable to search for efficient heuristics and algorithms to, at least, partially answer this question. Here we prove a general criterion-a sufficient condition-under which a local Hamiltonian is guaranteed to be frustration-free by lifting Shearer's theorem from classical probability theory to the quantum world. Remarkably, evaluating this condition proceeds via a fully classical analysis of a hardcore lattice gas at negative fugacity on the Hamiltonian's interaction graph, which, as a statistical mechanics problem, is of interest in its own right. We concretely apply this criterion to local Hamiltonians on various regular lattices, while bringing to bear the tools of spin glass physics that permit us to obtain new bounds on the satisfiable to unsatisfiable transition in random quantum satisfiability.We are then led to natural conjectures for when such bounds will be tight, as well as to a novel notion of universality for these computer science problems. Besides providing concrete algorithms leading to detailed and quantitative insights, this work underscores the power of marrying classical statistical mechanics with quantum computation and complexity theory.

KW - Critical exponents

KW - Hardcore lattice gas

KW - Local Hamiltonian

KW - Quantum satisfiability

KW - Universality

UR - http://www.scopus.com/inward/record.url?scp=84973308536&partnerID=8YFLogxK

U2 - 10.1073/pnas.1519833113

DO - 10.1073/pnas.1519833113

M3 - Article

C2 - 27199483

AN - SCOPUS:84973308536

SN - 0027-8424

VL - 113

SP - 6433

EP - 6437

JO - Proceedings of the National Academy of Sciences of the United States of America

JF - Proceedings of the National Academy of Sciences of the United States of America

IS - 23

ER -