# The local Hamiltonian problem on a line with eight states is QMA-complete

@article{Hallgren2013TheLH,
title={The local Hamiltonian problem on a line with eight states is QMA-complete},
author={Sean Hallgren and Daniel Nagaj and Sandeep Narayanaswami},
journal={Quantum Inf. Comput.},
year={2013},
volume={13},
pages={721-750}
}
• Published 1 September 2013
• Mathematics
• Quantum Inf. Comput.
The Local Hamiltonian problem is the problem of estimating the least eigenvalue of a local Hamiltonian, and is complete for the class QMA. The 1D problem on a chain of qubits has heuristics which work well, while the 13-state qudit case has been shown to be QMA-complete. We show that this problem remains QMA-complete when the dimensionality of the qudits is brought down to 8.

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## References

SHOWING 1-10 OF 18 REFERENCES
3-local Hamiltonian is QMA-complete
• Physics
Quantum Inf. Comput.
• 2003
The locality of the problem is reduced by showing that 3-local Hamiltonian is already QMA-complete, which means that the 5- local Hamiltonian problem is Q MA-complete.
The complexity of quantum spin systems on a two-dimensional square lattice
• Physics, Mathematics
Quantum Inf. Comput.
• 2008
It is obtained that quantum adiabatic computation using 2-local interactions restricted to a 2-D square lattice is equivalent to the circuitmodel of quantum computation.
Quantum SAT for a Qutrit-Cinquit Pair Is QMA1-Complete
• Physics
ICALP
• 2008
The main novel ingredient of the proof is a certain Hamiltonian construction named the Triangle Hamiltonian that allows to verify the application of a 2-qubit CNOT gate without generating explicitly interactions between pairs of workspace qubits.
3-local Hamitonian is QMA-complete
• Physics
• 2003
It has been shown by Kitaev that the 5-LOCAL HAMILTONIAN problem is QMA-complete. Here we reduce the locality of the problem by showing that 3-LOCAL HAMILTONIAN is already QMA-complete.
The Power of Quantum Systems on a Line
• Physics
48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)
• 2007
The proof of the QMA-completeness result requires an additional idea beyond the usual techniques in the area: Some illegal configurations cannot be ruled out by local checks, and are instead ruled out because they would, in the future, evolve into a state which can be seen locally to be illegal.
Efficient algorithm for a quantum analogue of 2-SAT
Complexity of a quantum analogue of the satisfiability problem is studied. Quantum k-SAT is a problem of verifying whether there exists n-qubit pure state such that its k-qubit reduced density
Fast universal quantum computation with railroad-switch local Hamiltonians
We present two universal models of quantum computation with a time-independent, frustration-free Hamiltonian. The first construction uses 3-local (qubit) projectors and the second one requires only
Quantum NP - A Survey
• Physics
• 2002
We describe Kitaev's result from 1999, in which he defines the complexity class QMA, the quantum analog of the class NP, and shows that a natural extension of 3-SAT, namely local Hamiltonians, is QMA
QMA-complete problems
An overview of the quantum computational complexity class QMA and a description of known QMA-complete problems to date are given and problems of interest to all of these professions can be found here.
Criticality without frustration for quantum spin-1 chains.
• Physics
Physical review letters
• 2012
This work proposes the first example of a FF translation-invariant spin-1 chain that has a unique highly entangled ground state and exhibits some signatures of a critical behavior.