791 research outputs found
A new construction for a QMA complete 3-local Hamiltonian
We present a new way of encoding a quantum computation into a 3-local
Hamiltonian. Our construction is novel in that it does not include any terms
that induce legal-illegal clock transitions. Therefore, the weights of the
terms in the Hamiltonian do not scale with the size of the problem as in
previous constructions. This improves the construction by Kempe and Regev, who
were the first to prove that 3-local Hamiltonian is complete for the complexity
class QMA, the quantum analogue of NP.
Quantum k-SAT, a restricted version of the local Hamiltonian problem using
only projector terms, was introduced by Bravyi as an analogue of the classical
k-SAT problem. Bravyi proved that quantum 4-SAT is complete for the class QMA
with one-sided error (QMA_1) and that quantum 2-SAT is in P. We give an
encoding of a quantum circuit into a quantum 4-SAT Hamiltonian using only
3-local terms. As an intermediate step to this 3-local construction, we show
that quantum 3-SAT for particles with dimensions 3x2x2 (a qutrit and two
qubits) is QMA_1 complete. The complexity of quantum 3-SAT with qubits remains
an open question.Comment: 11 pages, 4 figure
The power of quantum systems on a line
We study the computational strength of quantum particles (each of finite
dimensionality) arranged on a line. First, we prove that it is possible to
perform universal adiabatic quantum computation using a one-dimensional quantum
system (with 9 states per particle). This might have practical implications for
experimentalists interested in constructing an adiabatic quantum computer.
Building on the same construction, but with some additional technical effort
and 12 states per particle, we show that the problem of approximating the
ground state energy of a system composed of a line of quantum particles is
QMA-complete; QMA is a quantum analogue of NP. This is in striking contrast to
the fact that the analogous classical problem, namely, one-dimensional
MAX-2-SAT with nearest neighbor constraints, is in P. The proof of the
QMA-completeness result requires an additional idea beyond the usual techniques
in the area: Not all illegal configurations can be ruled out by local checks,
so instead we rule out such illegal configurations because they would, in the
future, evolve into a state which can be seen locally to be illegal. Our
construction implies (assuming the quantum Church-Turing thesis and that
quantum computers cannot efficiently solve QMA-complete problems) that there
are one-dimensional systems which take an exponential time to relax to their
ground states at any temperature, making them candidates for being
one-dimensional spin glasses.Comment: 21 pages. v2 has numerous corrections and clarifications, and most
importantly a new author, merged from arXiv:0705.4067. v3 is the published
version, with additional clarifications, publisher's version available at
http://www.springerlink.co
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