15 research outputs found
Extremal eigenvalues of local Hamiltonians
We apply classical algorithms for approximately solving constraint
satisfaction problems to find bounds on extremal eigenvalues of local
Hamiltonians. We consider spin Hamiltonians for which we have an upper bound on
the number of terms in which each spin participates, and find extensive bounds
for the operator norm and ground-state energy of such Hamiltonians under this
constraint. In each case the bound is achieved by a product state which can be
found efficiently using a classical algorithm.Comment: 5 pages; v4: uses standard journal styl
Computing the partition function of a polynomial on the Boolean cube
For a polynomial f: {-1, 1}^n --> C, we define the partition function as the
average of e^{lambda f(x)} over all points x in {-1, 1}^n, where lambda in C is
a parameter. We present a quasi-polynomial algorithm, which, given such f,
lambda and epsilon >0 approximates the partition function within a relative
error of epsilon in N^{O(ln n -ln epsilon)} time provided |lambda| < 1/(2 L
sqrt{deg f}), where L=L(f) is a parameter bounding the Lipschitz constant of f
from above and N is the number of monomials in f. As a corollary, we obtain a
quasi-polynomial algorithm, which, given such an f with coefficients +1 and -1
and such that every variable enters not more than 4 monomials, approximates the
maximum of f on {-1, 1}^n within a factor of O(sqrt{deg f}/delta), provided the
maximum is N delta for some 0< delta <1. If every variable enters not more than
k monomials for some fixed k > 4, we are able to establish a similar result
when delta > (k-1)/k.Comment: The final version of this paper is due to be published in the
collection of papers "A Journey through Discrete Mathematics. A Tribute to
Jiri Matousek" edited by Martin Loebl, Jaroslav Nesetril and Robin Thomas, to
be published by Springe
Beating the random assignment on constraint satisfaction problems of bounded degree
We show that for any odd and any instance of the Max-kXOR constraint
satisfaction problem, there is an efficient algorithm that finds an assignment
satisfying at least a fraction of
constraints, where is a bound on the number of constraints that each
variable occurs in. This improves both qualitatively and quantitatively on the
recent work of Farhi, Goldstone, and Gutmann (2014), which gave a
\emph{quantum} algorithm to find an assignment satisfying a fraction of the equations.
For arbitrary constraint satisfaction problems, we give a similar result for
"triangle-free" instances; i.e., an efficient algorithm that finds an
assignment satisfying at least a fraction of
constraints, where is the fraction that would be satisfied by a uniformly
random assignment.Comment: 14 pages, 1 figur
Hardness of Approximating Bounded-Degree Max 2-CSP and Independent Set on k-Claw-Free Graphs
We consider the question of approximating Max 2-CSP where each variable
appears in at most constraints (but with possibly arbitrarily large
alphabet). There is a simple -approximation algorithm for the
problem. We prove the following results for any sufficiently large :
- Assuming the Unique Games Conjecture (UGC), it is NP-hard (under randomized
reduction) to approximate this problem to within a factor of .
- It is NP-hard (under randomized reduction) to approximate the problem to
within a factor of .
Thanks to a known connection [Dvorak et al., Algorithmica 2023], we establish
the following hardness results for approximating Maximum Independent Set on
-claw-free graphs:
- Assuming the Unique Games Conjecture (UGC), it is NP-hard (under randomized
reduction) to approximate this problem to within a factor of .
- It is NP-hard (under randomized reduction) to approximate the problem to
within a factor of .
In comparison, known approximation algorithms achieve -approximation in polynomial time [Neuwohner, STACS 2021; Thiery
and Ward, SODA 2023] and -approximation in
quasi-polynomial time [Cygan et al., SODA 2013]
A quantum advantage over classical for local max cut
We compare the performance of a quantum local algorithm to a similar
classical counterpart on a well-established combinatorial optimization problem
LocalMaxCut. We show that a popular quantum algorithm first discovered by
Farhi, Goldstone, and Gutmannn [1] called the quantum optimization
approximation algorithm (QAOA) has a computational advantage over comparable
local classical techniques on degree-3 graphs. These results hint that even
small-scale quantum computation, which is relevant to the current state-of the
art quantum hardware, could have significant advantages over comparably simple
classical computation