Explore applications of the recent "Quantum Algorithm for Linear
Systems of Equations" by Harrow, Hassidim and Lloyd \cite{HHL}. This
algorithm can solve a linear system of the form $Ax = b$, in time
logarithmic in the dimensions of $A$, an exponential improvement over
classical algorithms. However, the quantum algorithm cannot output the
vector solution $x$ explicitly, but only give an approximation to the
value of $x^T M x$ for an input operator $M$.
Understand the construction and background for the gap example for the
semidefinite program for sparsest cut presented in \cite{DKSV}.
What does it say regarding directions for improved analysis?
Explore applications of the Multiplicative Weight Update method
to problems in quantum computation. Recently, this yielded the result
$\sc{QIP} = {\sc PSPACE}$ \cite{QIP}. Is it possible to say anything
about scenarios with multiple provers? Another direction is the
exploration of a possible connection between the update methods and
quantum evolution, and the Adiabatic Theorem.
A large effort has gone into finding
fast versions of the ARV algorithm. Currently, the best version only
needs a polylogarithmic number of s-t maxflow calls and runs in time
$\tilde{O}(n^{3/2}).$ We would like to explore the question of whether
this bound can be improved, in particular in light of the new
techniques introduced by Spielman and Teng. A couple of directions
are:
can the running time of s-t maxflow computations be improved?
more generally can we improve the number of iterations required by the interior point method for linear programming?
along a different direction, can we use electrical flows to
understand the ARV better?