24 research outputs found
A note on quantum algorithms and the minimal degree of epsilon-error polynomials for symmetric functions
The degrees of polynomials representing or approximating Boolean functions
are a prominent tool in various branches of complexity theory. Sherstov
recently characterized the minimal degree deg_{\eps}(f) among all polynomials
(over the reals) that approximate a symmetric function f:{0,1}^n-->{0,1} up to
worst-case error \eps: deg_{\eps}(f) = ~\Theta(deg_{1/3}(f) +
\sqrt{n\log(1/\eps)}). In this note we show how a tighter version (without the
log-factors hidden in the ~\Theta-notation), can be derived quite easily using
the close connection between polynomials and quantum algorithms.Comment: 7 pages LaTeX. 2nd version: corrected a few small inaccuracie
Solving the Shortest Vector Problem in Lattices Faster Using Quantum Search
By applying Grover's quantum search algorithm to the lattice algorithms of
Micciancio and Voulgaris, Nguyen and Vidick, Wang et al., and Pujol and
Stehl\'{e}, we obtain improved asymptotic quantum results for solving the
shortest vector problem. With quantum computers we can provably find a shortest
vector in time , improving upon the classical time
complexity of of Pujol and Stehl\'{e} and the of Micciancio and Voulgaris, while heuristically we expect to find a
shortest vector in time , improving upon the classical time
complexity of of Wang et al. These quantum complexities
will be an important guide for the selection of parameters for post-quantum
cryptosystems based on the hardness of the shortest vector problem.Comment: 19 page
Quantum Arthur-Merlin Games
This paper studies quantum Arthur-Merlin games, which are Arthur-Merlin games
in which Arthur and Merlin can perform quantum computations and Merlin can send
Arthur quantum information. As in the classical case, messages from Arthur to
Merlin are restricted to be strings of uniformly generated random bits. It is
proved that for one-message quantum Arthur-Merlin games, which correspond to
the complexity class QMA, completeness and soundness errors can be reduced
exponentially without increasing the length of Merlin's message. Previous
constructions for reducing error required a polynomial increase in the length
of Merlin's message. Applications of this fact include a proof that logarithmic
length quantum certificates yield no increase in power over BQP and a simple
proof that QMA is contained in PP. Other facts that are proved include the
equivalence of three (or more) message quantum Arthur-Merlin games with
ordinary quantum interactive proof systems and some basic properties concerning
two-message quantum Arthur-Merlin games.Comment: 22 page