Quantum Learning on Jice Lavocat

Impatient Learning and sub/sup Majority Problem

Wed, 16 Sep 2009 00:00:00 +0000

In this article I present some probabilities of 1-step learning optimized by Impatient Learning (http://www.citebase.org/abstract?id=oai:arXiv.org:quant-ph/0309059) for the sub/sup Majority Learning.

The (Sub/Sup) Majority Learning

Take a bitstring a, and an integer $$\theta$$. The oracle will respond 0 if $$d(a,x)\leq \theta$$ if queried with x and 1 otherwise. In other terms this oracle reply yes if two bitstrings agree on at least $$\theta$$ bit.

Here are the probability with a simple membership oracle (used with the usual phase kickback trick). Only even n gives an invertible matrix for the use of impatient learning.

Classical algorithm for the Majority Problem

Tue, 07 Jul 2009 00:00:00 +0000

Presentation of the problem :

The majority problem is equivalent to the perceptron learning. For each $$a \in \mathbb{Z}_n$$ define a function $$m_a : \mathbb{Z}_2^N \rightarrow \mathbb{Z}_2$$ :

$$m_a(x)= \begin{cases} 1 & \text{ if } wt(a-x)\leq n/2 \\0 & \text{ otherwise } \end{cases}$$

Where wt is the weight of a bit-string (number of 1).

Alternatively we can write : $$ m_a(x) = \Theta (n/2 – wt(x-a) )$$.

Some papers about Oracle identification

Wed, 01 Jul 2009 00:00:00 +0000

There is a bunch of interesting papers dealing with Oracle Identification.

Here is a list of the ones I found interesting to read :

Latex on my blog

Fri, 19 Jun 2009 00:00:00 +0000

WordPress permet d’inclure du Latex dans ses pages grace a un script hébergé sur ses propres serveur. Voila qui va me rendre heureux tout ca :