I want my Google+ account to be automatically updated when I post a tweet on Twitter.
Google+ API is not yet public, so there's no way to auto-update your Twitter posts to G+.
You can, however, do the other way around: post to Google+ and have it auto-update to Twitter. And Facebook too.
It's not possible (currently) to update Twitter from G+ without a browser extension. You should wait for the API.
The best way to update Twitter and Facebook from G+ that I have come across is to get your public RSS feed for your Google+ profile and then use a service like Twitterfeed to update Facebook and Twitter or use Manage Flitter. More detailed info here: http://iag.me/socialmedia/tools/how-to-update-twitter-and-facebook-with-your-google-posts/
You can even decide which posts you want to sync to Facebook and/or Twitter using tags (eg #fb or #tw)