How WordPress converts URL to $query_string

Check out WP::parse_request() – unfortunately, it was never really designed for public use. You could try this, not sure of the repercussions:

global $wp,
       $wp_the_query;

// Grab the request_uri component of the referer URL.
if ( ! $url = parse_url( $_GET['my_referer_parameter'] ) )
    return 'Oh dear, poorly formed referer.';

// Ewww! Trick WP::parse_request by spoofing REQUEST_URI. 
$_SERVER['REQUEST_URI'] = $url['path'] . "?{$url['query']}";

$wp->parse_request();
$wp->query_posts();

// $wp_the_query should now be populated with posts queried by your referer.
while ( $wp_the_query->have_posts() ) {

    $wp_the_query->the_post();
    the_title();

}