You can use the open standard called CardDav to synchronise your Zimbra contacts to your iPhone or iPad.
You are connecting to your Zimbra account via a special URL over which Zimbra allows connection to your contacts.
To configure this on your iPhone or iPad open iPhone Settings application and choose Mail, Contacts, Calendar:
Open iPhone Settings application and choose Mail, Contacts, Calendar:
Then select Add Account…:
Choose Add CardDAV Account:
Enter your Zimbra server hostname in the Server field,
provide your username and password then press Next:
On the account page, open Advanced Settings: