Tap water won't cycle a tank any faster then any kind of water. Tap water is normally not recommened because it can contain unwanted minerals. Distilled water, normally, is pure. If you nitrites in your pico are high then your tank isn't completely cycled. If your nitrates are high then it could be comming from your tap water. The best way to get rid of your nitrates is to do water changes and give your tank some time to level out.