Probably not...it is for strolling and observing.
I agree with the above post. We did not see any bikes while visiting there last year and frankly the layout is really not condusive to mixing vistors and bikes.
The paths are very narrow and some are quite steep. There are the occasional stairs. It's meant for strolling through the gardens. I'd be a little surprised if bicycles are allowed in although there is no mention on their official web site. It does say the paths are very steep and not suitable for wheelchairs. There are also shrubs that restrict the view in several places so it would be hard to ride a bicycle because you wouldn't know what was coming around the corner. Certainly, we've never see a bicycle in the gardens. They are lovely to walk, though, and the views from the top are just spectacular.
Third in agreement: it's really considered more museum than park (as it's not a "public" garden) and so I'm very sure they wouldn't let you bring a bike in.