I have had the white form of Dicentra formosa (var. alba, ht 30-45 cm) for as long as I can remember and I am pretty sure I grew it in my Ryde garden as well, nearly 20 years ago. It is related to the exquisite cool-climate Dicentra spectabilis (from Siberia, North China and Korea), often called 'bleeding heart', with its racemes of pink or white heart-shaped flowers - very difficult to grow in most parts of Sydney. I am content to grow its cousin from North America, which has clusters of nodding, heart-shaped flowers above dainty ferny leaves. It spreads by rhizomes to make a good sized-clump, dying down completely over winter.
More recently, I acquired the pink one (pictured at left), which is the original species, and it also seems to do reasonably well in Sydney. These plants do need a shaded, cool, moist position in the garden, being woodland plants in their native habitats. Give them humus-rich soil. The plants can be grown in a pot if a suitable position cannot be found in the garden. Propagation is by division of the rhizomes. Dicentra is related to a similar-looking plant: Corydalis flexuosa.
Postscript: the pink version didn't survive very long, but the white one still pops up each spring!