Which state in India is home to the famous ancient Jain rock-cut caverns of Udayagiri and Khandagiri?

The correct answer is Odisha.

Detailed Explanation:
The famous ancient rock-cut caverns of Udayagiri and Khandagiri are situated near the city of Bhubaneswar in the state of Odisha, India.

Key Facts about Udayagiri and Khandagiri Caves:
1. Historical Period: These caves were carved out mostly during the 2nd century BCE under the patronage of King Kharavela of the Mahameghavahana dynasty.
2. Religious Purpose: They were constructed as residential shelters (viharas) for Jain monks and ascetics.
3. Famous Inscriptions & Architecture: The group includes notable caves such as Hathigumpha (Elephant Cave), which contains the celebrated Hathigumpha Inscription written in Brahmi script, and Rani Gumpha (Queen's Cave), known for its intricate multi-storied relief sculptures.

Therefore, the state home to the Udayagiri and Khandagiri Jain rock-cut caverns is Odisha.