The Church of the East espouses the heresy of Nestorianism. They rejected the Council of Chalcedon and held to their own, heretical Christology.
However, the Church of the East and the Catholic Church have recently issued a joint declaration on Christology:
vatican.va/roman_curia/pontifical_councils/chrstuni/documents/rc_pc_chrstuni_doc_11111994_assyrian-church_en.html
Not anymore if ever. There was an Iraqi Assyrian in the same hostel as me at college, who showed me a prayer rug his mother gave him. It had a depection of the OLPH on it, complete with the Theotokos on it. It was a surprise to me as I thought they broke away because they refused to acknowledge Mary as the Mother of God.
Assyrians are the remnants of an ancient church founded by Apostles Thomas & Bartholomew that developed outside the Roman Empire and eventually evangelised much of Asia. Although they broke away ostensibly over a Christological dispute, faultlines already exists with Assyrians being largely Persian at that time and Catholic/Orthodox being in the Roman Empire They recognised only the first two Ecumenical Councils.
The most well known adherents, other than the remnants left in Iraq, are the Malabarese Christians in India, who were evanglised by St Thomas. At one point, they are over 600 bishops all over Asia. The Assyrians in Central Asia were very much decimated by the Mongols and Muslims. We do not have much records of how it expanded and why it collapsed elsewhere in East Asia.
Most of the Assyrians today live in US, following the recent exodus of Christians in Middle East. Since 16th century, there has been return of Assyrians to the Catholic Church and they have their own Rite under the Chaldean Patriach of Babylon and the Syro-Malabarese Major Archbishop. I believe the Catholics now outnumber the Assyrians.
Their liturgy is very ancient and is unique in having a Eucharistic Prayer that do not have explicitly have the words of the consecration. This Eucharistic Prayer known as the
Anaphora of Addai and Mari, is also used by Chaldean Catholics.
You can check out ‘Assyrian Church of the East’ in Wikipedia. If there is any quesiton, I can try to help unless there is a Assyrian, Chaldean or Syro-Malabarese around.