1. QMA-complete Problems Adam Bookatz December 12, 2012

2. Quantum-Merlin-Arthur (QMA) V 1=accept 0=reject

3. Q UANTUM C IRCUIT SAT (QCSAT) QMA-complete (by definition) V 1=accept 0=reject

4. Q UANTUM CHANNEL PROPERTY VERIFICATIONS QMA- COMPLETE problems

5. Recall… from class that Q UANTUM -k-SAT is QMA-complete We will now look at more general versions But we require a little bit of physics…

6. Hamiltonians

7. k-L OCAL H AMILTONIAN It is in P for k=1

8. k-L OCAL H AMILTONIAN There are a plethora of QMA-complete versions: Remoevd: bounded strength Hamiltonians (k ≥ 3) Density Functional Theory when only considering eigenvectors that are separable over a partition

9. Q UANTUM -k-SAT

10. L OCAL C ONSISTENCY OF D ENSITY M ATRICES QMA-complete for k ≥ 2 (Reduction from k- LOCAL H AMILTONIAN ) True also for bosonic and fermionic systems

11. Conclusion Not so many QMA-complete problems Contrast: thousands of NP-complete problems Most important problem is k- LOCAL H AMILTONIAN – Most research has focused on it and its variants There are a handful of other problems too – Verifying properties of quantum circuits/channels – LOCAL CONSISTENCY OF DENSITY MATRICES

12. C HANNEL P ROPERTY V ERIFICATION NON - IDENTITY CHECK NON - EQUIVALENCE CHECK QUANTUM CLIQUE NON - ISOMETRY TEST DETECT INSECURE Q. ENCRYPTION QUANTUM NON - EXPANDER TEST k- LOCAL CONSISTENCY [k ≥ 2] bosonic, fermionic QCSAT * for k ≥ 3

13. The End

15. Q UANTUM -k-SAT Equivalently, write it more SAT-like

16. Q UANTUM -k-SAT QMA 1 -complete