1. Adiabatic quantum computation and stoquastic Hamiltonians B.M. Terhal IQI, RWTH Aachen A review of joint work with S. Bravyi, D.P.DiVincenzo and R. Oliveira.

2. Overview

3. Stoquastic Hamiltonians Physicists have known for a long time that not all Hamiltonians are created equal. Some are more quantum than other’s. Physics lingo: amenable to Quantum Monte Carlo techniques without sign-problems, “Quantum ground-state problem mapped onto classical partition function problem” What is exactly the property of these Hamiltonians and what does this feature do to the complexity of determining the lowest- eigenvalue, the power of these Hamiltonians for implementing an adiabatic quantum computer, etc?

4. Stoquastic Hamiltonians

5. More examples of stoquastic Hamiltonians AQC).

6. Stoquastic Hamiltonians

7. Local Stoquastic Hamiltonians Remember k-local Hamiltonians are those that can be written as a sum of terms acting non-trivially on k qubits (qudits). Our results hold for local term-wise stoquastic Hamiltonians, each local term is stoquastic. Side remark: Local termwise-stoquastic is not necessarily the same as local stoquastic:  2-local on qubits: termwise-stoquastic is stoquastic.  3-local on qubits: there is a counter-example of a Hamiltonian which is stoquastic but not term-wise stoquastic (also allowing 1-local unitary transformations).

8. Why Complexity of Stoquastic Hamiltonians Understand power of stoquastic adiabatic computation. Does it go beyond classical computation? Remember general adiabatic quantum computation is universal for quantum computation. Common stoquastic physical systems which are studied by classical Monte Carlo techniques; does a quantum computer provide any advantage. Can we do a rigorous analysis of these heuristic techniques? Nonnegative sparse matrices such as G are not uncommon in the real world, e.g. adjacency matrix of internet graph of which Google’s Pagerank algorithm determines the largest eigenvector (see recent proposal for a stoquastic adiabatic algorithm to determine PageRank vector by Garnerone, Zanardi & Lidar)

9. Adiabatic Quantum Computation (AQC)

10. Quantum Simulated Annealing

11. A class of stoquastic frustration- free Hamiltonians

12. Technical Question

13. Technical Solution

14. Technical Solution

15. Conclusion for stoquastic frustration-free AQC

16. Questions for stoquastic Hamiltonians

17. Different Approach

18. Sampling from ground-state

19. Simulating stoquastic AQC?

20. Partition Function Estimation

21. Partition Function Estimation

22. Overview (if you like complexity classes)