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.

Overview

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?

Stoquastic Hamiltonians

More examples of stoquastic Hamiltonians AQC).

Stoquastic Hamiltonians

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).

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)

Adiabatic Quantum Computation (AQC)

Quantum Simulated Annealing

A class of stoquastic frustration- free Hamiltonians

Technical Question

Technical Solution

Technical Solution

Conclusion for stoquastic frustration-free AQC

Questions for stoquastic Hamiltonians

Different Approach

Sampling from ground-state

Simulating stoquastic AQC?

Partition Function Estimation

Partition Function Estimation

Overview (if you like complexity classes)