Quantum Complexity, Algorithms, and Primitives

Abstract

Abstract : The project undertook theoretical research in quantum algorithms, complexity of quantum computation, quantum primitives, and quantum communication protocols. In the area of complexity, it compared quantum computation models with classical ones, finding counting complexity classes between BQP and AWPP that are likely different from both. It investigated small-depth quantum circuits (both with and without unbounded fan-in gates such as quantum AND) and found lower and upper bounds on their power and complexity. In the area of new quantum primitives, the project found Hamiltonians for the quantum fan-out gate, based on spin-exchange interactions. In the area of quantum algorithms, the project showed that there are efficient quantum algorithms for various group theoretic problems, for example, group intersection and double coset membership for certain classes of solvable groups. It also found a network of efficient quantum reducibilities between these and other group-theoretic problems. These are the project's successes. The project was unsuccessful in some endeavors. It has so far failed to find natural problems in these intermediate classes between BQP and AWPP, or to isolate the more robust classes among these. It did not find further evidence that BQP does not contain NP. There was no significant progress on quantum communication protocols.