Edward Farhi

Quantum computation and decision trees

Many interesting computational problems can be reformulated in terms of decision trees. A natural classical algorithm is to then run a random walk on the tree, starting at the root, to see if the tree contains a node

Exponential algorithmic speedup by a quantum walk

n

An Example of the Difference Between Quantum and Classical Random Walks

level from the root. We devise a quantum-mechanical algorithm that evolves a state, initially localized at the root, through the tree. We prove that if the classical strategy succeeds in reaching level

A Quantum Algorithm for the Hamiltonian NAND Tree

n

Particle Production in the New Inflationary Cosmology

in time polynomial in

Robustness of adiabatic quantum computation

n,

Are the weak interactions strong

then so does the quantum algorithm. Moreover, we find examples of trees for which the classical algorithm requires time exponential in

Electroweak Fermion Number Violation and the Production of Stable Particles in the Early Universe

n,

Decoupling a fermion whose mass is generated by a Yukawa coupling: The general case☆

but for which the quantum algorithm succeeds in polynomial time. The examples we have so far, however, could also be solved in polynomial time by different classical algorithms.

Limit on the Speed of Quantum Computation in Determining Parity

We construct a black box graph traversal problem that can be solved exponentially faster on a quantum computer than on a classical computer. The quantum algorithm is based on a continuous time quantum walk, and thus employs a different technique from previous quantum algorithms based on quantum Fourier transforms. We show how to implement the quantum walk efficiently in our black box setting. We then show how this quantum walk solves our problem by rapidly traversing a graph. Finally, we prove that no classical algorithm can solve the problem in subexponential time.