1. Adiabatic Quantum Computing Josh Ball with advisor Professor Harsh Mathur Problems which are classically difficult to solve may be solved much more quickly by quantum computing. A new strategy for quantum computing, called adiabatic quantum computing, has been developed to solve a particularly hard problem called exact cover. Preliminary simulations suggest that the new quantum algorithm solves the problem much faster than a classical computer. We have investigated the algorithm and found small speedups, but could not find a case where the algorithm would fail. If the adiabatic quantum computing does in fact solve these problems quickly in all cases then a large class of problems could be solved quickly using this scheme. ABSTRACT The Adiabatic Theorem in Quantum Mechanics states that a system will stay in the n th eigenstate of the Hamiltonian if the Hamiltonian is changes slowly enough. Computations can be preformed by setting up the ground state of the initial Hamiltonian as the combination of all the possible solution. The ground state of the final Hamiltonian will be the solution. When the algorithm is run, the Hamiltonian slowly interpolates from the initial to the final Hamiltonian. Then we measure the state of the system and retrieve the answer with high probability. In the physical implementation of this algorithm, the initial ground state will be produced by magnetic fields in the x direction coupled to spins in an n-spin system. For an n spin system the Hamiltonian is given by a 2 n x 2 n matrix. The initial Hamiltonian can be written as a recursive formula which allows any such 2 n x 2 n matrix to be built up from σ x and the identity matrix. The final Hamiltonian is a diagonal matrix with elements corresponding to how many clauses a particular state fails to satisfy. We want the states which violate the most number of states to “cost” the most energy. In the USA case there should be a single diagonal element with value zero. This is the solution state which satisfies every clause, and is the ground state. ADIABATIC QUANTUM COMPUTING In the problem of exact cover we are given n bits (recall that a bit can either be zero or one) and a number of clauses. The causes each contain three of the bits and require that a one of the bits is one and the other two are zero. A solution to exact cover is a bit assignment which satisfies all of the clauses. With multiple clauses, the problem quickly becomes complicated. EXACT COVER (1, 2, 3) 0 / 1 Of the first three bits, one and only one can be a one, the other two must be zeros. The eigenvalues of the Hamiltonian change as the algorithm is run and the Hamiltonian interpolates. The important feature to see on the following graph is the lowest curve, which is the ground state. Note how this curve does not cross any of the other curves. This implies that when the algorithm is running, the system is not likely to “jump” up to another state. This is good, since we want the system to stay in the ground state. IMPROVING THE INTERPOLATION Instead of linearly interpolating between the initial and final Hamiltonians, the simulation produces better results when the rate of change of the interpolation is proportional to the difference between the lowest two eigenvalues of the current Hamiltonian. The rate of change should start out fast, and then slow down when the interpolation is about 70% done, then speed up again at the end. The graph clearly indicates a better performance for the improved interpolation. Notice how the effect is maximized for mid- ranged values of T. When T is very big, the probability of finding the correct answer tends towards 1. When T is near zero, the probability is near zero; it is equal to that of each of the other states. NON-USA CASES The Non-USA cases are those which do not have one solution. They are the instances of exact cover which may have multiple solutions or none at all. They were of particular interest, because we thought the algorithm could fail for some of these cases. If one case failed, the algorithm would have been shown to be faulty. In our investigation of over fifty Non-USA cases, we could not find a single instance where the algorithm failed. CASES WITH NO SOLUTIONS If the instance had no solution, the algorithm would produce the next best states, that is, the states which violated the minimum number of clauses. This is expected. When you measure the final state, it has to be something, so even when there is no solution you can measure the final state and look at the answer. This answer can be checked on a classical computer. Recall that finding a solution to exact cover is difficult, while checking the validity of an answer is easy. If the algorithm produces an ‘incorrect’ answer multiple times, one can conclude with increasing certainty, that the instance has no solutions. For the Non-USA cases with multiple solutions, the algorithm will end in a state which is a combination of these correct states. This result is unexpected for the instances where there are crossings in the eigenvalues. The algorithm still works in this case, producing the correct answers. It should be noted that the algorithm produces the correct answer with a slightly lower probability than it does for similar cases without crossings. CASES WITH TWO SOLUTIONS Here is an example of the algorithm’s output for a instance with no solutions. The plot is probability vs. time. The most likely states are those which violate the minimum number of clauses.