A quantum computer takes advantage of quantum entanglement manifest in the existence of a qubit.

From a many-universe perspective, quantum computation affords the possibility of massive parallelism by taking advantage of parallel universes instead of parallel processors.

The biggest challenge to QC is decoherence. The qubit must be complete isolated and that feat seems impossible.

1. Trapping ions looks promising technology as a foundation for quantum computing

2. One approach is to isolate quantum information in a quantum by cold electrodes and almost stationary atoms.

3. Christopher Monroe said, “Ion traps are widely regarded as the leading candidate for quantum computing”.

4. Quantum computing is useful for a few known algorithms.

5. Quantum information science is understanding some of the weird laws of nature.

6. Monroe’s team isolates individual trapped atoms in a vacuum chamber, and levitates them with electric fields produced from electrodes within 1/10 of a millimeter of the atoms. Lasers are used to probe the atom or push the atoms around Liquid helium is used to cool the probes.

1. The goal is to use quantum affects to create a form of computational power.

2. QC can make errors and algorithms need to be created that correct the errors.

3. Quantum mechanics is leading to a systematic exploitation of materials at a subatomic scale: transistor, laser, and solid state.

4. All proposals for doing quantum computing have the following sequence of steps: 1. Initialize the system 2. evolve it under some Hamiltonian 3. and then you measure something to get your answer.

5. Measurement steps in between could be model by Hamiltonian dynamics in a larger Hilbert space. So, everything in QC is creating a state, evolve, and then measure.

6. In Adiabatic QC, you evolve the system under a time-dependent Hamiltonian. Two constraints are your initial ground state and the Hamiltonian changes very slowly in time, so for all times your quantum state remains close to the ground state.

7. An Adiabatic QC algorithm may not be able to solve NP complete problems in less than exponential time.

1. The Hamiltonian forms an orthogonal basis for a Hilbert space.

2. Schroedinger equation in the box are sine waves with nodes at the two ends of the box. Any function that satisfies the Hamiltonian--that is, functions that have nodes at the ends of the box--can be decomposed as a series whose terms are the various pure sine wave solutions multiplied by the scalar product of those eigensolutions with the function to be decomposed.

3. A superposition of discrete energy states. The square of a particular coefficient is interpreted as the probability of measuring the particle in that particular eigenstate.

4. If you start with a superposition of states, you continue to have a superposition of states (unless there is some mechanism for energy loss) until you make an observation

5. The eigenvalues are the possible results of a measurement. The relative probabilities may be time-dependent, by the way, as they are for neutrino oscillations.

Hilbert Space

The Hamiltonian makes possible a time dependent probability of the states, hence the evolving nature of the final state.