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Unformatted text preview: CSE 599d  Quantum Computing One Qubit, Two Qubit Dave Bacon Department of Computer Science & Engineering, University of Washington Now that we’ve gone through the requisite review of linear algebra and Dirac notation, we can begin to consider how quantum mechanics allows us to build computing devices. In this lecture we will discuss a single qubit, then two qubits, and then a single qubit quantum algorithm, known as Deutsch’s algorithm. I. ONE QUBIT Suppose that we are dealing with a quantum machine with simply two configurations for each cell. Such a setup is called a qubit. The word “qubit” was first used in the scientific literature by Ben Schumacher. In fact the word was suggested, in jest, in a conversation between Ben Schumacher and William Wootters. The jest, I believe, was that qubit would be pronounced just like “cubit,” which is an ancient length about equal to the length of a forearm, something like 1822 inches (The Roman cubit was 17.4 inches; the Egyptian 20.64 inches.) Most people, I think, associate cubits with the Bible and if you ever get a chance to listen to Bill Cosby’s comedy routine about Noah’s ark, the word will hold a special place in your heart. I once missed a big opportunity when Ben Schumacher was visiting the Santa Fe Institute (SFI) in New Mexico where I was a postdoc. You see at SFI there was another fellow who had also invented a work that began with the letter “q” and was in the dictionary: Murray GellMann who invented the word “quark.” What opportunity did I miss? Well at the time my license plate read QUBITS and Murray’s read QUARKS. So think about it, I had the opportunity to get a picture taken of two people who invented two “q”words in the dictionary standing beside two cars with license plates both showing those words. I will never forgive myself for missing this opportunity. Okay, back to the issue at hand. A qubit, like we said, is a two configuration system. Physicists like to call this a twolevel system (TLS.) The quantum state of a two level system is given by  ψ i = α  i + β  1 i (1) where α,β ∈ C and since we require this to be a normalized state, we require that  α  2 +  β  2 = 1. One thing which I haven’t yet discussed is that for quantum states, a global phase for the state never has any observable consequences (and hence is irrelevant.) Thus the state  ψ i and e iγ  ψ i will, no matter what happens later, both produce the same observable consequences (to see this note that the phase does not effect measurement properties and that a phase commutes with any unitary operator.) Thus it is useful to factor out this global phase. In particular it is useful to always choose the global phase such that the coefficient of the  i ket is real an nonnegative. Using this and the normalization constraint, it is useful to not use α and β , but instead to use α = cos ( θ 2 ) and β = e iφ sin ( θ 2 ) where ≤ θ ≤ π and 0 ≤ φ < 2 π . When we do this, we see that we can map all of the single qubit states onto the surface of....
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 Fall '08
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 Computer Science, Quantum entanglement, Qubit, A. Deutsch

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