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Unformatted text preview: 6.080/6.089 GITCS May 6-8, 2008 Lecture 22/23 Lecturer: Scott Aaronson Scribe: Chris Granade 1 Quantum Mechanics 1.1 Quantum states of n qubits If you have an object that can be in two perfectly distinguishable states | or | 1 , then it can also be in a superposition of the | and | 1 states: | + | 1 where and are complex numbers such that: | | 2 + | | 2 = 1 For simplicity, lets restrict to real amplitudes only. Then, the possible states of this objectwhich we call a quantum bit, or qubit lie along a circle. Figure 1 : An arbitrary single-qubit state | drawn as a vector. If you measure this object in the standard basis, you see | with probability | | 2 and | 1 with probability | | 2 . Furthermore, the object collapses to whichever outcome you see. 1.2 Quantum Measurements Measurements (yielding | x with probability | x | 2 ) are irreversible, probabilistic, and discontinuous . As long as you dont ask specifically what a measurement is how the universe knows what constitutes a measurement and what doesntbut just assume it as an axiom, everything is well- defined mathematically. If you do ask, you enter a no-mans land. Recently theres been an important set of ideas, known as decoherence theory, about how to explain measurement as ordinary unitary interaction, but they still dont explain where the probabilities come from. 22/23-1 1.3 Unitary transformations But this is not yet interesting! The interesting part is what else we can do the qubit, besides measure it right away. It turns out that, by acting on a qubit in a suitable wayin the case of an electron, maybe shining a laser on itwe can effectively multiply the vector of amplitudes by any matrix that preserves the property that the probabilities sum to 1. By which I mean, any matrix that always maps unit vectors to other unit vectors. We call such a matrix a unitary matrix. Unitary transformations are reversible, deterministic, and continuous . Examples of unitary matrices: The identity I . 1 The NOT gate X = . 1 1 The phase- i gate . i 45-degree counterclockwise rotation. Physicists think of quantum states in terms of the Schr odinger equation, d | = iH | (perhaps dt the third most famous equation in physics after e = mc 2 and F = ma ). A unitary is just the result of leaving the Schr odinger equation on for a while. Q: Why do we use complex numbers? Scott: The short answer is that it works! A deeper answer is that if we used real numbers only, it would not be possible to divide a unitary into arbitrarily small pieces. For example, the NOT gate we saw earlier cant be written as the square of a real-valued unitary matrix. Well see in a moment that you can do this if you have complex numbers....
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