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Unformatted text preview: CSE 599d  Quantum Computing Introduction to Quantum Error Correction Dave Bacon Department of Computer Science & Engineering, University of Washington In the last lecture we saw that open quantum systems could interact with an environment and that this coupling could turn pure states into mixed states. As we have argued before, this is a bad process, because it can lessen or destroy the interference effects which are vital to distinguishing a quantum from a classical computer. This is called the decoherence problem . In this lecture we will begin to see how to deal with this problem. Note, however, right off the bat, that there are other problems besides the decoherence problem. For example we still havent address the fact that we dont have perfect control over our quantum system when attempting to perform unitary gates, preparation of states, or measurement. We will deal with all of these in good time, but for now we will focus on the problem of decoherence. I. SIMPLE CLASSICAL ERROR CORRECTION Well when we are stumped about what to do in the quantum world, it is often useful to look to the classical world to see if there is an equivalent problem, and if so, how that problem is dealt with. This leads us from quantum noise to classical noise. Suppose we have the following classical situation, we have a bit which we send through a channel and with probability 1 p nothing happens to our bit, but with probability p the bit is flipped. This channel is called a binary symmetric channel. We can describe it by one of our doubly stochastic matrices if we feel like it M = 1 p p p 1 p . (1) Now if we use this channel once, then the probability that we will receive the wrong bit is p . Is there a way to use this channel in such a way that we can decrease this probability? Yes, and it is rather simple. We just use the channel multiple times and use redundancy. In other words, if we want to send a 0, we use the encoding 0 000 and 1 111 and send each of these bits through the channel. Now of course there will still be errors on the channel. With probability (1 p ) 3 no errors occur on the bits. With probability 3(1 p ) 2 p one error occurs on the bits. With probability 3(1 p ) p 2 two error occur on the bits. And with probability p 3 three errors occur on the bits. Now assume that p is small for intuitions sake (we will calculate what small means in a second.) Notice that the three probabilities we have listed above will then be in decreasing order. In particular the probability of no or one error will be greater than there being two or three errors. But if a single error occurs on our bit, we can detect this and correct it. In particular if, on the other end of the channel we decode the states by { 000 , 001 , 010 , 100 } 0 and { 111 , 110 , 101 , 011 } 1, then in the first two cases we will have correctly transmitted the bit, even though a single error occurred on this bit. We can thus calculate the probability that this procedure, encoding, sending the bitserror occurred on this bit....
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 Fall '08
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 Computer Science

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