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Unformatted text preview: CSE 599d  Quantum Computing The Quantum Error Correcting Criteria Dave Bacon Department of Computer Science & Engineering, University of Washington Now that we have seen that quantum error correction is possible, it is interesting to try to formalize a criteria for why it was possible. In particular we are interested in understanding when it is possible to encode into a subspace such that, for certain errors on the quantum information, we can fix the quantum information from this error. One thing to note that is in the first lecture we discussed encoding quantum information from a bare, unencoded qubit, into a qubit encoded over the subspace. In practice we really want to never do this but instead we want to be able to prepare an encoded quantum state. We therefore wont spend much time discussing encoding into a quantum error correcting code. I. THE QUANTUM ERROR CORRECTING CRITERIA Suppose that we have a quantum system which evolves according to some error process which we will represent by the superoperator D . Now we will assume that this superoperator is given by some operator sum representation D [ ] = X k A k [ ] A k (1) Now, in general, our codes will not be able to reverse the effect of all errors on our system: the goal of quantum error correction is to make the probability of error so small that it is effectively zero, not to eliminate the possibility of error completely (although philosophers will argue about the difference between this two: Im talking to you Henry James (easy to pick on a dead guy.)) It is therefore useful to assume that the Kraus operators in the expansion for D are made up of some errors E i = A i , i S which we wish to correct. This will be a good assumption because the real error process will contain these terms, which we will then be certain we have fixed, plus the errors which we might not fix. Thus we may think about D as have Kraus operators, some of which are error E k and some of which are not. Define E as the operator, E [ ] = X i E i [ ] E i (2) Notice that E will not necessarily preserve the trace of a density matrix. This wont stop us from considering reversing its operation. Okay, so given E with some Kraus operators A k we can ask, under what conditions is it possible to design a quantum code and a recovery operations R such that R E [ C ] C (3) for C with support over the code subspace, H C H ? Why do we use here instead of =? Well because E is not trace preserving now. This means that there may be processes which are occurring in the full D which occur with some probability and we do not need to preserve on these errors. Lets call a basis for the code subspace  i i . H C = span { i i} . al We will show that a necessary and sufficient condition for the recovery operations to preserve the subspace is that h i  E k E l  j i = C kl ij (4) where C kl is a hermitian matrix. This equation is called the quantum error correcting criteria. It tells us when ouris a hermitian matrix....
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 Fall '08
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 Computer Science

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