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Unformatted text preview: CSE 599d  Quantum Computing Stabilizer Quantum Error Correcting Codes Dave Bacon Department of Computer Science & Engineering, University of Washington In the last lecture we learned of the quantum error correcting criteria and we discussed how it was possible for us to digitize quantum errors. But we didn’t talk about any concrete codes, the talk was in many ways very existential on these things called quantum error correcting codes. Of course we saw in Shor’s code that such codes could exist for single qubit errors. In this lecture we will introduce a very handy set of tools for describing a class of quantum error correcting codes. This formalism is called the stabilizer formalism. Further with stabilizer codes we can also begin to discuss an important topic, which is not just doing quantum error correction, but also we can begin to discuss quantum gates on these codes and this will eventually lead us to the issues of faulttolerant quantum computation. I. ANTICOMMUTING Suppose that we have a set of states  ψ i i which are +1 eigenstate of a hermitian operator S , S  ψ i i =  ψ i i . Further suppose that T is an operator which anticommutes with S , ST = TS ( T is not zero.) Then it is easy to see that S ( T  ψ i i ) = TS  ψ i i = ( T  ψ i i ). Thus the states T  ψ i are 1 eigenstates of S . Since the main idea of quantum error correction is to detect when an error has occurred on a code space, such pairs of operators S and T can be used in such a manner: if we are in the +1 eigenvalue subspace of S then an error of T on these subspaces vectors will move to a 1 eigenvalue subspace of S : we can detect that this error has occurred. In fact, we have already seen an example of this in the bit flip code. Recall that we noted that the code subspace for this code was spanned by  000 i and  111 i and that these two operators are +1 eigenstates of both S 1 = Z ⊗ Z ⊗ I and S 2 = Z ⊗ I ⊗ Z . Further note that ( X ⊗ I ⊗ I ) S 1 = S 1 ( X ⊗ I ⊗ I ) and ( X ⊗ I ⊗ I ) S 1 = S 2 ( X ⊗ I ⊗ I ). Thus if we start out in the +1 eigenvalue subspace of both S 1 and S 2 (like the bit flip code), then if a single bit flip occurs on the first qubit, we will now have a state which is in the 1 eigenvalue subspace of both S 1 and S 2 . This at least fulfills our requirement that our errors should take us to orthogonal subspaces. More generally, considering the following situation. Suppose that we have a set of operators S i such that our code space is defined by S i  ψ i =  ψ i for  ψ i in the code subspace. Now suppose that we have errors E i such that the products E † k E l always anticommutes with at least one S i . Recall the quantum error correcting criteria was h φ i  E † k E l  φ j i = C kl δ ij (1) Since the codewords are +1 eigenvalue eigenstates of S i , we find that h φ i  E † k E l  φ j i = h φ i  E † k E l S i  φ j i (2) Suppose that S i is one of the particular S i s that anticommute with...
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 Fall '08
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 Computer Science, Quantum computer, Stabilizer, Quantum information science, Quantum error correction, Pauli group, Stabilizer Group

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