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# quantum computation quantum error correcting codes and information theory Alstead, New Hampshire

D. The set of stabilizer codes is exactly the set of codes which can be created by a Clifford group encoder circuit using ∣0⟩ ancilla states. The salient point in these error-correction conditions is that the matrix element Cab does not depend on the encoded basis states i and j, which roughly speaking indicates that neither the S.

Asymptotically Good Quantum Codes. Because of the linearity of quantum mechanics, we can always take the set of errors E to be a linear space: If a QECC corrects Ea and Eb, it will also A short review of whats known about the different kinds of quantum capacities. The first demonstration was with NMR qubits.[4] Subsequently, demonstrations have been made with linear optics,[5] trapped ions,[6][7] and superconducting (transmon) qubits.[8] Other error correcting codes have also been implemented, such as

In either case, the final state still tells us nothing about the data beyond the eigenvalue of M. If we only wish to detect errors, a distance d code can detect errors on up to d − 1 qubits. Quantum Error Correction Via Codes Over GF(4). M.

The advantage of this procedure is that it measures just M and nothing more. Definition 3 Let S ⊂ Pn be an Abelian subgroup of the Pauli group that does not contain  − 1 or  ± i, and let C(S) = {∣ψ⟩ s.t. Instead of the unencoded ∣ + ⟩ state, we must use a more complex ancilla state ∣00…0⟩ + ∣11…1⟩ known as a 'cat' state. An operation consisting only of single-qubit gates is automatically transversal.

Fuchs, and J. The latter is counter-intuitive at first sight: Since noise is arbitrary, how can the effect of noise be one of only few distinct possibilities? w ⋅ v = 0 for all v ∈ C. Cerf and U.

Schaetz, M. D. Shor, Scheme for reducing decoherence in quantum memory'' A. Using this or another verification procedure, we can check a non-fault-tolerant construction.

S. A procedure due to Steane uses (forCSS codes) one ancilla in a logical $\left|\overline{0}\right\rangle$ state of the same code and one ancilla in a logical $\left|\overline{0}\right\rangle + \left|\overline{1}\right\rangle$ state. Nigg, L. C.

Lett. 81, 2152–2155 (1998), doi:10.1103/PhysRevLett.81.2152 ^ T. Cambridge University Press. ^ W.Shor, Peter (1995). "Scheme for reducing decoherence in quantum computer memory". Mikael Lassen, Metin Sabuncu, Alexander Huck, Julien Niset, Gerd Leuchs, Nicolas J. Raymond Laflamme and collaborators found a class of 5-qubit codes which do the same, which also have the property of being fault-tolerant.

Then the bit flip code from above can recover | ψ ⟩ {\displaystyle |\psi \rangle } by transforming into the Hadamard basis before and after transmission through E phase {\displaystyle E_{\text{phase}}} They would therefore appear to be those errors which cannot be detected by the code. The 7-qubit code is much studied because its properties make it particularly well-suited to fault-tolerant quantum computation. Unfortunately, the Clifford group by itself does not have much computational power: it can be efficiently simulated on a classical computer.

R. On the other hand, the fault-tolerant protocol is larger, requiring more qubits and more time to do each operation, and therefore providing more opportunities for errors. If the state is a  + 1 eigenvector of M, the ancilla will be ∣ + ⟩, and if the state is a  − 1 eigenvector, the ancilla will be ∣ − ⟩. About 15 years ago Peter Shor invented a quantum circuit (see figure above) consisting of 9 qubits (subsequently improved to 5 qubits) which allows for perfect reconstruction of 1 qubit of

B. Conversely, we are also interested in the problem of setting upper bounds on achievable values of (logK)/n and d/n. Of these, only the assumption of independent errors is at all necessary, and that can be considerably relaxed to allow short-range correlations and certain kinds of non-Markovian environments. For H2, we perform the same procedure, but each 1 is instead replaced by X.

Frank Gaitan (2008). "Quantum Error Correction and Fault Tolerant Quantum Computing". Quantum information is stored in a coherent state of a quantum system. C. Smolin, Quantum error-correcting codes need not completely reveal the error syndrome,'' quant-ph/9604006.

Unfortunately, the practical requirements for this result are not nearly so good.