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In this talk I'll discuss recent joint work with Raymond Laflamme, David Poulin and Maia Lesosky in which a unified approach to quantum error correction is presented, called "operator quantum error correction". This scheme relies on a generalized notion of noiseless subsystems and includes the known techniques for the error correction of quantum operations --i.e., the standard model, the method of decoherence-free subspaces, and the noiseless subsystem method--as special cases. Correctable codes in this approach take the form of operator algebras and operator semigroups. The condition from the standard model is shown to be necessary for all of the known methods of error correction, and we'll see this as part of a discussion on conditions that characterize correctability for the general case.

Superconducting circuits based on Josephson junctions are promising candidates for the implementation of solid-state qubits. In most of the recent experiments on these circuits, the qubits are controlled by a classical field containing a large number of photons. The possibility of coherently coupling these systems to a single photon has been recently suggested, opening the possibility to study analogs of quantum optics in condensed matter systems. I will review one of these proposals based on a superconducting charge qubit fabricated inside a high quality transmission line resonator and will describe its recent experimental realization. When the qubit is brought into resonance with the resonator, vacuum Rabi splitting is observed indicating that the regime of strong coupling has been reached. When the qubit is detuned from the cavity, I will explain how quantum non-demolition measurement can be realized. I will discuss how the measurement process can be quantitatively understood in this regime allowing us to explore the effect of measurement back-action on the qubit and to extract, for the first time in superconducting qubits, large visibility in Rabi oscillations.

One of the central critical results in the theory of fault-tolerant quantum computation is that arbitrarily long reliable computation is possible provided the error rate per gate and per time step is below some threshold value. This was proved by a number of groups, but the detailed published proofs are complex and furthermore only hold for concatenation of quantum error-correcting codes able to correct 2 errors per block, while typically the best estimates of the threshold value are based on the 7-qubit code, which only corrects 1 error per block. I will describe recent work by Panos Aliferis, John Preskill, and myself which substantially simplifies existing proofs and applies as well to the concatenated 7-qubit code. The new proof also provides a nice framework in which to attempt to prove relatively high values of the threshold, which so far have only emerged as estimates from simulations