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I will introduce Kitaev's suface codes as a block quantum error-correcting code. Recovery procedures will be described in the case of imperfect syndrome measurements. More might be covered if time permits.
Imagine that Alice and Bob share a quantum state, from which they want to distill something useful like entanglement or secret key. For this they need to communicate classically and they want to do this by one way communication from Alice to Bob. For some states, it might happen that the state is a part of a tripartite state shared with Charlie, which is invariant if Bob's and Charlie's systems are switched. Such a state is called a symmetric extension, and if it exists Alice and Bob have no chance of distilling key or entanglement by one way communication.
Traditionally, we use the quantum Fourier transform circuit (QFT) in order to perform quantum phase estimation, which has a number of useful applications. The QFT circuit for a binary field generally consists controlled-rotation gates which, when removed, yields the lower-depth approximate QFT circuit. It is known that a logarithmic-depth approximate QFT circuit is sufficient to perform phase estimation with a degree of accuracy negligibly lower than that of the full QFT.
Proving the additivity of the classical capacity of quantum channels is a major open problem in quantum information. This problem is related to the multiplicativity of certain norms with respect to the tensor product. These problems are introduced and some approached to resolving them are discussed. Several special cases that have been solved are also mentioned.
Any implementation of a quantum computer will require the ability to reset qubits to a pure input state, both to start the computation and more importantly to implement fault-tolerant operations. Even if we cannot reset to a perfectly pure state, heat-bath algorithmic cooling provides a method of purifying mixed states. By combining the ability to pump entropy out of the system through a controllable interaction with a heat bath and coherent control of the qubits, we are able to cool a subset of the qubits far below the heat bath temperature.
In this study, we are interested in the practical question of how many times a quantum directional reference frame (i.e., a spin-J system) can be used to perform a certain task with a given probability of success, under the assumption that the quantum directional reference frame evolves under a map that is covariant under rotations in SU(2). Our main theorem restricts the form of the state of the quantum reference frame as a function of how many times the covariant map was applied to it. Our results are a generalization of the paper of Bartlett el al.