ASSESSMENT OF CURRENT RESULTS AND OUTLOOK ON FUTURE EFFORTS
QUANTUM INFORMATION SCIENCE THEORY
QUANTUM ERROR CORRECTION & PURIFICATION
The ability to carry out coherent quantum operation even in the presence of inevitable noise is a key requirement for quantum information processing. To cope with this decoherence problem, active strategies (quantum error correcting codes) as well as passive ones (error avoiding codes) have been developed.
Error correcting codes allow one to reduce errors by suitable encoding of logical qubits into larger systems. It has been shown that, with operations of accuracy above some threshold, the ideal quantum algorithms can be implemented. Recent ideas involving error correcting teleportation have made the threshold estimate more favorable by several orders of magnitude. This path has to be continued and adapted to realistic error models and to alternative models of quantum computation like the adiabatic model or the cluster model (see section 4.3.3).
In error avoiding codes, no active monitoring/intervention on the system is in principle necessary, since errors are simply circumvented. Error avoiding is based on the symmetry structure of the system-environment interaction that in some circumstances allows for the existence of decoherence-free subspaces (DFS), i.e., subspaces of the system Hilbert state-space over which the dynamics is still unitary. The prototype noise model for which this situation occurs is provided by the so-called collective decoherence, where all the qubits are affected by the environment in the same way. For encoding a single logical noiseless qubit for general collective decoherence (dephasing), four (two) physical qubits are needed. DFSs have been experimentally demonstrated in a host of physical systems, and their scope extended by generalizing the idea of symmetry-aided protection to noiseless subsystems.
A fruitful connection with the theory of entanglement purification, which has been developed primarily in the context of quantum communication, and has been used in protocols such as the quantum repeater, is also emerging,. Entanglement purification is a method to “distill” from a large ensemble of impure (low-fidelity) entangled states a smaller ensemble of pure (high-fidelity) entangled states. It seems that appropriately generalized procedures can be employed also in general quantum computation (e.g. for quantum gate purification, or for the generation of high fidelity resource states) while benefiting from the relaxed thresholds that exist for entanglement purification.
Key references
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[2] J. Preskill, ‘‘Fault-tolerant quantum computation’’, in ‘Introduction to quantum computation and information’, (H.K. Lo, S. Popescu, T. Spiller, eds.) pp. 213-269, World Scientific, Singapore (1998).
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[5] D. Deutsch, A. Ekert, R. Josza, C. Macchiavello, S. Popescu, and A. Sanpera, ‘‘Quantum privacy amplification and the security of quantum cryptography over noisy channels’’, Phys. Rev. Lett. 77, 2818 (1996).
[6] H.-J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, ‘‘Quantum repeaters: The role of imperfect local operations in quantum communication’’, Phys. Rev. Lett. 81, 5932 (1998).
[7] A.M. Steane, ‘‘Overhead and noise threshold of fault-tolerant quantum error correction’’, Phys. Rev. A 68, 042322 (2003).
[8] E. Knill, ‘‘Quantum computing with very noisy devices’’, quant-ph/0410199, http://www.arxiv.org.