Quantum Error Correction and Fault-Tolerant Quantum Computation
We have learned now that quantum resources can provide substantial benefits in practical tasks like safe communication, computation and metrology. A goal for quantum error correction is to protect the fragile quantum effects from desctruction so that we can benefit from them in practise. And the desctruction, in a process called decoherence, can occure both from unwanted interaction with environment, as well as from a necesarry interface between operating quantum systems and macro-scale world. Fault-tolerance of the error correcting schemes is a further demand - that error-correction fulfills its task even under influence of noise.

KCIK reserchers have contributed to the field by analyzing stability of toric codes against thermal noise, showing that 2D quantum memory is unstable [1], while 4D is stable [2]. This was achieved by bounding spectral gap of Liouvillian.

We have also proved possibility of long distance quantum communication in two-dimensional networks with local resources [3] and fault-tolerant algorithms for encoding unknown quantum states into quantum error correction codes, which may be utilised to perform universal quantum computation [4].

[1] R. Alicki, M. Fannes, M. Horodecki, On thermalization in Kitaev's 2D model, J. Phys. A: Math. Theor. 42 (2009) 065303.
[2] R. Alicki, M. Horodecki, P. Horodecki, R. Horodecki, On thermal stability of topological qubit in Kitaev's 4D model, Open Syst. Inf. Dyn. 17 (2010) 1.
[3] P. Mazurek, A. Grudka, M. Horodecki, P. Horodecki, J. Łodyga, Ł. Pankowski, A. Przysiężna, Long-distance quantum communication over noisy networks without long-time quantum memory, Phys. Rev. A 90, 062311 (2014).
[4] J. Łodyga, P. Mazurek, A. Grudka, M. Horodecki, Simple scheme for encoding and decoding a qubit in unknown state for various topological codes, Scientific Reports 5, 8975 (2015).