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Open Access Publications from the University of California

The Toric Code at Finite Temperature

  • Author(s): Freeman, Christian Daniel
  • Advisor(s): Whaley, Birgitta
  • Moore, Joel
  • et al.

Alexei Kitaev's toric code is a rich model, that has birthed and stimulated the development of topological quantum computing, error correction, and field theory. It was also the first example of a quantum error correcting code that could resiliently store quantum information in certain types of noisy environments without the need for active error correction. Unfortunately, the toric code loses much of its power as a noise-resilient quantum memory at any finite temperature. Many of the problems with the toric code at finite temperature are likewise shared among its cousin stabilizer codes. These problems can be traced to the proliferation of stringlike ``defects'' in these codes at finite temperature. The aim of this thesis is then twofold.

First, I characterize both numerically and theoretically the failure modes of the toric code at finite temperature with fairly modest bath assumptions. I achieve this by numerically sampling the nonequilibrium dynamics of the toric code using a continuous time monte carlo algorithm. From this analysis, I was able to derive an exact expression for the low and high temperature dynamics of the toric code, as well as a regime over which the toric code is ``most'' stable to noise.

Secondly, by leveraging the results of this analysis, I construct algorithms that aim to suppress these noise channels. These algorithms are broadly separated into measurement-free and few-measurement protocols. The measurement-free algorithms are reminiscent of dynamical-decoupling pulse sequences, applied in parallel or in serial to a target stabilizer code, and are entirely unitary. I find that measurement-free algorithms can provide a constant factor increase to the lifetime of a large class of stabilizer codes at finite temperature. The few-measurement protocols are more complex, but I provide evidence that they can provide a threshold for the toric code with asymptotically fewer measurements than was previously known to be achievable.

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