Celebratio Mathematica

article P. Bonderson, M. Freedman, and C. Nayak: “Measurement-only topological quantum computation,” Phys. Rev. Lett. 101 : 1 (2008), pp. 010501. MR 2429542 Zbl 1228.81121 ArXiv 0802.0279

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article P. Bonderson, M. Freedman, and C. Nayak: “Measurement-only topological quantum computation via anyonic interferometry,” Ann. Physics 324 : 4 (2009), pp. 787–826. MR 2508474 Zbl 1171.81004 ArXiv 0808.1933

We describe measurement-only topological quantum computation using both projective and interferometrical measurement of topological charge. We demonstrate how anyonic teleportation can be achieved using “forced measurement” protocols for both types of measurement. Using this, it is shown how topological charge measurements can be used to generate the braiding transformations used in topological quantum computation, and hence that the physical transportation of computational anyons is unnecessary. We give a detailed discussion of the anyonics for implementation of topological quantum computation (particularly, using the measurement-only approach) in fractional quantum Hall systems.

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techreport P. Bonderson, S. Das Sarma, M. Freedman, and C. Nayak: A blueprint for a topologically fault-tolerant quantum computer. Preprint, March 2010. ArXiv 1003.2856

The advancement of information processing into the realm of quantum mechanics promises a transcendence in computational power that will enable problems to be solved which are completely beyond the known abilities of any “classical” computer, including any potential non-quantum technologies the future may bring. However, the fragility of quantum states poses a challenging obstacle for realization of a fault-tolerant quantum computer. The topological approach to quantum computation proposes to surmount this obstacle by using special physical systems — non-Abelian topologically ordered phases of matter — that would provide intrinsic fault-tolerance at the hardware level. The so-called “Ising-type” non-Abelian topological order is likely to be physically realized in a number of systems, but it can only provide a universal gate set (a requisite for quantum computation) if one has the ability to perform certain dynamical topology-changing operations on the system. Until now, practical methods of implementing these operations were unknown. Here we show how the necessary operations can be physically implemented for Ising-type systems realized in the recently proposed superconductor-semiconductor and superconductor-topological insulator heterostructures. Furthermore, we specify routines employing these methods to generate a computationally universal gate set. We are consequently able to provide a schematic blueprint for a fully topologically-protected Ising based quantum computer using currently available materials and techniques. This may serve as a starting point for attempts to construct a fault-tolerant quantum computer, which will have applications to cryptanalysis, drug design, efficient simulation of quantum many-body systems, solution of large systems of linear equations, searching large databases, engineering future quantum computers, and — most importantly — those applications which no one in our classical era has the prescience to foresee.

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Michael H. Freedman

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