Alicea et al.

[3] described a method to accomplish this, as shown in Figure 5.

First, one attaches a perpendicular nanowire to form a T-junction.

Then, using precisely controlled electric fields, one can transport one of the MFs onto the bottom of the T, slide the second MF across the T, and transport the first MF to the opposite side.

By having a network of these T-junctions, one can perform arbitrary braid operations on the MFs.

Figure 5: Procedure of braiding using a T-junction.

Error Correction with Topological Quantum ComputingAs mentioned earlier in this study, we noted that the reason why topological quantum computing is so appealing is that the local perturbations and noise are less likely to affect the state of the system, meaning it is less likely for decoherence to occur.

This is mathematically stated by the threshold theorem.

The threshold theorem states that given that the probability of a quantum error per gate is low enough (lower than a threshold value pc), we are able to use the system for quantum computation for an indefinite amount of time with accuracy.

Traditionally, the values of pc needed are quite low, on the order of 10–5.

However, with topological quantum computing using error correction, the values of pc are on the order of 10–2 [5].

In 2012, Yao et al have been able to experimentally show topological error corrections to work in cluster state computing.

The group used six photons structured in a 3D cell structure known as a “cell complex” that has redundant topological correlations.

An illustration of the cell complex is shown in Figure 6.

These correlation values are always one in an undisturbed system and under specific errors, follow the scheme described by the table in Figure 7.

Through this correlation scheme, we can determine if there is an error and on what qubit the error occurs on.

Figure 6: Photons in this cluster state have topological propertiesFigure 7: Redundancy of information coming from the cluster state allows us to observe errors when the correlations are offRunning the experiments on an intricate system of lasers gives us the resulting expectation values that confirm the theoretical understanding of the topological structure.

The graph in Figure 8 shows the experimental results.

Figure 8: Experimental results that show error detectionCurrent State of the Art ResearchCurrently, there are many institutions that are betting on topological quantum computing on becoming realized and they are putting a lot of resources in doing so.

One group, in particular, is Microsoft’s Station Q, a group dedicated to research in this field.

They have many sites that are working on separate yet interconnected pieces of the topological quantum computing puzzle.

The teams are listed below.

● Santa Barbara: Founded by Dr.

Michael Freedman, a Field Medal winner.

Researching the topological phases of matter, specifically Majorana zero modes.

Practical implementation of topological qubits.

● Delft: Testing the stability of topological quantum gates and working on the demonstration of the failure of decoherence in topologically protected states.

● Redmond: Developing real-world quantum algorithms and understanding their implications, as well as designing software architecture.

● Sydney: Addressing the challenges of scaling up quantum computing.

Interfacing between quantum systems and working on classical control and readout.

● Copenhagen: Studying the use of and properties of the Majorana fermions and building systems that can support such a particle.

● Purdue: Studying quantum mechanical properties of electrons in ultra-high purity III-V semiconductor devices.

Heavily focused on molecular beam epitaxy (MBE), which allows them to build structures one atomic layer at a time.

● ETH Zurich: Looking for the right materials that permit quantum computers to exist.

Recently, researchers from Zurich, Intel, and Redmond published a paper that described emulation of quantum circuits on classical machines to test theoretical quantum algorithms.

The emulations are much faster in performance than simulations of quantum circuits.

Current simulations compute slow, large matrix-vector products in computing each quantum gate in a classical computer.

Instead, the emulations reduce the amount of complicated linear algebra and rely on using classical algorithms, such as the fast Fourier transform, to run the quantum programs [6].

Currently, the field of topological quantum computation is still in its infancy, compared to more conventional quantum computation, such as with trapped ions or with superconducting circuits.

However, the recent incremental progress, both in the theoretical and experimental aspects, indicates that the future of the field is promising.

AcknowledgmentAlexander Shengzhi Li, Andre He, Dennis Feng, Jeffrey Zhang worked on this study together as part of a quantum computing course at the University of California, Berkeley Computer Science Department with Professor Umesh Vazirani.

The study would not be completed without the feedback of the professor or the class TAs.

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