Quantum error correction and quantum hard drives in four dimension. Part IV of my attempt to explain one of my main research interests in quantum computing:

Prior parts: Part I, Part II, Part III.

### Quantum Error Correction

Classical error correction worked by encoding classical information across multiple systems and thus protecting the information better than if it was encoded just locally. Fault-tolerant techniques extend these results to the building of actual robust classical computers. Given that quantum theory seems to be quite different from classical theory, an important question to ask is whether the same can be achieved for information encoded in a quantum manner. The answer to this question, of whether quantum information can be successfully protected even when the quantum system being used is exposed to unwanted evolutions, is one of the great discoveries of quantum computing. In 1995, Peter Shor and Andrew Steane showed that, by using some clever tricks, one could perform quantum error correction on quantum systems and therefore preserve quantum information by suitably encoding the quantum information across multiple independently erred systems. This was a remarkable result, and was the beginning of a series of important discoveries about quantum information which showed how a reliable quantum computer was possible, in spite of the seemingly odd nature of quantum information.

The threshold theorem for fault-tolerant quantum computing moves the model of quantum computation from totally crazy to fundable |

The culmination of research in quantum error correction is usually expressed in terms of a result known as the threshold theorem for fault-tolerant quantum computing. This result states that if a quantum system can be controlled with enough precision, and does not enact with its environment too strongly (both below a threshold) then long quantum computations can be enacted with a cost which scales efficiently with the desired computation being enacted. The threshold theorem essentially states that the model of quantum computing is emphatically not the model of an analog computer, and that, assuming we understand how quantum theory and physics works, a large scale quantum computer is possible. That being said, the conditions for the threshold theorem for quantum computation are severe. While small scale quantum computers of a few qubits have successfully been demonstrated, none of the systems has been easily scaled up to a large scale system needed to test the threshold theorem of quantum computing, nor are all of the quantum computers demonstrated below the threshold values in terms of control of the quantum system.

### Quantum Hard Drives in Four Dimensions

Given that the quantum error correction is possible, a natural question to ask is whether, like in classical computers, there exist, or whether we can engineer, quantum systems which robustly store quantum information via the physics of these quantum systems. The first to suggest that this might be a viable path toward constructing a quantum computer was Alexei Kitaev. Kitaev suggested that there were certain physical systems, related to topological field theories, where one could encode quantum information into their ground states, and an energetic gap would protect this quantum information from certain errors. The model Kitaev considered could be made into robust storage devices, but were not, by their physics alone, fault-tolerant. Thus, while enacting a computation, Kitaev’s original models were not robust to error. A way around this, however, was found: if instead of using Kitaev’s model in the two spatial dimensions originally considered, one looked at these models in four spatial dimensions, then the resulting physical system would be self-correcting and fully fault-tolerant due to the physics of these devices. This model, considered by Dennis, Landahl, Kitaev, and Preskill, was, in essence a recipe for constructing a four dimensional quantum hard drive. However, unfortunately, we do not live in a four dimensional world, so this model is not realistic.

### Toward Quantum Hard Drives in Real Systems

A potential quantum memory |

So, given that we know that fault-tolerant quantum computation is possible, under reasonable assumptions, and we know that there exists models of physical systems which can enact these ideas in a natural setting, an important, and vexing problem, is whether we can engineer realistic physical systems which enact these ideas and which don’t have bad properties, like only existing in four spatial dimensions. This is the focus of my own research on “self-correcting” quantum computers: to develop techniques for building quantum computers whose physical dynamics enacts quantum error correction and which therefore don’t need an active quantum error correcting control system. One of our proposed systems, is the three dimensional system seen to the right of this picture. For details on this models see arXiv:quant-ph/0506023.

So, is it possible that building a quantum computer will be done via self-correcting quantum computers? At this point we don’t know the answer. We have a few good examples of such self-correcting systems, but none of them, as of yet, are completely reasonable. But if such systems exist with reasonable interactions/geometries this might present a different method for building a quantum computer than the path pursued by the majority of the quantum computing community. In other words, high risk and high reward.