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IOP Publishing

New J. Phys. 18 (2016) 021006

doi:10.1088/1367-2630/18/2/021006

New Journal of Physics

The open access journal atthe forefront of physics

Deutsche Physikalische Gesellschaft DPG

IOP Institute of Physics

Published in partnership with: Deutsche Physikalische Gesellschaft and the Institute of Physics

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PUBLISHED

5 February 2016

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Fractal hard drives for quantum information

James R Wootton

Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland Keywords: quantum computation, topological order, quantum error correction, quantum memory

Abstract

A quantum hard drive, capable of storing qubits for unlimited timescales, would be very useful for quantum computation. Unfortunately, the most ideal solutions currently known can only be built in a universe of four spatial dimensions. In a recent publication (Brell 2016 New J. Phys. 18 013050), Brell introduces a new family of models based on these ideal solutions. These use fractal lattices, and result in models whose Hausdorff dimension is less than 3. This opens a new avenue of research towards a quantum hard drive that can be build in our own 3D universe.

A very useful feature of our computers is the hard drive. With it we can save a file one day and expect it to still be there the next, or even in a year. Despite the ravages of time, with the magnetic film within the drive being constantly subjected to thermal noise and fluctuating magnetic fields, we can be confident that the information stored can be reliably retrieved. The hard drive does not need to remain powered in order to achieve this. It does not need to be constantly checking itself for errors and undoing any damage. Instead it can rely on the natural robustness ofthe magnetic materials used to store the data.

Within a quantum computer we would similarly need to store quantum information. This is made up of quantum bits or 'qubits'. Rather than a simple 0 or 1, these can be in any arbitrary quantum superposition of the two. Unfortunately, current storage times are extremely short. For a qubit, even a second is nearly an eternity. Finding ways to extend their lifetimes is therefore a highly important part of developing a quantum computer.

The most promising proposals are those in which qubits are not stored in single particles, but their states are instead spread over multipartite systems. This additional redundancy prevents single particle errors from affecting the qubit, and allows the possibility for errors to be detected and corrected before they can do any harm.

A quantum hard drive, often known as a self correcting quantum memory, would perform this correction passively [2]. It would require no constant influx of resources or external help. Instead it's energy landscape would ensure that errors are energetically suppressed, and are far more likely to be decay than grow into something dangerous for the qubit.

One proposal for a such a quantum hard drive has been known for around a decade. It is almost perfect, promising a lifetime for the qubit that grows exponentially with the system size. However, it has one fatal flaw. In order for its many body interactions to be realistic, particles must only be required to interact with their near neighbours. But, given the nature ofthe required interactions, this would require the particles to be arranged in a four dimensional lattice. Unfortunately, this is one more spatial dimension than we have access to. This model, known as the 4D toric code, is a tantalising example of what might be possible [3].

In the last few years there has been many studies into what is and isn't possible for quantum hard drives. Mutiple proof-of-principle systems have been proposed [4-6], and some may well satisfy our practical needs within a quantum computer. But none yet have all the theoretical perfection of the 4D toric code. Can this model, or one like it, be realised in our three dimensional universe? Important progress towards answering this question has now been made in [1] by using a different perspective on the problem. Rather than directly asking whether it can be embedded in our universe of three Euclidean dimensions, it asks whether it is possible for lattice with Hausdorffdimension less than three.

The Hausdorff dimension is a concept from the study of fractals, generalising the notion of dimension from the Euclidean spaces that we are familiar with. It still assigns a dimension of two to a square, and three to a cube, but it is also defined in a way that can account for the more complex fractal structures. For these the dimension

© 2016 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft

Publishing New J. Phys. 18 (2016) 021006

JRWootton

need not be an integer. The relationship between the Euclidean and Hausdorff dimensions is not straightforward: it may not be possible for a structure with Hausdorff dimension less than 3 to be built in our 3D universe, for example. Nevertheless, they are very much related.

To construct models with favourable properties, we can often try to modify or combine existing ones. The homological product is one method to do this [7]. It takes two existing models and creates a more complex one, typically defined on a lattice in a larger number of spatial dimensions. It turns out that the 4D toric code can be thought of as the homological product of a pair of systems, each defined on a two dimensional square lattice. In order to alter the outcome of the product, Brell alters the input by using systems based on a Sierpinski carpet graph instead. Unlike the two dimensional square lattice, the Hausdorff dimension of this is less than two. Indeed, with a suitable choice of the fractal, any Hausdorff dimension between 1 and 2 can be achieved. This reduction in dimensionality results in the final homological product code having a Hausdorff dimension that is not only less than four, but is even less than three. Whether this translates to a model that can be built in our Eulcidean universe remains to seen, but it is a major step towards the goal.

It is important to note that it is not only the dimensionality of the model that has changed. Its ability to resist thermal noise and preserve our qubit cannot be taken for granted. To demonstrate this, Brell shows that the partition function of the new code corresponds to that of a simpler and well-studied model: the Ising model on a Sierpinski carpet. This is well known to have a magnetically ordered phase below a finite temperature phase transition [8-10]. For the Brell's code, this corresponds to a phase in which the qubit is protected from the noise caused by temperature.

Further important properties are still yet to be explored, such as robustness against noise caused by Hamiltonian perturbations and how easy it is to measure the qubit when we want to. However, even if the current code is found wanting, Brell's work has introduced us to the new landscape of fractal product codes. Perhaps our quantum hard drive lies somewhere therein.

References

[1] BrellCG2016 New J. Phys. 18 013050

[2] Brown B J, Loss D, Pachos J K, Self C N and Wootton J R 2014 arXiv:1411.6643

[3] Dennis E, Kitaev A, Landahl A and Preskill J 2002 J. Math. Phys. 43 4452

[4] Bravyi S and Haah J 2013 Phys. Rev. Lett. 111 200501

[5] Pedrocchi F L, Hutter A, Wootton J Rand Loss D 2013 Phys. Rev. A 88 062313

[6] Hutter A, Pedrocchi F L, Wootton J Rand Loss D 2014 Phys. Rev. A 90 012321

[7] Bravyi Sand Hastings MB 2014 Proc. ofthe XLVI Annual ACM Symposium on Theory of Computing, STOC vol 14 (New York: ACM)

273-82

[8] Shinoda M 2002 J.Appl.Prob. 39 1

[9] Vezzani A 2003 J. Phys. A: Math. Gen. 261593

[10] Campari R and Cassi D 2010 Phys. Rev. E 81 021108