A family of stabilizer codes for $D({{\mathbb{Z}}_{2}})$ anyons and Majorana modes

Wootton, James R.

doi:10.1088/1751-8113/48/21/215302

Cited by 20 publications

(31 citation statements)

References 26 publications

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“…We now study crossing domain walls and the twists which terminate them. This is relevant when twists are moved over domain walls using, for instance, code deformations [12,17,21,44,88]. To figure out how the twist changes after crossing a domain wall we consider closed contractible loops engulfing the crossing point of the domain walls.…”

Section: Interplay Between Twists and Domain Wallsmentioning

confidence: 99%

“…Hyperbolic codes [92][93][94] exceed the bound given in Eq. (21), but are expected to come along with significant experimental challenges, as they can not be embedded in two spatial dimensions while maintaining the locality of stabilizers.…”

Section: Stellated Color Codesmentioning

confidence: 99%

“…Indeed, among the leading architectures that have been proposed for scalable quantum-information processing are topological quantum error-correcting codes. The physics of these phases is enriched further when their low-energy excitations are connected by symmetries, in which case they give rise to non-trivial domain walls and pointlike topological defects [10][11][12][13][14][15][16] that have been utilized in a number of instances [12,[17][18][19][20][21][22][23][24][25][26][27] to improve protocols for topological quantum computation and to reduce overhead requirements. Advancing the study of topological defects and their corresponding symmetries gives us new tools to improve architectures for topological quantum computation and, moreover, extends our understanding of the underlying topological phases that support them.…”

mentioning

confidence: 99%

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The boundaries and twist defects of the color code and their applications to topological quantum computation

Kesselring

Pastawski

Eisert

et al. 2018

Quantum

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The color code is both an interesting example of an exactly solvable topologically ordered phase of matter and also among the most promising candidate models to realize faulttolerant quantum computation with minimal resource overhead. The contributions of this work are threefold. First of all, we build upon the abstract theory of boundaries and domain walls of topological phases of matter to comprehensively catalog the objects as they are realizable in color codes. Together with our classification we also provide lattice representations of these objects which include three new types of boundaries as well as a generating set for all 72 color code twist defects. Our work thus provides an explicit toy model that will help to better understand the abstract theory of domain walls. Secondly, building upon the established framework, we discover a number of interesting new applications of the cataloged objects for devising quantum information protocols. These include improved methods for performing quantum computations by code deformation, a new four-qubit error-detecting code, as well as families of new quantum errorcorrecting codes we call stellated color codes, which encode logical qubits at the same distance as the next best color code, but using approximately half the number of physical qubits. To the best of our knowledge, our new topological codes have the highest encoding rate of local stabilizer codes with boundedweight stabilizers in two dimensions. Finally, we show how the boundaries and twist defects of the color code are represented by multiple copies of other phases. Indeed, in addition to the well studied comparison between the color code and two copies of the surface code, we also compare the color code to two copies of the three-fermion model. In particular, we find that this analogy offers a very clear lens through which we can view the symmetries of the color code which gives rise to its multitude of domain walls.

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Section: Interplay Between Twists and Domain Wallsmentioning

confidence: 99%

Section: Stellated Color Codesmentioning

confidence: 99%

mentioning

confidence: 99%

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The boundaries and twist defects of the color code and their applications to topological quantum computation

Kesselring

Pastawski

Eisert

et al. 2018

Quantum

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“…Alternatively, braiding can be done using either interaction-based proposals [31,32] or a measurement-only approach [33][34][35] without physically moving anyons. These two approaches are shown to be equivalent [36] and they can avoid diabatic errors associated with moving anyons.…”

Section: Introductionmentioning

confidence: 99%

Measurement-only topological quantum computation without forced measurements

2017

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We investigate the measurement-only topological quantum computation (MOTQC) approach proposed by Bonderson et al (2008 Phys. Rev. Lett. 101 010501) where the braiding operation is shown to be equivalent to a series of topological charge 'forced measurements' of anyons. In a forced measurement, the charge measurement is forced to yield the desired outcome (e.g. charge 0) via repeatedly measuring charges in different bases. This is a probabilistic process with a certain success probability for each trial. In practice, the number of measurements needed will vary from run to run. We show that such an uncertainty associated with forced measurements can be removed by simulating the braiding operation using a fixed number of three measurements supplemented by a correction operator. Furthermore, we demonstrate that in practice we can avoid applying the correction operator in hardware by implementing it in software. Our findings greatly simplify the MOTQC proposal and only require the capability of performing charge measurements to implement topologically protected transformations generated by braiding exchanges without physically moving anyons.

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“…[8][9][10][11] for reviews. Ising anyons also appear as excitations [12][13][14][15] or ends of defect lines [16][17][18][19] in several spin-lattice models.…”

Section: Introductionmentioning

confidence: 99%

Continuous error correction for Ising anyons

Hutter

Wootton

2016

Phys. Rev. A

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Quantum gates in topological quantum computation are performed by braiding non-Abelian anyons. These braiding processes can presumably be performed with very low error rates. However, to make a topological quantum computation architecture truly scalable, even rare errors need to be corrected. Error correction for non-Abelian anyons is complicated by the fact that it needs to be performed on a continuous basis and further errors may occur while we are correcting existing ones. Here, we provide the first study of this problem and prove its feasibility, establishing non-Abelian anyons as a viable platform for scalable quantum computation. We thereby focus on Ising anyons as the most prominent example of non-Abelian anyons and show that for these a finite error rate can indeed be corrected continuously. There is a threshold error rate pc > 0 such that for all error rates p < pc the probability of a logical error per time-step can be made exponentially small in the distance of a logical qubit.

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A family of stabilizer codes for $D({{\mathbb{Z}}_{2}})$ anyons and Majorana modes

Cited by 20 publications

References 26 publications

The boundaries and twist defects of the color code and their applications to topological quantum computation

The boundaries and twist defects of the color code and their applications to topological quantum computation

Measurement-only topological quantum computation without forced measurements

Continuous error correction for Ising anyons

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