Anyonic entanglement and topological entanglement entropy

Bonderson, Parsa; Knapp, Christina; Patel, Kaushal

doi:10.1016/j.aop.2017.07.018

Cited by 24 publications

(35 citation statements)

References 108 publications

(244 reference statements)

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“…The topological entanglement entropy is defined using von Neumann entanglement entropies for the subsystems [24,25]. In the case of Abelian topological order (such as the toric code), the same equation holds when the von Neumann entropies are replaced by second Rényi entropies [41,42]. This equivalence is helpful when investigating larger system sizes, as we can extract the second Rényi entropies from the statistical correlations of the subsystems using the technique of randomized measurement (RM) [28][29][30].…”

Section: Measuring Topological Entanglement Entropymentioning

confidence: 99%

Realizing topologically ordered states on a quantum processor

Satzinger

Liu

Smith

et al. 2021

Science

301

131

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The discovery of topological order has revolutionized the understanding of quantum matter in modern physics and provided the theoretical foundation for many quantum error correcting codes. Realizing topologically ordered states has proven to be extremely challenging in both condensed matter and synthetic quantum systems. Here, we prepare the ground state of the toric code Hamiltonian using an efficient quantum circuit on a superconducting quantum processor. We measure a topological entanglement entropy near the expected value of ln 2, and simulate anyon interferometry to extract the braiding statistics of the emergent excitations. Furthermore, we investigate key aspects of the surface code, including logical state injection and the decay of the non-local order parameter. Our results demonstrate the potential for quantum processors to provide key insights into topological quantum matter and quantum error correction.

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Section: Measuring Topological Entanglement Entropymentioning

confidence: 99%

Realizing topologically ordered states on a quantum processor

Satzinger

Liu

Smith

et al. 2021

Science

301

131

View full text Add to dashboard Cite

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“…[37]. The universal statistical properties of anyons are encoded in their topological quantum field theory (TQFT) [38][39][40] . Our subsequent symmetry decomposition of entanglement relies on the concept of "topological charge" or "total fusion channel", a conserved quantity in anyonic systems.…”

Section: A Non-abelian Anyonsmentioning

confidence: 99%

“…2(a) (here, we utilize the diagrammatic notations by Kitaev 38 as presented in Ref. [39]; see Appendix A). Consequently, the density matrix of the full system is…”

Section: Boundary-anyons Entanglementmentioning

confidence: 99%

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Entanglement spectroscopy of non-Abelian anyons: Reading off quantum dimensions of individual anyons

et al. 2019

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We study the entanglement spectrum of topological systems hosting non-Abelian anyons. Akin to energy levels of a Hamiltonian, the entanglement spectrum is composed of symmetry multiplets. We find that the ratio between different eigenvalues within one multiplet is universal and is determined by the anyonic quantum dimensions. This result is a consequence of the conservation of the total topological charge. For systems with non-Abelian topological order, this generalizes known degeneracies of the entanglement spectrum, which are hallmarks of topological states. Experimental detection of these entanglement spectrum signatures may become possible in Majorana wires using multicopy schemes, allowing the measurement of quantum entanglement and its symmetry resolution.arXiv:1810.01853v2 [cond-mat.str-el]

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“…where N ϕ ϕ,a tells us if the anyon a can localize at the twist ϕ. In addition to calculating the quantum dimension of a twist defect directly from its fusion rules, we also point out work where, like with non-abelian anyons [83,84], the quantum dimension of twist defects can be evaluated by studying the entanglement entropy of the ground state of topological phases that include twist defects [85][86][87].…”

Section: Anyon Localization and The Quantum Dimensions Of Twistsmentioning

confidence: 99%

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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Anyonic entanglement and topological entanglement entropy

Cited by 24 publications

References 108 publications

Realizing topologically ordered states on a quantum processor

Realizing topologically ordered states on a quantum processor

Entanglement spectroscopy of non-Abelian anyons: Reading off quantum dimensions of individual anyons

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

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