Knots, links, and long-range magic

Fliss, Jackson R.

doi:10.48550/arxiv.2011.01962

Cited by 4 publications

(9 citation statements)

References 54 publications

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“…It would be of interest to investigate how our notion of complexity might be related to other geometric notions of complexity in that context. We note the recent related paper [31].…”

Section: Summary and Discussionmentioning

confidence: 94%

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Complexity for link complement states in Chern-Simons theory

Leigh

Pai²

2021

Phys. Rev. D

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We study notions of complexity for link complement states in Chern Simons theory with compact gauge group G. Such states are obtained by the Euclidean path integral on the complement of ncomponent links inside a 3-manifold M 3 . For the Abelian theory at level k we find that a natural set of fundamental gates exists and one can identify the complexity as differences of linking numbers modulo k. Such linking numbers can be viewed as coordinates which embeds all link complement states into Z ⊗n(n−1)/2 k and the complexity is identified as the distance with respect to a particular norm. For non-Abelian Chern Simons theories, the situation is much more complicated. We focus here on torus link states and show that the problem can be reduced to defining complexity for a single knot complement state. We suggest a systematic way to choose a set of minimal universal generators for single knot complement states and then evaluate the complexity using such generators. A detailed illustration is shown for SU (2) k Chern Simons theory and the results can be extended to general compact gauge group.

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“…It would be of interest to investigate how our notion of complexity might be related to other geometric notions of complexity in that context. We note the recent related paper [31].…”

Section: Summary and Discussionmentioning

confidence: 94%

“…The order in (31) does not matter since their commutator is higher order. Comparing (30) and (31) we get…”

Section: The Path Of Minimal Complexitymentioning

confidence: 99%

Complexity for link complement states in Chern-Simons theory

Leigh

Pai²

2021

Phys. Rev. D

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“…Fliss [19] has studied knot and link states of SU(2) d Chern-Simons theory, and has shown that knot and link states are generically magical. However for U(1) d , magic is absent for all knot and link states.…”

Section: Discussionmentioning

confidence: 99%

“…However for U(1) d , magic is absent for all knot and link states. Since U(1) d is level-rank dual to SU(d) 1 , the knot and link states for this theory also have zero magic [19][20][21], [22].…”

Section: Discussionmentioning

confidence: 99%

Hypergraph States in SU(N)1, N odd prime, Chern-Simons Theory

Schnitzer¹

2021

Preprint

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Graph states and hypergraph states can be constructed from products of basic operations that appear in SU(N ) 1 . The level-rank dual of a theorem of Salton, Swingle, and Walter implies that these operations can be prepared topologically in the n-torus Hilbert space of Chern-Simons theory for N = 5 mod 4.For SU(N ) 1 , N = 5 mod 4, only stabilizer states can be prepared on the n-torus Hilbert space, which restricts the construction to graph states.

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“…[18][19][20][21][22]. Qudits also naturally arise in topological field theories -the Hilbert space of an SU (2) k Chern-Simons theory on a torus is k + 1-dimensional [23] -see, e.g., [24,25] for discussion related to quantum computing.…”

Section: Introductionmentioning

confidence: 99%

A Normal Form for Single-Qudit Clifford+$T$ Operators

Jain¹,

Kalra²,

Prakash³

2020

Preprint

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We propose a normal form for single-qudit gates composed of Clifford and T -gates for qudits of odd prime dimension p ≥ 5. We prove that any single-qudit Clifford+T operator can be re-expressed in this normal form in polynomial time. We also provide strong numerical evidence that this normal form is unique. Assuming uniqueness, we are able to use this normal form to provide an algorithm for exact synthesis of any single-qudit Clifford+T operator with minimal T -count.

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Knots, links, and long-range magic

Cited by 4 publications

References 54 publications

Complexity for link complement states in Chern-Simons theory

Complexity for link complement states in Chern-Simons theory

Hypergraph States in SU(N)1, N odd prime, Chern-Simons Theory

A Normal Form for Single-Qudit Clifford+$T$ Operators

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