Fault-tolerant error correction for non-abelian anyons=1[1]

57 Pages·2017·1.89 MB·English

Non-abeliananyonsandquantuminformation Errorcorrectionforabeliananyons Errorcorrectionfornon-abeliananyons
Fault-tolerant error correction for non-abelian
anyons1
Guillaume Dauphinais and David Poulin
Institutquantique&d´epartementdephysique,Universit´edeSherbrooke
QIP
January 18, 2017
1arXiv:1607.02159
GuillaumeDauphinaisandDavidPoulin Institutquantique&d´epartementdephysique,Universit´edeSherbrooke
Fault-toleranterrorcorrectionfornon-abeliananyons1
Non-abeliananyonsandquantuminformation Errorcorrectionforabeliananyons Errorcorrectionfornon-abeliananyons
Outline
1 Non-abelian anyons and quantum information
2 Error correction for abelian anyons
3 Error correction for non-abelian anyons
GuillaumeDauphinaisandDavidPoulin Institutquantique&d´epartementdephysique,Universit´edeSherbrooke
Fault-toleranterrorcorrectionfornon-abeliananyons1
Each excitation is described by a unique label, called its
topological charge from a finite set {a,b,c,...}
We can imagine bringing 2 excitation together (a and b), and
ask what is their total charge c.
The possible outcomes are given by the fusion rules:
(cid:88)
a×b = Nc c
ab
c
Non-abeliananyonsandquantuminformation Errorcorrectionforabeliananyons Errorcorrectionfornon-abeliananyons
What are anyons1 ?
Localized gapped excitations living on a 2-dimensional surface
1A. Kitaev, Annals Phys. 321, 2-111 (2006)
GuillaumeDauphinaisandDavidPoulin Institutquantique&d´epartementdephysique,Universit´edeSherbrooke
Fault-toleranterrorcorrectionfornon-abeliananyons1
We can imagine bringing 2 excitation together (a and b), and
ask what is their total charge c.
The possible outcomes are given by the fusion rules:
(cid:88)
a×b = Nc c
ab
c
1arXiv:1607.02159
Non-abeliananyonsandquantuminformation Errorcorrectionforabeliananyons Errorcorrectionfornon-abeliananyons
What are anyons1 ?
Localized gapped excitations living on a 2-dimensional surface
Each excitation is described by a unique label, called its
topological charge from a finite set {a,b,c,...}
1A. Kitaev, Annals Phys. 321, 2-111 (2006)
GuillaumeDauphinaisandDavidPoulin Institutquantique&d´epartementdephysique,Universit´edeSherbrooke
Fault-toleranterrorcorrectionfornon-abeliananyons1
The possible outcomes are given by the fusion rules:
(cid:88)
a×b = Nc c
ab
c
1arXiv:1607.02159
Non-abeliananyonsandquantuminformation Errorcorrectionforabeliananyons Errorcorrectionfornon-abeliananyons
What are anyons1 ?
Localized gapped excitations living on a 2-dimensional surface
Each excitation is described by a unique label, called its
topological charge from a finite set {a,b,c,...}
We can imagine bringing 2 excitation together (a and b), and
ask what is their total charge c.
1A. Kitaev, Annals Phys. 321, 2-111 (2006)
GuillaumeDauphinaisandDavidPoulin Institutquantique&d´epartementdephysique,Universit´edeSherbrooke
Fault-toleranterrorcorrectionfornon-abeliananyons1
1arXiv:1607.02159
Non-abeliananyonsandquantuminformation Errorcorrectionforabeliananyons Errorcorrectionfornon-abeliananyons
What are anyons1 ?
Localized gapped excitations living on a 2-dimensional surface
Each excitation is described by a unique label, called its
topological charge from a finite set {a,b,c,...}
We can imagine bringing 2 excitation together (a and b), and
ask what is their total charge c.
The possible outcomes are given by the fusion rules:
(cid:88)
a×b = Nc c
ab
c
1A. Kitaev, Annals Phys. 321, 2-111 (2006)
GuillaumeDauphinaisandDavidPoulin Institutquantique&d´epartementdephysique,Universit´edeSherbrooke
Fault-toleranterrorcorrectionfornon-abeliananyons1
Whereas for non-abelian anyons, the fusion rules are in
general probabilistic, for example with Fibonacci anyons:
τ ×τ = 1+τ.
A Hilbert space is associated to each fusion/splitting process.
Fusing two anyons a and a collapses the wavefunction into a
1 2
definite super-selection sector, with probability given by Born’s
rule:
P(c) = (cid:104)ψ|Πa1a2|ψ(cid:105). (1)
c
Non-abeliananyonsandquantuminformation Errorcorrectionforabeliananyons Errorcorrectionfornon-abeliananyons
Abelian vs non-abelian anyons
The fusion rules for abelian anyons are deterministic and
unique, as for excitations in the toric code:
e ×e = 1, m×m = 1, e ×m = em, ...
GuillaumeDauphinaisandDavidPoulin Institutquantique&d´epartementdephysique,Universit´edeSherbrooke
Fault-toleranterrorcorrectionfornon-abeliananyons1
A Hilbert space is associated to each fusion/splitting process.
Fusing two anyons a and a collapses the wavefunction into a
1 2
definite super-selection sector, with probability given by Born’s
rule:
P(c) = (cid:104)ψ|Πa1a2|ψ(cid:105). (1)
c
1arXiv:1607.02159
Non-abeliananyonsandquantuminformation Errorcorrectionforabeliananyons Errorcorrectionfornon-abeliananyons
Abelian vs non-abelian anyons
The fusion rules for abelian anyons are deterministic and
unique, as for excitations in the toric code:
e ×e = 1, m×m = 1, e ×m = em, ...
Whereas for non-abelian anyons, the fusion rules are in
general probabilistic, for example with Fibonacci anyons:
τ ×τ = 1+τ.
GuillaumeDauphinaisandDavidPoulin Institutquantique&d´epartementdephysique,Universit´edeSherbrooke
Fault-toleranterrorcorrectionfornon-abeliananyons1
Fusing two anyons a and a collapses the wavefunction into a
1 2
definite super-selection sector, with probability given by Born’s
rule:
P(c) = (cid:104)ψ|Πa1a2|ψ(cid:105). (1)
c
1arXiv:1607.02159
Non-abeliananyonsandquantuminformation Errorcorrectionforabeliananyons Errorcorrectionfornon-abeliananyons
Abelian vs non-abelian anyons
The fusion rules for abelian anyons are deterministic and
unique, as for excitations in the toric code:
e ×e = 1, m×m = 1, e ×m = em, ...
Whereas for non-abelian anyons, the fusion rules are in
general probabilistic, for example with Fibonacci anyons:
τ ×τ = 1+τ.
A Hilbert space is associated to each fusion/splitting process.
GuillaumeDauphinaisandDavidPoulin Institutquantique&d´epartementdephysique,Universit´edeSherbrooke
Fault-toleranterrorcorrectionfornon-abeliananyons1
1arXiv:1607.02159
Non-abeliananyonsandquantuminformation Errorcorrectionforabeliananyons Errorcorrectionfornon-abeliananyons
Abelian vs non-abelian anyons
The fusion rules for abelian anyons are deterministic and
unique, as for excitations in the toric code:
e ×e = 1, m×m = 1, e ×m = em, ...
Whereas for non-abelian anyons, the fusion rules are in
general probabilistic, for example with Fibonacci anyons:
τ ×τ = 1+τ.
A Hilbert space is associated to each fusion/splitting process.
Fusing two anyons a and a collapses the wavefunction into a
1 2
definite super-selection sector, with probability given by Born’s
rule:
P(c) = (cid:104)ψ|Πa1a2|ψ(cid:105). (1)
c
GuillaumeDauphinaisandDavidPoulin Institutquantique&d´epartementdephysique,Universit´edeSherbrooke
Fault-toleranterrorcorrectionfornon-abeliananyons1