SLIDE 1 Universal Topological Phase of 2D Stabilizer Codes
Héctor Bombín Perimeter Institute
- H. Bombin, G. Duclos-Cianci, D. Poulin arXiv:1103.4606
- H. Bombin arXiv:1107.2707
SLIDE 2
collaborators
David Poulin Guillaume Duclos-Cianci Université de Sherbrooke
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topological codes
general structure?
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translational invariance
classify the bulk!
SLIDE 5 topological order
- gapped excitations
- GS degeneracy
- locally indistinguishable GS
SLIDE 6 topological order
- gapped excitations
- GS degeneracy
- locally indistinguishable GS
topological order describes the equivalent classes defined by local unitary evolutions
(Chen, Gu, Wen ’10)
SLIDE 7 topological order
- they argue that two gapped GSs…
…are in the same phase …can be connected adiabatically without closing the gap …are connected by a local unitary transformation …are connected by a quantum circuit of constant depth
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classify topological codes/order = classify long-range entanglement patterns
topological order
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goals
structure of 2D topological stabilizer (subsystem) codes equivalence up to local transformations
SLIDE 10
- utline
- anyons & toric code
- universality
- subsystem codes
- conclusions
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anyons in 2D systems
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abelian anyons
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toric code
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strings & charges
SLIDE 15 mutual statistics
SLIDE 16 self-statistics
- e, m are bosons, their composite is fermionic:
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anyon model
SLIDE 18
- utline
- anyons & toric code
- universality
- subsystem codes
- conclusions
SLIDE 19 topological stabilizer groups
- local and translationally invariant (LTI) generators
SLIDE 20 topological stabilizer groups
local undetectable errors do not affect encoded states
- local and translationally invariant (LTI) generators
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topological stabilizer groups
SLIDE 22
topological stabilizer groups
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coarse graining LTI Clifford mapping add/remove disentangled qubits
local equivalence
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- same anyon model ↔ equivalent!
- ruling out chiral anyons (Kitaev ‘06):
every 2D TSG is locally equivalent to a finite number of copies of the toric code
universality result
SLIDE 25
- 1. 2D TSGs admit LTI independent generators
- 2. # charges < ∞
- 3. anyon model from string operators
- 4. string segment framework, plaquette stabilizers
- 5. other stabilizers have no charge
- 6. map: string segments <--> string segments,
uncharged stabilizers <--> single qubit stabilizers
proof outline
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- coarse grain:
- till each site holds any charge
- till excitation pairs of same charge can be removed
with string-like operators
string operators
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anyon model
SLIDE 30 anyon model
1 2 k k 1 2 b b b b b/f b/f j i j i i j
SLIDE 31 anyon model
1 2 k k 1 2 b b b b b/f b/f toric code chiral case
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- model independent commutation relations
string framework
1 2 1 2
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mapping
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- utline
- anyons & toric code
- universality
- subsystem codes
- conclusions
SLIDE 35
subsystem codes
SLIDE 36
subsystem codes
stabilizer group gauge group
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topological subsystem codes
SLIDE 38
topological subsystem codes
local undetectable errors do not affect encoded states
SLIDE 39
topological subsystem codes
SLIDE 40
topological subsystem codes
SLIDE 41
topological subsystem codes
SLIDE 42 generalized charge
morphisms commutators
=
SLIDE 43 generalized charge
morphisms commutators
=
SLIDE 44
topological ‘interactions’
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charge morphism
SLIDE 46
- gauge charge generators:
- dual stabilizer charge generators:
canonical generators
1 k 1 k 1 2 l b b b b b/f b/f b/f
non-interacting
1 k 1 k 1 2 l
SLIDE 47 canonical generators
- all possible anyon models are combinations of
toric code: topological subsystem color codes: subsystem toric code: honeycomb subsystem code:
b b f f b f
SLIDE 48
string framework
SLIDE 49 string framework
- gauge generators (non-trivial charge)
- stabilizer generators (non-trivial charge)
SLIDE 50 every 2D TSC has a structure based
string framework
SLIDE 51 conclusions & questions
- the long-range entanglement pattern of toric codes is
universal for 2D topological stabilizer models
- all 2D TSCs are anyon based
- the same approach could be used for boundaries or point
defects…
- more general 2D models?
- what about 3D?
