Universal Topological Phase of 2D Stabilizer Codes

Transcription

1 H. Bombin, G. Duclos-Cianci, D. Poulin arxiv: H. Bombin arxiv: Universal Topological Phase of 2D Stabilizer Codes Héctor Bombín Perimeter Institute

2 collaborators David Poulin Guillaume Duclos-Cianci Université de Sherbrooke

3 topological codes general structure?

4 translational invariance classify the bulk!

5 topological order gapped excitations GS degeneracy locally indistinguishable GS

6 topological order gapped excitations topological order describes the equivalent GS degeneracy classes defined by local unitary evolutions locally indistinguishable GS (Chen, Gu, Wen 10)

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

8 topological order classify topological codes/order = classify long-range entanglement patterns

9 goals structure of 2D topological stabilizer (subsystem) codes equivalence up to local transformations

10 outline anyons & toric code universality subsystem codes conclusions

11 anyons in 2D systems

12 abelian anyons

13 toric code

14 strings & charges

15 semionic interactions: mutual statistics

16 self-statistics e, m are bosons, their composite is fermionic:

17 anyon model

18 outline anyons & toric code universality subsystem codes conclusions

19 topological stabilizer groups local and translationally invariant (LTI) generators

20 topological stabilizer groups local and translationally invariant (LTI) generators local undetectable errors do not affect encoded states

21 topological stabilizer groups

22 topological stabilizer groups

23 local equivalence coarse graining LTI Clifford mapping add/remove disentangled qubits

24 universality result 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

25 proof outline 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

26 coarse grain: string operators till each site holds any charge till excitation pairs of same charge can be removed with string-like operators

27 anyon model

28 well defined: anyon model

29 well defined: anyon model

30 anyon model charge generators: k k b b b b b/f b/f i j i j i j

31 anyon model charge generators: k k b b b b b/f b/f toric code chiral case

32 string framework model independent commutation relations

33 mapping

34 outline anyons & toric code universality subsystem codes conclusions

35 subsystem codes

36 subsystem codes stabilizer group gauge group

37 topological subsystem codes

38 topological subsystem codes local undetectable errors do not affect encoded states

39 topological subsystem codes

40 topological subsystem codes

41 topological subsystem codes

42 generalized charge Stabilizer charge = morphisms commutators

43 generalized charge Gauge charge = morphisms commutators

44 topological interactions

45 charge morphism

46 canonical generators gauge charge generators: non-interacting 1 1 k k 1 2 l b b b/f b/f b b dual stabilizer charge generators: b/f 1 1 k k 1 2 l

47 canonical generators all possible anyon models are combinations of toric code: b b topological subsystem color codes: f f subsystem toric code: honeycomb subsystem code: b f

48 string framework

49 string framework gauge generators (non-trivial charge) stabilizer generators (non-trivial charge)

50 string framework every 2D TSC has a structure based on an anyon model

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?