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Topological Quantum Error Correcting Codes Kasper Duivenvoorden JARA IQI, RWTH Aachen Topological Phases of Quantum Matter 4 September 2014 Topology Physical errors are local Store information globally Gapped topological phase with


  1. Topological Quantum Error Correcting Codes Kasper Duivenvoorden JARA IQI, RWTH Aachen Topological Phases of Quantum Matter – 4 September 2014

  2. Topology Physical errors are local  Store information globally Gapped topological phase with ground space degeneracy: Definition: locally ground states are the same B A Use ground space manifold of a topological phase to encode quantum information

  3. Overview • Homological codes A formalism using emphasizing topology • Fractal codes Possibly thermally stable codes • Chamon code Work in progress

  4. Example: Toric Code Ground states: Logicals: preserves the code space

  5. Homological codes Lattice  CW-complex Point  0-cells  1-cells Line Surface  2-cells `  k-cells ... Boundary map Co-Boundary map = = = =

  6. Homological codes Boundary map Co-Boundary map = = Properties 1: 2:

  7. Homological codes Step 1: CW-complex Step 2: Spins on every k-cells Example: k=1 Step 3: Hamiltonian z = z z z x = x x x

  8. Homological codes = = Logicals

  9. Homological codes = = Logicals

  10. Toric Codes D – dimensional lattice, qubits on k cells  k and D-k dimensional logicals Dimensionality of logicals Z X 2D 1 1 3D 1 2 4D 2 2

  11. Thermal Stability E Movement of anyons At T=0 At T>0 Tunneling Thermal excitation p = exp(-L) p = exp(-E/kT) No stability!

  12. Thermal Stability E At T>0 Thermal excitation p = exp(-E/kT) Stability! Chesi, Loss, Bravyi, Terhal, 2010

  13. Toric Codes D – dimensional lattice, qubits on k cells  k and D-k dimensional logicals Dim. of logicals Stable Memory Z X 2D 1 1 No 3D 1 2 4D 2 2 Quantum

  14. Toric Codes D – dimensional lattice, qubits on k cells  k and D-k dimensional logicals 2D Drawbacks - Constant information - We only have 3 dimensions 3D  Fractal codes 4D

  15. Overview • Homological codes A formalism using emphasizing topology • Fractal codes Possibly thermally stable codes • Chamon code Work in progress Yoshida, 2013

  16. Fractal Codes Algebraic representation of Z-type operators • Generalize to higher dimensions: y z x • More spins per site: Yoshida, 2013

  17. Fractal Codes Commutation relations Interpretation: shift X[g] by 0 , Z[f] and X[g] anticommute shift X[g] by 2 , Z[f] and X[g] anticommute Yoshida, 2013

  18. Fractal Codes Hamiltonian Example: y Z Z Z Z x Logical X[g]  g(1+fy) = 0 modulo 2 Assumptions:

  19. Fractal Codes Logical: X[g] Hamiltonian terms: Z Z Z Z X X X X X X X X 1 2 3 2 1 X X X X X X X X X X X X X X X y X x Set of Logicals: fractal like point like

  20. Fractal Codes Z Z Z Z Excitations X X X X X X X X X X X X Compare with Toric code:  Possible energy barrier Yoshida, 2013

  21. Quantum Fractal Codes Now in 3 dimensions: Terms commute since: Fractal logicals in x-y plane and in x-z plane Yoshida, 2013

  22. Quantum Fractal Codes Excitations: Example: X X X First spin propagation in y direction X X X X Second spin propagation in z direction Yoshida, 2013

  23. Quantum Fractal Codes Excitations: y second cancellation X X X X X second X X X X z X X X X X X first first Excitations can propagate freely in the z-y direction Due to algebraic dependence of g and f Yoshida, 2013

  24. Quantum Fractal Codes y No algebraic dependence second cancellation X X X X  (at least) logarithmic energy barrier X Bravyi, Haah, 2011 X z X E ≈ Log(L) p ≈ exp( -E/kT) X first Memory time  Polynomial rate  Optimal system size Bravyi, Haah, 2013 System size

  25. Overview • Homological codes A formalism using emphasizing topology • Fractal codes Possibly thermally stable codes • Chamon code Work in progress

  26. Chamon Code X ↔ Z Z z x X X z z x x  z x Z Chamon, 2005

  27. Chamon Code x-X-X-X-X-x Nog Bravyi, Leemhuis, Terhal, 2011

  28. Chamon Code Error threshold Decode  p failure Encode Noise (p) p failure p failure  Increased system size p p Ben-Or, Aharonov, 1999

  29. Chamon Code Error threshold Decode  p failure Encode Noise (p) Can be related to percolation

  30. Chamon Code Memory Time Decode  p succes Encode Bath (T) Time (t) 

  31. Future Work • Homological codes Determine a more general condition for thermal stability in terms of Hamiltonian properties • Fractal codes Understand relation between fractal codes and topological order • Chamon code Consider better (but computational more demanding) decoders Conclustion: Existence of a quantum memory in 3D is still open

  32. Fractal Codes Logicals: Z Logicals: X Similarly: Yoshida, 2013

  33. Fractal Codes Logicals: Z Logicals: X Commutation relations Both fractal like! Yoshida, 2013

  34. Error Correction Communication noise

  35. Error Correction Storage time noise space

  36. Error Correction noise Solution: Build in Redundancy 1 11111 11001 1 Encoding Decoding

  37. Gottesman, PRA 1996 Stabilizer Codes Stabilizers: Ground states: Error: Exitated states: • Allow for fault tollerant computations: errors do not accumalate when correcting • Overhead independent of computational time Logicals / Symmetries: Gottesman, PRA 1998 Gottesman, 1310.2984 Trade off Memory time Information k Stability d (number of qubits) (weight of a logical)

  38. Relation to topological order Topological order at T > 0 ↔ Stable quantum memory at T > 0 Topological Entropy Adiabatic Evolution 2D  No quantum memory in 2D 3D 4D Mazac, Hamma, 2012 Caselnovo, Chamon , 2007/2008 Hastings, 2011

  39. Homological codes What can we learn k: stored information, related to genus d: distance, related to systole n: number of qubits, related to volume (sys) 2 Intuitively... ≤ volume genus x (sys) 2 ... or better ≤ volume Gromov, 1992 In general genus / log 2 (genus) x (sys) 2 ≤ volume Delfosse, 1301.6588 k / log 2 (k) x (d) 2 ≤ n

  40. Example: Toric Code Ground states: Error:  Excitations

  41. Example: Toric Code Plaquette: Star:

  42. Fractal Codes Algebraic representation of Hamiltonian Example: toric code Plaquette: Star: y x Yoshida, 2013

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