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Decoding problem for topological quantum codes Guillaume Duclos-Cianci Dpartement de Physique Universit de Sherbrooke Joint work with: David Poulin and Hector Bombin Second International Conference on Quantum Error Correction University of

  1. Kitaev’s toric code Error chains Error chains are attached to a pair of particles. X X 1 The syndrome configuration on the X endpoint doesn’t depend on the X 1 X X X geometry (path, length) of the string. Error chains can be stretched freely: constant energy cost. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 8 / 44

  2. Kitaev’s toric code Particle annihilation An error can annihilate two 1 X X X particles. X 1 X X X The particle’s worldline is left behind after fusion. Particle fusion can leave behind a worldline corresponding to a logical operation. X X X X X X X X Memory corruption Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 9 / 44

  3. Kitaev’s toric code Particle annihilation An error can annihilate two particles. X X X X X X X 1 1 X X X X The particle’s worldline is left behind after fusion. Particle fusion can leave behind a worldline corresponding to a logical X X X operation. X X X X X X X X X Memory corruption Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 9 / 44

  4. Kitaev’s toric code Particle annihilation An error can annihilate two particles. X X X X X X X 1 1 X X X X The particle’s worldline is left behind after fusion. Particle fusion can leave behind a worldline corresponding to a logical X X X operation. X X X X X X X X X Memory corruption Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 9 / 44

  5. Kitaev’s toric code Site particles The same story holds for σ z errors Z Z 1 These will create site particles Z located at the lattice’s vertices Z (plaquette of dual lattice). Z Z Z 1 Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 10 / 44

  6. Decoding problem Outline Kitaev’s toric code 1 Decoding problem 2 Renormalization Group Decoder 3 Results for Kitaev’s code 4 5 Extension to other codes Fault-tolerance 6 7 2D Fault-Tolerant Quantum Cellular Automaton Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 11 / 44

  7. Decoding problem Error model Depolarizing error model Independent on every qubit. No error with probability 1 − p . Error X , Y , or Z with probability p / 3. Bit-flip error model Independent on every qubit. No error with probability 1 − p . Error X with probability p . Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 12 / 44

  8. Decoding problem Error syndrome & decoding 15 % Noise rate An error produces defects (error syndrome) 1 Y 1 1 Measure particle position, 1 Z 1 1 1 1 Z X but not worldline. 1 1 Y 1 1 1 Y 1 X 1 Y 1 1 1 Many worldlines consistent 1 1 1 Z 1 Y 1 with defects. 1 Y 1 X 1 1 1 1 1 X 1 1 Worldline with different Z 1 1 homologies have different 1 X 1 X 1 X 1 1 1 Y 1 1 1 effect on ground space: 1 1 1 Y Z 1 Y MUST be distinguished. 1 1 Z 1 1 1 1 1 1 Z Z 1 1 Y 1 X Decoding 1 Y 1 1 1 1 Infer worldline homology from 1 particle location. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 13 / 44

  9. Decoding problem Error syndrome & decoding 15 % Noise rate An error produces defects (error syndrome) 1 1 1 Measure particle position, 1 1 1 1 1 but not worldline. 1 1 1 1 1 1 1 1 1 1 Many worldlines consistent 1 1 1 1 1 1 1 1 1 with defects. 1 1 1 1 1 Worldline with different 1 1 homologies have different 1 1 1 1 1 1 1 1 1 effect on ground space: 1 1 1 1 MUST be distinguished. 1 1 1 1 1 1 1 1 1 1 1 Decoding 1 1 1 1 1 Infer worldline homology from 1 particle location. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 13 / 44

  10. Decoding problem Error syndrome & decoding 15 % Noise rate An error produces defects (error syndrome) 1 1 1 Measure particle position, 1 1 1 1 1 but not worldline. 1 1 1 1 1 1 1 1 1 1 Many worldlines consistent 1 1 1 1 1 1 1 1 1 with defects. 1 1 1 1 1 Worldline with different 1 1 1 1 homologies have different 1 1 1 1 1 1 1 effect on ground space: 1 1 1 1 MUST be distinguished. 1 1 1 1 1 1 1 1 1 1 1 Decoding 1 1 1 1 1 Infer worldline homology from 1 particle location. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 13 / 44

  11. Decoding problem Error syndrome & decoding 15 % Noise rate An error produces defects (error syndrome) 1 1 1 Measure particle position, 1 1 1 1 1 but not worldline. 1 1 1 1 1 1 1 1 1 1 Many worldlines consistent 1 1 1 1 1 1 1 1 1 with defects. 1 1 1 1 1 Worldline with different 1 1 1 1 homologies have different 1 1 1 1 1 1 1 effect on ground space: 1 1 1 1 MUST be distinguished. 1 1 1 1 1 1 1 1 1 1 1 Decoding 1 1 1 1 1 Infer worldline homology from 1 particle location. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 13 / 44

  12. Decoding problem Error syndrome & decoding 15 % Noise rate An error produces defects (error syndrome) 1 1 Measure particle position, 1 1 1 1 1 but not worldline. 1 1 1 1 1 1 1 1 1 1 Many worldlines consistent 1 1 1 1 1 1 1 1 1 with defects. 1 1 1 1 1 Worldline with different 1 1 homologies have different 1 1 1 1 1 1 1 1 1 effect on ground space: 1 1 1 1 MUST be distinguished. 1 1 1 1 1 1 1 1 1 1 1 Decoding 1 1 1 1 1 Infer worldline homology from 1 particle location. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 13 / 44

  13. Decoding problem Error syndrome & decoding 15 % Noise rate An error produces defects (error syndrome) 1 1 Measure particle position, 1 1 1 1 1 but not worldline. 1 1 1 1 1 1 1 1 1 1 Many worldlines consistent 1 1 1 1 1 1 1 1 1 with defects. 1 1 1 1 1 Worldline with different 1 1 homologies have different 1 1 1 1 1 1 1 1 1 effect on ground space: 1 1 1 1 MUST be distinguished. 1 1 1 1 1 1 1 1 1 1 1 Decoding 1 1 1 1 1 Infer worldline homology from 1 particle location. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 13 / 44

  14. Decoding problem Existing methods Energy Minimization Find shortest path connecting all defects. Equivalent to minimizing energy of random bond Ising model. Edmonds’ perfect matching algorithm: O ( ℓ 6 ) Polynomial complexity, but still prohibitive, O ( ℓ 6 ) . Recent progress : average O ( 1 ) decoding time with marginal losses (Fowler et al. 2011). Sub-optimal: Does not take into account the homological equivalence of errors. Does not take into account correlations between site and plaquette defects. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 14 / 44

  15. Decoding problem Existing methods Energy Minimization Find shortest path connecting all defects. Equivalent to minimizing energy of random bond Ising model. Edmonds’ perfect matching algorithm: O ( ℓ 6 ) Polynomial complexity, but still prohibitive, O ( ℓ 6 ) . Recent progress : average O ( 1 ) decoding time with marginal losses (Fowler et al. 2011). Sub-optimal: Does not take into account the homological equivalence of errors. Does not take into account correlations between site and plaquette defects. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 14 / 44

  16. Decoding problem Existing methods Energy Minimization Find shortest path connecting all defects. Equivalent to minimizing energy of random bond Ising model. Edmonds’ perfect matching algorithm: O ( ℓ 6 ) Polynomial complexity, but still prohibitive, O ( ℓ 6 ) . Recent progress : average O ( 1 ) decoding time with marginal losses (Fowler et al. 2011). Sub-optimal: Does not take into account the homological equivalence of errors. Does not take into account correlations between site and plaquette defects. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 14 / 44

  17. Decoding problem Existing methods Energy Minimization Find shortest path connecting all defects. Equivalent to minimizing energy of random bond Ising model. Edmonds’ perfect matching algorithm: O ( ℓ 6 ) Polynomial complexity, but still prohibitive, O ( ℓ 6 ) . Recent progress : average O ( 1 ) decoding time with marginal losses (Fowler et al. 2011). Sub-optimal: Does not take into account the homological equivalence of errors. Does not take into account correlations between site and plaquette defects. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 14 / 44

  18. Decoding problem Existing methods Energy Minimization Find shortest path connecting all defects. Equivalent to minimizing energy of random bond Ising model. Edmonds’ perfect matching algorithm: O ( ℓ 6 ) Polynomial complexity, but still prohibitive, O ( ℓ 6 ) . Recent progress : average O ( 1 ) decoding time with marginal losses (Fowler et al. 2011). Sub-optimal: Does not take into account the homological equivalence of errors. Does not take into account correlations between site and plaquette defects. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 14 / 44

  19. Decoding problem Existing methods Energy Minimization Find shortest path connecting all defects. Equivalent to minimizing energy of random bond Ising model. Edmonds’ perfect matching algorithm: O ( ℓ 6 ) Polynomial complexity, but still prohibitive, O ( ℓ 6 ) . Recent progress : average O ( 1 ) decoding time with marginal losses (Fowler et al. 2011). Sub-optimal: Does not take into account the homological equivalence of errors. Does not take into account correlations between site and plaquette defects. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 14 / 44

  20. Decoding problem Minimum distance vs Degeneracy Two possible pairings with different homologies 1 First one has lower weight 1 (Energy). Second one is highly 1 degenerate (Entropy). 1 1 Optimal decoding 1 Homology class with lowest free energy F = E − TS . 1 Nishimori T − 1 = ln 3 ( 1 − p ) 1 . p Sum over all equivalent errors. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 15 / 44

  21. Decoding problem Minimum distance vs Degeneracy Two possible pairings with different homologies 1 First one has lower weight 1 (Energy). Second one is highly 1 degenerate (Entropy). 1 1 Optimal decoding 1 Homology class with lowest free energy F = E − TS . 1 Nishimori T − 1 = ln 3 ( 1 − p ) 1 . p Sum over all equivalent errors. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 15 / 44

  22. Decoding problem Minimum distance vs Degeneracy Two possible pairings with different homologies 1 First one has lower weight (Energy). 1 Second one is highly 1 degenerate (Entropy). 1 Optimal decoding 1 Homology class with lowest free 1 energy F = E − TS . 1 Nishimori T − 1 = ln 3 ( 1 − p ) . 1 p Sum over all equivalent errors. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 15 / 44

  23. Decoding problem Minimum distance vs Degeneracy Two possible pairings with different homologies 1 First one has lower weight (Energy). 1 Second one is highly 1 degenerate (Entropy). 1 Optimal decoding 1 Homology class with lowest free 1 energy F = E − TS . 1 Nishimori T − 1 = ln 3 ( 1 − p ) . 1 p Sum over all equivalent errors. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 15 / 44

  24. Decoding problem Minimum distance vs Degeneracy Two possible pairings with different homologies 1 First one has lower weight (Energy). 1 Second one is highly 1 degenerate (Entropy). 1 Optimal decoding 1 Homology class with lowest free 1 energy F = E − TS . 1 Nishimori T − 1 = ln 3 ( 1 − p ) . 1 p Sum over all equivalent errors. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 15 / 44

  25. Decoding problem Minimum distance vs Degeneracy Two possible pairings with different homologies 1 First one has lower weight (Energy). 1 Second one is highly 1 degenerate (Entropy). 1 Optimal decoding 1 Homology class with lowest free 1 energy F = E − TS . 1 Nishimori T − 1 = ln 3 ( 1 − p ) . 1 p Sum over all equivalent errors. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 15 / 44

  26. Decoding problem Plaquette-Site string correlations 1 1 1 1 Two possible pairings with 1 1 different homologies Both seemingly have same 1 1 weight 1 1 A Y error has same weight as X and Z : overcounting. 1 1 Site Z and plaquette X errors are not independent. 1 1 1 1 1 1 Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 16 / 44

  27. Decoding problem Plaquette-Site string correlations 1 1 1 1 Two possible pairings with 1 1 different homologies Both seemingly have same 1 1 weight 1 1 A Y error has same weight as X and Z : overcounting. 1 1 Site Z and plaquette X errors are not independent. 1 1 1 1 1 1 Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 16 / 44

  28. Decoding problem Plaquette-Site string correlations 1 1 1 1 Two possible pairings with 1 1 different homologies Both seemingly have same 1 1 weight 1 1 A Y error has same weight as X and Z : overcounting. 1 1 Site Z and plaquette X errors are not independent. 1 1 1 1 1 1 Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 16 / 44

  29. Decoding problem Plaquette-Site string correlations 1 1 1 1 Two possible pairings with 1 1 different homologies Both seemingly have same 1 1 weight 1 1 A Y error has same weight as X and Z : overcounting. 1 1 Site Z and plaquette X errors are not independent. 1 1 1 1 1 1 Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 16 / 44

  30. Decoding problem Plaquette-Site string correlations 1 1 Y Y 1 1 Two possible pairings with 1 1 different homologies Both seemingly have same 1 1 weight Y 1 1 Y A Y error has same weight as X and Z : overcounting. 1 1 Site Z and plaquette X errors are not independent. 1 1 Y Y 1 1 1 1 Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 16 / 44

  31. Decoding problem Plaquette-Site string correlations 1 1 Y Y 1 1 Two possible pairings with 1 1 different homologies Both seemingly have same 1 1 weight Y 1 1 Y A Y error has same weight as X and Z : overcounting. 1 1 Site Z and plaquette X errors are not independent. 1 1 Y Y 1 1 1 1 Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 16 / 44

  32. Renormalization Group Decoder Outline Kitaev’s toric code 1 Decoding problem 2 Renormalization Group Decoder 3 Results for Kitaev’s code 4 5 Extension to other codes Fault-tolerance 6 7 2D Fault-Tolerant Quantum Cellular Automaton Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 17 / 44

  33. Renormalization Group Decoder Scale invariance Original B p checks Basis change (row operations on C ) Obtain scale invariant generators Structure similar to a concatenated code. Soft-decode each small block. Pass information to next encoding level. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 18 / 44

  34. Renormalization Group Decoder Scale invariance Original B p checks Basis change (row operations on C ) Obtain scale invariant generators Structure similar to a concatenated code. Soft-decode each small block. Pass information to next encoding level. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 18 / 44

  35. Renormalization Group Decoder Scale invariance Original B p checks Basis change (row operations on C ) Obtain scale invariant generators Structure similar to a concatenated code. Soft-decode each small block. Pass information to next encoding level. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 18 / 44

  36. Renormalization Group Decoder Scale invariance Original B p checks Basis change (row operations on C ) Obtain scale invariant generators Structure similar to a concatenated code. Soft-decode each small block. Pass information to next encoding level. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 18 / 44

  37. Renormalization Group Decoder Scale invariance Original B p checks Basis change (row operations on C ) Obtain scale invariant generators Structure similar to a concatenated code. Soft-decode each small block. Pass information to next encoding level. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 18 / 44

  38. Renormalization Group Decoder Scale invariance Original B p checks Basis change (row operations on C ) Obtain scale invariant generators Structure similar to a concatenated code. Soft-decode each small block. Pass information to next encoding level. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 18 / 44

  39. Renormalization Group Decoder Scale invariance Original B p checks Basis change (row operations on C ) Obtain scale invariant generators Structure similar to a concatenated code. Soft-decode each small block. Pass information to next encoding level. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 18 / 44

  40. Renormalization Group Decoder Scale invariance Original B p checks Basis change (row operations on C ) Obtain scale invariant generators Structure similar to a concatenated code. Soft-decode each small block. Pass information to next encoding level. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 18 / 44

  41. Renormalization Group Decoder Concatenated code Decoding | 0 � P ( L ) = � ′ U E P ( E ) where | 0 � | 0 � | 0 � U U Sum over E equivalent to L | 0 � | 0 � | 0 � and with right syndrome. U | 0 � P ( E ) given by error model. | 0 � U | ψ � | 0 � | 0 � | 0 � | 0 � U U U | 0 � | 0 � | 0 � Decoding | 0 � U | 0 � Compute error probability for | 0 � U | 0 � each encoded qubit. | 0 � | 0 � U U | 0 � | 0 � Pass that probability to the | 0 � U | 0 � next level up. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 19 / 44

  42. Renormalization Group Decoder Concatenated code Decoding P ( L ) = � ′ E P ( E ) where Sum over E equivalent to L and with right syndrome. P ( E ) given by error model. | ψ � | 0 � U | 0 � Decoding Compute error probability for each encoded qubit. Pass that probability to the next level up. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 19 / 44

  43. Renormalization Group Decoder Concatenated code Decoding P ( L ) = � ′ E P ( E ) where Sum over E equivalent to L and with right syndrome. L P ( E ) given by error model. † Decoding U Compute error probability for each encoded qubit. Pass that probability to the next level up. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 19 / 44

  44. Renormalization Group Decoder Concatenated code Decoding P ( L ) = � ′ E P ( E ) where Sum over E equivalent to L and with right syndrome. L P ( E ) given by error model. † Decoding U Compute error probability for each encoded qubit. Pass that probability to the next level up. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 19 / 44

  45. Renormalization Group Decoder Concatenated code Decoding P ( L ) = � ′ E P ( E ) where L Sum over E equivalent to L and with right syndrome. † U P ( E ) given by error model. Decoding † † † U U U Compute error probability for each encoded qubit. † † † † † † † † † U U U U U U U U U Pass that probability to the next level up. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 19 / 44

  46. Renormalization Group Decoder Renormalization: Information coarse graining Think of Kitaev’s code as a concatenated code: It is made up of a bunch of small (open boundary) topological codes, joined into larger topological codes, etc. Logical operators are strings going across the codes. Given the particle configuration (syndrome) in a unit cell, compute the prob that the a string type went through. There are 16 possibilities corresponding to { I , X , Y , Z } 2 , 2 qubits. This is done by brute force: sum over all worldline configurations. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 20 / 44

  47. Renormalization Group Decoder Renormalization: Information coarse graining Think of Kitaev’s code as a concatenated code: It is made up of a bunch of small (open boundary) topological codes, joined into larger topological codes, etc. Logical operators are strings going across the codes. Given the particle configuration (syndrome) in a unit cell, compute the prob that the a string type went through. There are 16 possibilities corresponding to { I , X , Y , Z } 2 , 2 qubits. This is done by brute force: sum over all worldline configurations. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 20 / 44

  48. Renormalization Group Decoder Renormalization: Information coarse graining Think of Kitaev’s code as a concatenated code: It is made up of a bunch of small (open boundary) topological codes, joined into larger topological codes, etc. Logical operators are strings going across the codes. Given the particle configuration (syndrome) in a unit cell, compute the prob that the a string type went through. There are 16 possibilities corresponding to { I , X , Y , Z } 2 , 2 qubits. This is done by brute force: sum over all worldline configurations. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 20 / 44

  49. Renormalization Group Decoder Renormalization: Information coarse graining Think of Kitaev’s code as a concatenated code: It is made up of a bunch of small (open boundary) topological codes, joined into larger topological codes, etc. Logical operators are strings going across the codes. Given the particle configuration (syndrome) in a unit cell, compute the prob that the a string type went through. There are 16 possibilities corresponding to { I , X , Y , Z } 2 , 2 qubits. This is done by brute force: sum over all worldline configurations. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 20 / 44

  50. Renormalization Group Decoder Overlaping cells Topological codes are NOT concatenated codes. Cannot break lattices into constant-size cells in such a way that each stabilizer overlaps with a single region. Use overlapping cells instead. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 21 / 44

  51. Renormalization Group Decoder Overlaping cells Topological codes are NOT concatenated codes. Cannot break lattices into constant-size cells in such a way that each stabilizer overlaps with a single region. Use overlapping cells instead. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 21 / 44

  52. Renormalization Group Decoder Overlaping cells Topological codes are NOT concatenated codes. Cannot break lattices into constant-size cells in such a way that each stabilizer overlaps with a single region. Use overlapping cells instead. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 21 / 44

  53. Renormalization Group Decoder Self-consistency 1 1 Boundary qubits treated as independent variables on neighboring unit cells. Probabilities assigned by different cells to a given qubit differ. Impose mean-field consistencies conditions on marginal probabilities. Solve by belief propagation. Complexity O ( ℓ 2 ) parallelizable to constant time. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 22 / 44

  54. Renormalization Group Decoder Self-consistency 1 1 Boundary qubits treated as independent variables on neighboring unit cells. Probabilities assigned by different cells to a given qubit differ. Impose mean-field consistencies conditions on marginal probabilities. Solve by belief propagation. Complexity O ( ℓ 2 ) parallelizable to constant time. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 22 / 44

  55. Renormalization Group Decoder Self-consistency 1 1 Boundary qubits treated as independent variables on neighboring unit cells. Probabilities assigned by different cells to a given qubit differ. Impose mean-field consistencies conditions on marginal probabilities. Solve by belief propagation. Complexity O ( ℓ 2 ) parallelizable to constant time. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 22 / 44

  56. Renormalization Group Decoder Self-consistency P L,i ( X ) � = P R,i ( X ) Boundary qubits treated as independent variables on neighboring unit cells. Probabilities assigned by different cells to a given qubit differ. Impose mean-field consistencies conditions on marginal probabilities. Solve by belief propagation. Complexity O ( ℓ 2 ) parallelizable to constant time. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 22 / 44

  57. Renormalization Group Decoder Self-consistency P � L,i ( X ) = P � R,i ( X ) Boundary qubits treated as independent variables on neighboring unit cells. Probabilities assigned by different cells to a given qubit differ. Impose mean-field consistencies conditions on marginal probabilities. Solve by belief propagation. Complexity O ( ℓ 2 ) parallelizable to constant time. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 22 / 44

  58. Renormalization Group Decoder Self-consistency P � L,i ( X ) = P � R,i ( X ) Boundary qubits treated as independent variables on neighboring unit cells. Probabilities assigned by different cells to a given qubit differ. Impose mean-field consistencies conditions on marginal probabilities. Solve by belief propagation. Complexity O ( ℓ 2 ) parallelizable to constant time. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 22 / 44

  59. Renormalization Group Decoder Self-consistency P � L,i ( X ) = P � R,i ( X ) Boundary qubits treated as independent variables on neighboring unit cells. Probabilities assigned by different cells to a given qubit differ. Impose mean-field consistencies conditions on marginal probabilities. Solve by belief propagation. Complexity O ( ℓ 2 ) parallelizable to constant time. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 22 / 44

  60. Results for Kitaev’s code Outline Kitaev’s toric code 1 Decoding problem 2 Renormalization Group Decoder 3 Results for Kitaev’s code 4 5 Extension to other codes Fault-tolerance 6 7 2D Fault-Tolerant Quantum Cellular Automaton Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 23 / 44

  61. Results for Kitaev’s code RG decoder with 3 BP rounds 1 l =8 l =16 l =32 Probabilite d’erreur du decodeur l =64 0.1 0.01 12 13 14 15 16 17 Force du canal depolarizant , p (%) Threshold ≈ 15 % , compared to 15 . 5 % for PMA. O ( log ℓ ) time complexity with marginal performance loss. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 24 / 44

  62. Results for Kitaev’s code Smaller unit cell, 2 × 1 1 l =8 l =16 l =32 Probabilite d’erreur du decodeur l =64 l =128 l =256 l =512 0.1 l =1024 Failure probability 0.01 0.001 0.0001 2 4 6 8 10 20 30 40 50 60 70 80 90 100 Depolarization Strength % Force du canal Bit-Flip , p (%) Bit-flip threshold ≈ 8 . 2 % , compared to 10 . 3 % for PMA. Much faster even without parallelization (10 6 sites). Illustrates flexibility. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 25 / 44

  63. Results for Kitaev’s code Higher threshold 1 l =8 l =16 l =32 Probabilite d’erreur du decodeur l =64 l =128 0.1 PMA 15 15.5 16 16.5 17 17.5 18 Force du canal depolarizant , p (%) Use of additional belief propagation. Threshold ≈ 16 . 5 % , compared to 15 . 5 % for PMA. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 26 / 44

  64. Extension to other codes Outline Kitaev’s toric code 1 Decoding problem 2 Renormalization Group Decoder 3 Results for Kitaev’s code 4 5 Extension to other codes Fault-tolerance 6 7 2D Fault-Tolerant Quantum Cellular Automaton Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 27 / 44

  65. Extension to other codes Topological color code Qubits located on vertices Plaquette stabilizers S p = � j ∈ ∂ p σ j , for σ = σ x and σ z . Same particle types and statistics as 2 copies of Kitaev’s code Efficient decoding algorithm for this code? Sarvepalli, Raussendorf 2011 : Adaptation of the RG scheme to the 6.6.6. color code → many subtleties : many qubits shared between cells, better to work on the dual lattice. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 28 / 44

  66. Extension to other codes Topological color code X X X X Qubits located on vertices X X X X Plaquette stabilizers S p = � j ∈ ∂ p σ j , for σ = σ x and σ z . Z Z Z Z Z Z Same particle types and statistics as Z Z 2 copies of Kitaev’s code Efficient decoding algorithm for this code? Sarvepalli, Raussendorf 2011 : Adaptation of the RG scheme to the 6.6.6. color code → many subtleties : many qubits shared between cells, better to work on the dual lattice. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 28 / 44

  67. Extension to other codes Topological color code X X X X Qubits located on vertices X X X X Plaquette stabilizers S p = � j ∈ ∂ p σ j , for σ = σ x and σ z . Same particle types and statistics as Z Z Z Z 2 copies of Kitaev’s code Z Z Z Z Efficient decoding algorithm for this code? Sarvepalli, Raussendorf 2011 : Adaptation of the RG scheme to the 6.6.6. color code → many subtleties : many qubits shared between cells, better to work on the dual lattice. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 28 / 44

  68. Extension to other codes Topological color code Qubits located on vertices X X X X Plaquette stabilizers S p = � j ∈ ∂ p σ j , Z for σ = σ x and σ z . Z Z Z Same particle types and statistics as 2 copies of Kitaev’s code Efficient decoding algorithm for this code? Sarvepalli, Raussendorf 2011 : Adaptation of the RG scheme to the 6.6.6. color code → many subtleties : many qubits shared between cells, better to work on the dual lattice. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 28 / 44

  69. Extension to other codes Topological color code Qubits located on vertices X X X X Plaquette stabilizers S p = � j ∈ ∂ p σ j , Z for σ = σ x and σ z . Z Z Z Same particle types and statistics as 2 copies of Kitaev’s code Efficient decoding algorithm for this code? Sarvepalli, Raussendorf 2011 : Adaptation of the RG scheme to the 6.6.6. color code → many subtleties : many qubits shared between cells, better to work on the dual lattice. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 28 / 44

  70. Extension to other codes Topological color code Qubits located on vertices X X X X Plaquette stabilizers S p = � j ∈ ∂ p σ j , Z for σ = σ x and σ z . Z Z Z Same particle types and statistics as 2 copies of Kitaev’s code Efficient decoding algorithm for this code? Sarvepalli, Raussendorf 2011 : Adaptation of the RG scheme to the 6.6.6. color code → many subtleties : many qubits shared between cells, better to work on the dual lattice. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 28 / 44

  71. Extension to other codes Topological color code Qubits located on vertices X X X X Plaquette stabilizers S p = � j ∈ ∂ p σ j , Z for σ = σ x and σ z . Z Z Z Same particle types and statistics as 2 copies of Kitaev’s code Efficient decoding algorithm for this code? Sarvepalli, Raussendorf 2011 : Adaptation of the RG scheme to the 6.6.6. color code → many subtleties : many qubits shared between cells, better to work on the dual lattice. Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 28 / 44

  72. Extension to other codes Equivalence of Topological codes Every 2D, translationally invariant, non-chiral stabilizer code with local generators and macroscopic minimal distance is locally equivalent to a finite number of copies of Kitaev’s code. Topological Color code equivalent to two copies of Kitaev’s code : a local clifford map exists Do the mapping and operate decoding on KTCs instead. Local Clifford Map Noise model remains Pauli and local Map for syndrome bits (particles) Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 29 / 44

  73. Extension to other codes Equivalence of Topological codes Every 2D, translationally invariant, non-chiral stabilizer code with local generators and macroscopic minimal distance is locally equivalent to a finite number of copies of Kitaev’s code. Topological Color code equivalent to two copies of Kitaev’s code : a local clifford map exists Do the mapping and operate decoding on KTCs instead. Local Clifford Map Noise model remains Pauli and local Map for syndrome bits (particles) Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 29 / 44

  74. Extension to other codes Equivalence of Topological codes Every 2D, translationally invariant, non-chiral stabilizer code with local generators and macroscopic minimal distance is locally equivalent to a finite number of copies of Kitaev’s code. Topological Color code equivalent to two copies of Kitaev’s code : a local clifford map exists Do the mapping and operate decoding on KTCs instead. Local Clifford Map Noise model remains Pauli and local Map for syndrome bits (particles) Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 29 / 44

  75. Extension to other codes Local Clifford Map KTC 1 KTC 2 KTC 1 KTC 2 TCC TCC → → → → → → → → → → Local : 1 body →≤ 3 body Preserves commutation relations Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 30 / 44

  76. Extension to other codes Charge Map = ⇒ Z Z → Z → 1 Z 1 = ⇒ → Z → Z Map of hopping operators ⇒ map of the charges Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 31 / 44

  77. Extension to other codes Results for Topological color code 1 l =16 l =32 l =64 l =128 l =256 Decoding error probability 0.1 0.01 7 7.5 8 8.5 9 9.5 10 10.5 11 Bit-Flip channel strength p % Threshold ≈ 8 . 7 % compared to ∼ 11 % . First efficient decoder for this code Guillaume Duclos-Cianci (Sherbrooke) Decoding Problem QEC’11 USC 32 / 44

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