φ -Automaton Decoders Dynamic Setting Outlook & Conclusion Cellular-automaton decoders for topological quantum memories arXiv:1406.2338 Michael Herold 1 Earl T. Campbell 1 , 2 Jens Eisert 1 Michael J. Kastoryano 1 , 3 1 Freie Universität Berlin 2 University of Sheffield 3 University of Copenhagen Third International Conference on Quantum Error Correction Zürich, December 2014 Michael Herold Freie Universität Berlin
φ -Automaton Decoders Dynamic Setting Outlook & Conclusion Outline ◮ φ -Automaton Decoders 2D ∗ -decoder 3D-decoder ◮ Dynamic Setting ◮ Outlook & Conclusion Michael Herold Freie Universität Berlin
φ -Automaton Decoders Dynamic Setting Outlook & Conclusion New decoders for the 2D toric code? ◮ Sophisticated decoders exist Realspace RG Decoder O ( log L ) 1 MWPM O ( L 2 ) 2 ◮ Parallelization does not imply locality Hidden communication costs Not necessarily suited for embedded hardware ◮ Question we address Natural parallelization without hidden costs Connecting decoding with physical systems 1 G. Duclos-Cianci and D. Poulin, Phys. Rev. Lett. 104 , 050504 (2010) 2 A. G. Fowler, A. C. Whiteside, and L. C. L. Hollenberg, Phys. Rev. Lett. 108 , 180501 (2012) Michael Herold Freie Universität Berlin
φ -Automaton Decoders Dynamic Setting Outlook & Conclusion φ -Automaton Decoders Michael Herold Freie Universität Berlin
φ -Automaton Decoders Dynamic Setting Outlook & Conclusion 2D toric code ◮ Consider X -errors with probability p ◮ Consider plaquette operators ( Z � ) Michael Herold Freie Universität Berlin
φ -Automaton Decoders Dynamic Setting Outlook & Conclusion Setting: fields for the anyons Michael Herold Freie Universität Berlin
φ -Automaton Decoders Dynamic Setting Outlook & Conclusion Setting: fields for the anyons Michael Herold Freie Universität Berlin
φ -Automaton Decoders Dynamic Setting Outlook & Conclusion Setting: fields for the anyons ◮ Implementation as classical cellular automaton? Michael Herold Freie Universität Berlin
φ -Automaton Decoders Dynamic Setting Outlook & Conclusion Local field generation ∇ 2 Φ( r ) = q ( r ) Differential equation discretization # „ r ∈ R D � x ∈ Z D � Set of linear equations − 2 D φ ( x ) + φ ( y ) = q ( x ) � y , x � Jacobi method # „ φ ( x ) � φ t + 1 ( x ) φ ( y ) � φ t ( y ) Interation φ t + 1 ( x ) = avg φ t ( y ) + q ( x ) � y , x � Michael Herold Freie Universität Berlin
φ -Automaton Decoders Dynamic Setting Outlook & Conclusion Local field generation ∇ 2 Φ( r ) = q ( r ) Differential equation discretization # „ r ∈ R D � x ∈ Z D � Set of linear equations − 2 D φ ( x ) + φ ( y ) = q ( x ) � y , x � Jacobi method # „ φ ( x ) � φ t + 1 ( x ) φ ( y ) � φ t ( y ) Interation φ t + 1 ( x ) = avg φ t ( y ) + q ( x ) � y , x � Michael Herold Freie Universität Berlin
φ -Automaton Decoders Dynamic Setting Outlook & Conclusion Local field generation ∇ 2 Φ( r ) = q ( r ) Differential equation discretization # „ r ∈ R D � x ∈ Z D � Set of linear equations − 2 D φ ( x ) + φ ( y ) = q ( x ) � y , x � Jacobi method # „ φ ( x ) � φ t + 1 ( x ) φ ( y ) � φ t ( y ) Interation φ t + 1 ( x ) = avg φ t ( y ) + q ( x ) � y , x � Michael Herold Freie Universität Berlin
φ -Automaton Decoders Dynamic Setting Outlook & Conclusion 1. Field-updates ( φ -automaton) ◮ Every cell stores a field value φ ( x ) ◮ Update rule: Average of neighboring fields + charge Michael Herold Freie Universität Berlin
φ -Automaton Decoders Dynamic Setting Outlook & Conclusion 2. Anyon-updates ◮ Anyons move via X -flips on crossed edges ◮ Update rule: Move to the neighbor cell with maximal φ -value Michael Herold Freie Universität Berlin
φ -Automaton Decoders Dynamic Setting Outlook & Conclusion The decoding algorithm Sequence ◮ c × field-update ◮ 1 × anyon-update ◮ Algorithm: Repeat sequence until all anyons are fused Field velocity c Parameter for the ‘speed of field propagation’ Michael Herold Freie Universität Berlin
φ -Automaton Decoders Dynamic Setting Outlook & Conclusion The 2D ∗ -decoder Sequence ∗ ◮ c × field-update ◮ 1 × anyon-update c → c + 0 . 2 ◮ ◮ Algorithm: Repeat sequence ∗ until all anyons are fused Field velocity c Parameter for the ‘speed of field propagation’ Michael Herold Freie Universität Berlin
φ -Automaton Decoders Dynamic Setting Outlook & Conclusion 2D ∗ -decoder: numerical analysis Exponential suppression Recovery threshold 8 . 2 % 0.5 p = 7 % S YS . SIZE L 100 D ECODER FAIL RATE D ECODER FAIL RATE 0.4 10 − 1 200 300 400 500 0.3 0.2 0.1 10 − 2 8 % 8.1 % 8.2 % 8.3 % 8.4 % 0 20 40 60 80 100 P HYSICAL ERROR RATE p S YSTEM SIZE ℓ ◮ Runtime: O ( log 7 . 5 L ) ◮ Resembles Wootton’s decoder 3 3 J. R. Wootton, A simple decoder for topological codes , arXiv:1310.2393 Michael Herold Freie Universität Berlin
φ -Automaton Decoders Dynamic Setting Outlook & Conclusion Avoiding sequence dependence ◮ Finding the right field c → ∞ must not be a problem Before: Poisson’s equation in 2D − log r profile Idea: Try fields of the form φ ( r ) ∼ 1 r α ◮ Simulation with explicit fields, no cellular automaton Anyons move towards max. field (as before) Michael Herold Freie Universität Berlin
φ -Automaton Decoders Dynamic Setting Outlook & Conclusion Avoiding sequence dependence ◮ Finding the right field c → ∞ must not be a problem Before: Poisson’s equation in 2D − log r profile Idea: Try fields of the form φ ( r ) ∼ 1 r α ◮ Simulation with explicit fields, no cellular automaton Anyons move towards max. field (as before) Michael Herold Freie Universität Berlin
φ -Automaton Decoders Dynamic Setting Outlook & Conclusion Avoiding sequence dependence ◮ Idea: Try fields of the form φ ( r ) ∼ 1 / r α ◮ Conjecture: At α = 1, transition from not-decoding to decoding regime 100 S YSTEM SIZE L 75 50 0.01 0.1 25 D ECODER FAIL RATE 0 0.5 1 1.5 2 L ONG RANGE PARAMETER α Michael Herold Freie Universität Berlin
φ -Automaton Decoders Dynamic Setting Outlook & Conclusion Avoiding sequence dependence ◮ Conjecture: At α = 1, transition from not-decoding to decoding regime ◮ ⇒ φ ( r ) ∼ 1 / r should work as decoder 0.6 S YS . SIZE L 100 0.5 D ECODER FAIL RATE 200 300 400 0.4 500 0.3 0.2 0.1 6.1 % 6.2 % 6.3 % 6.4 % 6.5 % P HYSICAL ERROR RATE p Michael Herold Freie Universität Berlin
φ -Automaton Decoders Dynamic Setting Outlook & Conclusion 3D φ -automaton L Michael Herold Freie Universität Berlin
φ -Automaton Decoders Dynamic Setting Outlook & Conclusion 3D φ -automaton L Michael Herold Freie Universität Berlin
φ -Automaton Decoders Dynamic Setting Outlook & Conclusion 3D φ -automaton ◮ Sufficient field convergence if c ( L ) ∼ log 2 L L Michael Herold Freie Universität Berlin
φ -Automaton Decoders Dynamic Setting Outlook & Conclusion 3D decoder: numerical analysis Exponential suppression Recovery threshold: 6 . 1 % 0.5 p = 4 % S YS . SIZE L 10 − 1 25 D ECODER FAIL RATE D ECODER FAIL RATE 0.4 50 75 100 125 0.3 10 − 2 0.2 10 − 3 0.1 5.8 % 6 % 6.2 % 6.4 % 6.6 % 0 10 20 30 40 50 60 70 80 90 P HYSICAL ERROR RATE p S YSTEM SIZE ℓ ◮ Runtime: O ( log 3 L ) ◮ Fundamentally new working principle Michael Herold Freie Universität Berlin
φ -Automaton Decoders Dynamic Setting Outlook & Conclusion Dynamic Setting Michael Herold Freie Universität Berlin
φ -Automaton Decoders Dynamic Setting Outlook & Conclusion Static setting Extract error syndrome Apply corrections Run decoder Dynamic setting Extract error syndrome · · · #„ #„ #„ #„ #„ #„ #„ Run decoder · · · #„ #„ #„ #„ #„ #„ #„ Apply corrections · · · #„ #„ #„ #„ #„ #„ #„ ◮ Decoder has to take new information into account ◮ 3D decoder has no ‘time zero’ Promising candidate to work in dynamic setting Michael Herold Freie Universität Berlin
φ -Automaton Decoders Dynamic Setting Outlook & Conclusion Static setting Extract error syndrome Apply corrections Run decoder Dynamic setting Extract error syndrome · · · #„ #„ #„ #„ #„ #„ #„ Run decoder · · · #„ #„ #„ #„ #„ #„ #„ Apply corrections · · · #„ #„ #„ #„ #„ #„ #„ ◮ Decoder has to take new information into account ◮ 3D decoder has no ‘time zero’ Promising candidate to work in dynamic setting Michael Herold Freie Universität Berlin
φ -Automaton Decoders Dynamic Setting Outlook & Conclusion Static setting Extract error syndrome Apply corrections Run decoder Dynamic setting Extract error syndrome · · · #„ #„ #„ Field-updates · · · #„ #„ #„ #„ #„ #„ #„ Anyon-updates · · · #„ #„ #„ ◮ Decoder has to take new information into account ◮ 3D decoder has no ‘time zero’ Promising candidate to work in dynamic setting Michael Herold Freie Universität Berlin
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