1. Using surface code experimental output correctly and effectively
Austin Fowler
Google Inc.

2. Overview Conclusions Gate and state definitions Error propagation
Error detection
Classical processing
Fast feedforward
Parameter tuning
Error model tuning

3. Conclusions
Classical processing for error correction cannot be broken into independent rounds as this is not fault-tolerant
Corrections should not be applied to the quantum system, as they are constantly revised
Fast feedforward primarily required in a large quantum computer rapidly executing a nontrivial quantum algorithm, and not for error correction
Errors that have already occurred do not become more dangerous as additional errors occur

4. Gate and state definitions
Computational basis:
Initialization to
Unitary gates:
Measurement in computational basis:

5. Error propagation
Above identities can be proven via matrix multiplication of the definitions on the previous slide

6. Error propagation
Surface code data qubits (circles) constrained to be eigenstate of certain operators (stabilizers)
Z-stabilizers enable detection of X errors
Keep things simple and focus on a slice of the surface code

7. Error propagation

8. Error propagation X 1 1
Change in measured value indicates endpoint of error chain
Most likely error chain simply connects to nearest boundary
Record in software that we believe the top data qubit is associated with an X error

9. Error propagation Two error chain endpoints observed1 1 X 1 1
Two error chain endpoints observed
Most likely pattern of error chains is a single chain connecting the two endpoints
Record in software that we believe the middle data qubit is associated with an X error

10. Error propagation Two error chain endpoints observed1 X
Two error chain endpoints observed
Most likely pattern of error chains is a single chain connecting the two endpoints
Flip the classical measurement value

11. Error propagation Two error chain endpoints observed1 1 X 1
Two error chain endpoints observed
Most likely pattern of error chains is a single chain connecting the two endpoints
Record in software that we believe the middle data qubit is associated with an X error and flip the second classical measurement value on the lower measurement qubit
Round by round processing fails to correctly identify the above error

12. Classical processing
10 data qubits
One detection event

13. Classical processing 10 data qubits One detection event
Explore uniformly, boundary found

14. Classical processing 10 data qubits One detection eventX
10 data qubits
One detection event
Explore uniformly, boundary found
Match detection event to boundary, record belief that X error present

15. Classical processing X
Two more detection events

16. Classical processing Two more detection eventsX
Two more detection events
Pick one, explore, current time boundary encountered

17. Classical processing Two more detection eventsX
Two more detection events
Pick one, explore, current time boundary encountered
Explore around other, exploratory regions touch

18. Classical processing Two more detection eventsX
Two more detection events
Pick one, explore, current time boundary encountered
Explore around other, exploratory regions touch
Match, record belief that two more X errors present X X

19. Classical processing X
One more detection event X X

20. Classical processing One more detection eventX
One more detection event
Explore, current time boundary encountered, must wait for more data X X

21. Classical processing One more detection eventX
One more detection event
Explore, current time boundary encountered, must wait for more data
Explore further, boundary encountered X X

22. Classical processing One more detection eventX X X
One more detection event
Explore, current time boundary encountered, must wait for more data
Explore further, boundary encountered
Match, record belief that two more X errors present X X

23. Classical processing One more detection eventX
One more detection event
Explore, current time boundary encountered, must wait for more data
Explore further, boundary encountered
Match, record belief that two more X errors present
Cancel double error
Don’t apply physical corrections X X

24. Classical processing X X X

25. Classical processing X X X

26. Classical processing X X X X X

27. Classical processing X X X

28. Classical processing X X X

29. Classical processing X X X

30. Classical processing X X X X X

31. Classical processing X X X

32. Classical processing X X X

33. Classical processing X X X X X X X

34. Classical processing X X X X X

35. Classical processing X X X X X

36. Classical processing X X X X X

37. Classical processing X X X X X

38. Classical processing X X X X X

39. Classical processing X X X X X

40. Classical processing X X X X X

41. Classical processing X X X X X X X X

42. Classical processing X

43. Fast feedforward
Could be useful if measurement is QND and the |0> state is more robust than |1> and the qubits really are qubits and no better reset gate is available
Really need good reset gate in any system with leakage

44. Fast feedforward
Fast feedforward is necessary only at the error corrected logical gate level where it is a critical ingredient in fast implementations of quantum algorithms
T gates are probabilistic and 50% of the time require S gate corrections
Prior logical measurements determine the basis of future logical measurements
At the moment the speed at which this can occur is entirely limited by the speed classical processing

45. Parameter tuning
Suppose have continuously running surface code quantum computer
Every measurement qubit can be associated with the detection event rate
Choose array of parameters
Adjust parameter
If detection event rate increases reverse direction of adjustment
Designed to keep gates tuned up while long algorithm runs
J. Kelly et al. in preparation

46. Error model tuning Error model is essentially a weighted graph
Can track the relative number of times each geometrically distinct edge is observed
Enables better matching

47. Conclusions
Classical processing for error correction cannot be broken into independent rounds as this is not fault-tolerant
Corrections should not be applied to the quantum system, as they are constantly revised
Fast feedforward primarily required in a large quantum computer rapidly executing a nontrivial quantum algorithm, and not for error correction
Errors that have already occurred do not become more dangerous as additional errors occur