1. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. A New Scheduling Problem Motivated by Quantum Computation Robert Carr Anand Ganti Cynthia A. Phillips Sandia National Laboratories

2. Slide 2 Quantum Computation Use a machine motivated by quantum mechanics to solve problems that are difficult for traditional computers Known benefits include faster: Factoring Search Simulating quantum physics To date, theoretical algorithms and a few early physical experiments

3. Slide 3 Sandia National Laboratories Project Sandia basic quantum information sciences –Advanced computing architectures –Future engineered systems will require increased understanding of quantum effects. Current three-year project to –Build physical qubit Will test current understanding of quantum mechanics –Design a logical qubit There are scheduling problems critical for quantum architecture design

4. Slide 4 Quantum Bits Classical bits: 0 or 1 Quantum bits (qubits): Superposition Measurement destroys superposition, makes

5. Slide 5 Gates (examples) 1-bit gates: 2-bit gates:

6. Slide 6 Quantum Errors Interaction with environment  decoherence Errors act like X,Y,Z gates Errors are continuous

7. Slide 7 Quantum Error Correction Consider just flip errors Idea similar to classical error correction –Encode a single bit with more bits –Define a set of legal codewords –Ensure that all illegal codewords that result from a single error are closest to unique legal codeword Simple example: Use majority to correct any single flip error. Real Example Steane [7,3,3], Calderbank-Shor-Steane codes

8. Slide 8 Quantum Complication 1 Have to encode as without knowing  or . –Only 2 of the 8 possible states have positive probability This circuit creates the appropriate (entangled) states: }

9. Slide 9 Quantum Complication 2 Measurement destroys information Ancilla bits –Interact with real qbits –Pattern of ancilla values encodes single errors uniquely –Measure the ancilla

10. Slide 10 Quantum Error Correction Critical for quantum computing –Cannot completely isolate qubits from the world (e.g. components of the computer itself) Error correction happens often –Essentially after every operation –Error correction vastly dominates operations Error correction is worth doing quickly/well –Throughput –Error threshold Burn error correction into silicon, kind of like microcode The precise nature depends on –General quantum architecture –Precise code

11. Slide 11 Our Architecture: Bilinear Array Hollenberg et al Rail } Gate Gate entry node Gate node Measurement Gate = location that can hold a qubit/information

12. Slide 12 Bilinear Array: Legal Movement Move wherever there is an edge, including across gate Multiple possible transport mechanisms such at CTAP (teleportation) One edge per step (full to empty) Bits cannot pass through each other

13. Slide 13 Error Correction is a Program Three types of operations Single bit 2 bit Measurement PREPAREPLUS 7 CNOT(7,9) MEASUREX 8 MEASUREZ 9 CNOT (0,3) CNOT (3,8) … } Executed in gates

14. Slide 14 Scheduling Problem Select initial placement (cyclic) Schedule location and timing of operations Schedule legal movements Obey precedence constraints –(Usually) two operations that share a bit done serially Possible parallelism limits Minimize makespan Avoid unnecessary movement

15. Slide 15 Example 3 encoding bits, 2 ancilla 4 measurements, 4 CNOTs (2-bit gates)

16. Slide 16 Example m Step 0

17. Slide 17 Example Step 1 m CNOT

18. Slide 18 Example Step 2 CNOT

19. Slide 19 Example Step 3

20. Slide 20 Example CNOT Step 4

21. Slide 21 Example CNOT Step 5

22. Slide 22 Example Step 6

23. Slide 23 Example Step 7

24. Slide 24 Example Step 8 m m

25. Slide 25 Integer Programming Variables x bnt, binary, 1 if bit b in node n at start of time t y (1) git binary, 1 if 1-bit instruction i executes in gatenode g, time t y (2) git binary, 1 if 2-bit instruction i executes at full gate g, time t y (2f) git same as y (2) git but flip control bit top to bottom y (m) mit binary, 1 if measurement instruction executes in measurement gate m at time t f bvwt implicit binary flow variables. Bit b moves v->w during time t

26. Slide 26 Some simple Special Ordered Sets Bit locations (0 is empty) Performing all operations

27. Slide 27 Movement Control Flow conservation Full->empty Cyclic

28. Slide 28 Precedence Constraints 9 sets depending on i,j in I 1, I 2, I m  = minimum time between operations (usually 1) Enforce only for nearest neighbors EST = earliest start time LAST = last start time

29. Slide 29 Matching Computation with Transportation c i = control bit d i = data bit g 1 = top gatenode of gate g g 2 = bottom gatenode of gate g

30. Slide 30 Stronger Transportation/Computation Coupling If a bit is not in a gatenode at the proper time, none of the associated gate-firing variables can be 1. Over 20x faster (similar constraints for bottom gates and measurement gates)

31. Slide 31 Objective Generally none. Can add a relaxation variable z, relaxing all coupling constraints: Minimize z Strange phenomenon: When z is integral, cplex 11 can require 4x as long to solve as when z and y’s are continuous. When y’s are integral, having no z is better (tiny examples)

32. Slide 32 LP cheating Half-bits can pass each other m m CNOT Steps 0 and 3 Steps 1 and 2

33. Slide 33 Comments and Issues LP example motivates forcing initial placements –Considerably faster –Have to enumerate over placements Need to understand structure How to determine time? Number of rails –Recursive doubling –Better to understand/compute bounds –LP time grows quickly with both Heuristics –LP based? –Constraint programming?

34. Slide 34 Extra Slides

35. Slide 35 Error Corrected Logical Qubit

36. Slide 36 Example m m CNOT Step 0 Step 1 Step 2

37. Slide 37 Example CNOT Step 3 Step 4 Step 5

38. Slide 38 Example Step 6 Step 7 Step 8 m m