Embedding Logical Qubits into the D-Wave Hardware Graph

1. Embedding Logical Qubits into the D-Wave Hardware Graph Keith Britt keithbritt@utk.edu February 26th, 2014

4. Agenda • A Little Bit About Quantum Computing • A Little Bit About D-Wave • The D-Wave Hardware • Transforming the Hardware into a Graph • Embedding Literature “Review” • Embedding Limits • Example Embedding(s) • Homework Problems

5. Quantum Computing • In the Quantum World, classical physics give way to 2 phenomena: • Superposition • Entanglement • We can use these phenomena to massively parallelize computation and also do some crazy stuff that is otherwise impossible (truly random numbers, data teleportation, etc.)

6. Quantum Computing • Parallelism Scales Exponentially with the Number of Qubits () • Several Models of Quantum Computation • Gate/Circuit Model • Adiabatic Model • Topological model (and more…) • The First Commercially Available Quantum Computer (D-Wave) Uses the Adiabatic Model

7. D-Wave • Canadian Company Founded by Physicists and Material Scientists • Created a Quantum Chip Using Adiabatic Model Running at Extremely Cold Temperatures and Minimal Interference • Very Controversial… Is the Computer Really Quantum? Is It Worth It? • Simulations Beat It • Classical Adiabatic Computer Matched It

8. Ising Model : State (Spin/Value) of the Qubit ( ) : Set of Qubits : Strength of the Qubit Value : Strength of the Qubit-Qubit Interaction : The Total Energy of the System http://dwave.files.wordpress.com/2010/12/weightedmaxsat_v2.pdf

9. D-Wave Hardware http://dwave.wordpress.com/2011/12/01/vesuvius-a-closer-look-512-qubit-processor-gallery/

10. So is Hardware Just a Line Graph? Not Valid

11. Unit Cell Graph Representation • Each Qubit Becomes a Vertex • Each Coupler Becomes an Edge Between Vertices Valid

12. Ising Model : State (Spin/Value) of the Qubit ( ) : Set of Qubits : Strength of the Qubit Value : Strength of the Qubit-Qubit Interaction : The Total Energy of the System Vertex Weights Edge Weights http://dwave.files.wordpress.com/2010/12/weightedmaxsat_v2.pdf

13. Linking Unit Cells

14. Linking Unit Cells Not Valid

15. Linking Unit Cells Valid

16. 512 Vertices (Qubits) • 16 Intra Unit Cell Edgesper Unit Cell • 8 – 16 Inter Unit Cell Edges per Unit Cell

18. Literature Review • Choi, Vicky. Minor-Embedding in Adiabatic Quantum Computation: I. The Parameter Setting Problem. Quantum Information Processing, 7, pp 193 – 209, 2008.  arXiv:0804.4884v1 [quant-ph]. • Choi, Vicky. Minor-embedding in adiabatic quantum computation: II. Minor-universal graph design. Quantum Information Processing: Volume 10, Issue 3 (2011), Page 343.  arXiv:1001.3116v2 [quant-ph]. • Klymko, C., Sullivan, B., Humble, T. Adiabatic quantum programming: minor embedding with hard faults. Quantum Information Processing: Volume 13, Issue 3 (2014), pp 709 – 729.  arXiv:1210.8395v2 [quant-ph].

19. D-Wave Embedding Limits • Logical Qubit is Not the Same as a Physical Qubit • Can Always Embed N Logical Qubits onto Physical Qubits • This is Not a Tight Upper-bound, Even for Complete Graphs • Choi’s Triad Technique Able to Embed on 128-Qubit Processor ()

20. Logical to Physical • A Logical Qubit can be Spread Over Multiple Physical Qubits as Long as There is a Common Path Between All The Physical Qubits VALID NOT VALID

21. Something We’re Not Going Over and values determine a lot about how aligned qubits are with one another. This gets a bit too outside of graph theory, but there’s an easy to read explanation at: http://www.dwavesys.com/sites/default/files/Map%20Coloring%20WP2.pdf Vertex Weights Edge Weights

22. An Example • Embed into a Unit Cell

23. http://arxiv.org/pdf/1001.3116v2.pdf

24. http://arxiv.org/pdf/1001.3116v2.pdf

25. Embedding • Start by Embedding in a Unit Cell

26. Embedding • Next Embed the Other in a Unit Cell

27. Embedding • Next Embed in a Unit Cell

28. Embedding • Tada! is embedded in the D-Wave Hardware

29. Homework Questions • Embed into the D-Wave Hardware Graph in 6 or Fewer Unit Cells.

30. Homework Questions • Embed the Peterson Graph into the D-Wave Hardware Graph (Diamond Isomorphism) in 6 or Fewer Unit Cells.