1. Copyright© 2012, D-Wave Systems Inc. 1 Quantum Boltzmann Machine Mohammad Amin D-Wave Systems Inc.

2. Copyright© 2012, D-Wave Systems Inc. 2 Does D-Wave Return Boltzmann Samples? Copyright© 2015, D-Wave Systems Inc. Correlation between D-Wave and SA equilibrated at  Hen et al., arXiv:1502.01663

3. Copyright© 2012, D-Wave Systems Inc. 3 Copyright© 2015, D-Wave Systems Inc. Quantum Machine Learning Can we do with D-Wave

4. Copyright© 2012, D-Wave Systems Inc. 4 Copyright© 2015, D-Wave Systems Inc. Jason Rolfe Emile Hoskinson Trevor Lanting Yuki Sato Monte Carlo Roger Melko Bohdan Kulchytskyy Collaborators Brandon Denis Evgeny Andriyash Quantum Machine Learning

5. Copyright© 2012, D-Wave Systems Inc. 5 Boltzmann Machine Copyright© 2015, D-Wave Systems Inc. z i z visible hidden z a   z   z i  Boltzmann distribution: Ising Hamiltonian:

6. Copyright© 2012, D-Wave Systems Inc. 6 Training Ising Hamiltonian Parameters Copyright© 2015, D-Wave Systems Inc. Clamped averageUnclamped average Gradients can be estimated using sampling!

7. Copyright© 2012, D-Wave Systems Inc. 7 Question: Copyright© 2015, D-Wave Systems Inc. Ising Hamiltonian Transverse Ising Hamiltonian

8. Copyright© 2012, D-Wave Systems Inc. 8 Moving to Quantum Copyright© 2015, D-Wave Systems Inc. Hamiltonian (Energy):  

9. Copyright© 2012, D-Wave Systems Inc. 9 Matrix Representation Copyright© 2015, D-Wave Systems Inc. Partition function:  

10. Copyright© 2012, D-Wave Systems Inc. 10 Matrix Representation Copyright© 2015, D-Wave Systems Inc. Boltzmann probability:  

11. Copyright© 2012, D-Wave Systems Inc. 11 Matrix Representation Copyright© 2015, D-Wave Systems Inc. Boltzmann probability: visibles = v visibles ≠ v Projection operator Identity matrix

12. Copyright© 2012, D-Wave Systems Inc. 12 Transverse Ising Hamiltonian Copyright© 2015, D-Wave Systems Inc. non-diagonal matrix classical Ising Hamiltonian (diagonal matrix)

13. Copyright© 2012, D-Wave Systems Inc. 13 Quantum Boltzmann Distribution Copyright© 2015, D-Wave Systems Inc. Boltzmann distribution: Projection operator Identity matrix

14. Copyright© 2012, D-Wave Systems Inc. 14 Gradient Descent - Classical Copyright© 2015, D-Wave Systems Inc. Clamped average Unclamped average  = Classically:

15. Copyright© 2012, D-Wave Systems Inc. 15 Gradient Descent - Quantum Copyright© 2015, D-Wave Systems Inc. ≠ Gradient cannot be estimated using sampling! Clamped average Unclamped average ≠

16. Copyright© 2012, D-Wave Systems Inc. 16 Two Useful Properties of Trace Copyright© 2015, D-Wave Systems Inc. Golden-Thompson inequality: For Hermitian matrices A and B

17. Copyright© 2012, D-Wave Systems Inc. 17 Finding lower bounds Copyright© 2015, D-Wave Systems Inc. Golden-Thompson inequality

18. Copyright© 2012, D-Wave Systems Inc. 18 Finding lower bounds Copyright© 2015, D-Wave Systems Inc. Golden-Thompson inequality Lower bound for log-likelihood

19. Copyright© 2012, D-Wave Systems Inc. 19 Calculating the Gradients Copyright© 2015, D-Wave Systems Inc. Minimize the upper bound ???? Unclamped average

20. Copyright© 2012, D-Wave Systems Inc. 20 Clamped Hamiltonian Copyright© 2015, D-Wave Systems Inc. Infinite energy penalty for states different from v for Visible qubits are clamped to their classical values given by the data

21. Copyright© 2012, D-Wave Systems Inc. 21 Estimating the Steps Copyright© 2015, D-Wave Systems Inc. Clamped averageUnclamped average We can now use sampling to estimate the steps

22. Copyright© 2012, D-Wave Systems Inc. 22 Training  a Copyright© 2015, D-Wave Systems Inc. for all visible qubits, thus cannot be estimated from measurements Two problems: Minimizing the upper bound:  cannot be trained using the bound

23. Copyright© 2012, D-Wave Systems Inc. 23 Example: 10-Qubit QBM Copyright© 2015, D-Wave Systems Inc. Graph: fully connected (K10), fully visible

24. Copyright© 2012, D-Wave Systems Inc. 24 Example: 10-Qubit QBM Copyright© 2015, D-Wave Systems Inc. Training set: M -modal distribution

25. Copyright© 2012, D-Wave Systems Inc. 25 Example: 10-Qubit QBM Copyright© 2015, D-Wave Systems Inc. Training set: M -modal distribution Random spin orientation Single qubit: = 90% aligned = 10% not aligned

26. Copyright© 2012, D-Wave Systems Inc. 26 Example: 10-Qubit QBM Copyright© 2015, D-Wave Systems Inc. Training set: M -modal distribution Random spin orientation Single mode: Hamming distance between v and S k Bernoulli distribution

27. Copyright© 2012, D-Wave Systems Inc. 27 Example: 10-Qubit QBM Copyright© 2015, D-Wave Systems Inc. Training set: M -modal distribution Random spin orientation Multi-mode: We use p = 0.9, M = 8

28. Copyright© 2012, D-Wave Systems Inc. 28 Exact Diagonalization Results Copyright© 2015, D-Wave Systems Inc. Classical BM Bound gradient  Exact gradient  is trained   final  KL-divergence:

29. Copyright© 2012, D-Wave Systems Inc. 29 Training Trajectories Copyright© 2015, D-Wave Systems Inc.

30. Copyright© 2012, D-Wave Systems Inc. 30 Scaling with Size Copyright© 2015, D-Wave Systems Inc. KL classical  KL quantum averaged over 100 problems

31. Copyright© 2012, D-Wave Systems Inc. 31 Adding Hidden Variables Copyright© 2015, D-Wave Systems Inc. Clamped averageUnclamped average Computationally expensive for large training sets

32. Copyright© 2012, D-Wave Systems Inc. 32 Quantum RBM Copyright© 2015, D-Wave Systems Inc. Clamped averageUnclamped average Can be easily calculated

33. Copyright© 2012, D-Wave Systems Inc. 33 Quantum RBM Copyright© 2015, D-Wave Systems Inc. Effective bias applied to the hiddens:

34. Copyright© 2012, D-Wave Systems Inc. 34 Example: 10-Qubit QRBM Copyright© 2015, D-Wave Systems Inc. Graph: 8 visibles fully connected (K8) 2 hiddens unconnected

35. Copyright© 2012, D-Wave Systems Inc. 35 Exact Diagonalization Results Copyright© 2015, D-Wave Systems Inc. Classical positive phase Quantum positive phase

36. Copyright© 2012, D-Wave Systems Inc. 36 Training Trajectories Copyright© 2015, D-Wave Systems Inc.

37. Copyright© 2012, D-Wave Systems Inc. 37 Sampling from Conditional Probability Copyright© 2015, D-Wave Systems Inc. Classical BM:  Joint distribution

38. Copyright© 2012, D-Wave Systems Inc. 38 Sampling from Conditional Probability Copyright© 2015, D-Wave Systems Inc. Classical BM: x clamped to data  Conditional distribution

39. Copyright© 2012, D-Wave Systems Inc. 39 Sampling from Conditional Probability Copyright© 2015, D-Wave Systems Inc. QBM: ≠ x clamped to data  Conditional distribution

40. Copyright© 2012, D-Wave Systems Inc. 40 Conditional Distribution Copyright© 2015, D-Wave Systems Inc. projection operators Clamped Hamiltonian Classical BM:

41. Copyright© 2012, D-Wave Systems Inc. 41 Conditional Distribution Copyright© 2015, D-Wave Systems Inc. projection operators Clamped Hamiltonian QBM: ≠ ? Generative supervised learning can be challenging ≠

42. Copyright© 2012, D-Wave Systems Inc. 42 Example: 11-Qubit QBM Copyright© 2015, D-Wave Systems Inc. Graph (K11): 8 input qubits 3 output qubits

43. Copyright© 2012, D-Wave Systems Inc. 43 Exact Diagonalization Results Copyright© 2015, D-Wave Systems Inc. QBM trained by joint distribution Joint distribution

44. Copyright© 2012, D-Wave Systems Inc. 44 Exact Diagonalization Results Copyright© 2015, D-Wave Systems Inc. Conditional distribution QBM trained by joint distribution

45. Copyright© 2012, D-Wave Systems Inc. 45 Using D-Wave as a QBM Copyright© 2015, D-Wave Systems Inc. Amin, PRA 92, 052323 (2015) Boltzmann distribution Open quantum simulation of 16 qubit QA

46. Copyright© 2012, D-Wave Systems Inc. 46 Residual Energy vs Annealing Time Copyright© 2015, D-Wave Systems Inc. Frustrated loops (  ) 50 random problems, 100 samples per problem per annealing time Bimodal ( J , h  )

47. Copyright© 2012, D-Wave Systems Inc. 47 Conclusions: Copyright© 2015, D-Wave Systems Inc. A QBM can use quantum Boltzmann distribution for machine learning Supervised learning with QBM should be done with care A quantum annealer can provide fast samples for QBM training A QBM can be trained by sampling

48. Copyright© 2012, D-Wave Systems Inc. 48 Copyright© 2015, D-Wave Systems Inc. Please talk to any of us or visit www.dwavesys.com