1. Quantum Boltzmann Machine
Mohammad Amin
D-Wave Systems Inc. 1

2. Not the only use of QA
Maybe not the best use of QA

3. Adiabatic Quantum Computation
H(t) = (1-s)HD + sHP , s = t/tf
energy levels
gmin
tf ~ (1/gmin)2
Solution
Initial state s 1

4. Thermal Noise System Bath Interaction Dynamical freeze-out P0 s 1
energy levels
kBT
Dynamical freeze-out P0 s 1

5. Open quantum calculations of a 16 qubit random problem
Classical energies

6. Equilibration Can Cause Correlation
Correlation with simulated annealing
Hen et al., PRA 92, (2015)

7. Equilibration Can Cause Correlation
Boixo et al., Nature Phys. 10, 218 (2014)
Correlation with Quantum Monte Carlo

8. Equilibration Can Cause Correlation
Correlation with spin vector Monte Carlo
Shin et al., arXiv:
SVMC
SVMC

9. Equilibration Can Mask Quantum Speedup
Brooke et al., Science 284, 779 (1999)
Quantum advantage is expected to be dynamical

10. Equilibration Can Mask Quantum Speedup
Ronnow et al., Science 345, 420 (2014)
Hen et al., arXiv:
King et al., arXiv:
Equilibrated probability!!!
Computation time is independent of dynamics!

11. Residual Energy vs Annealing Time
50 random problems, 100 samples per problem per annealing time
Bimodal (J=-1, +1 , h=0)
Mean residual energy
Lowest residual energy
Annealing time (ms)

12. Residual Energy vs Annealing Time
50 random problems, 100 samples per problem per annealing time
Frustrated loops (a=0.25)
Bimodal (J=-1, +1 , h=0)
Annealing time (ms)
Annealing time (ms)

13. Boltzmann sampling is #P Quantum Boltzmann Distribution?
harder than NP
What can we do with a
Quantum Boltzmann Distribution?

14. arXiv:1601.02036 Evgeny Andriyash Jason Rolfe Bohdan Kulchytskyy
Roger Melko

15. Machine Learning in our Daily Life

16. Introduction to Machine Learning
Data
Model 3
Model
Unseen data

17. Data Model Probabilistic Models Training: Tune q such that
Probability distribution
Data
Variables
Parameters q
Model
Training: Tune q such that

18. Boltzmann distribution (b =1)
Boltzmann Machine
Data
Variables
Parameters q
Model
Boltzmann distribution (b =1)

19. Boltzmann Machine
Ising model:
spins
parameters

20. za Fully Visible BM Hz only has O(N2) parameters
needs O(2N) parameters to be fully described

21. Adding Hidden Variables
z i zn
visible hidden
za = (zn , zi)
hidden
visible

22. We need an efficient way to calculate
Training a BM
Tune such that
We need an efficient way to calculate
Maximize log-likelihood:
Or minimize:
gradient descent
technique
training rate

23. Calculating the Gradient
Average with clamped visibles
Unclamped average

24. Training Ising Hamiltonian Parameters
Clamped average
Unclamped average
Gradients can be estimated using sampling!

25. Is it possible to train a quantum Boltzmann machine?
Question:
Is it possible to train a
quantum Boltzmann machine?
Ising Hamiltonian
Transverse Ising Hamiltonian

26. Transverse Ising Hamiltonian

27. Quantum Boltzmann Distribution
Boltzmann probability distribution:
Density matrix:
Projection operator
Identity matrix

28. Gradient Descent =
Classically:
Clamped average
Unclamped average =

29. Calculating the Gradient
Gradient cannot be estimated using sampling! ≠ ≠
Clamped average
Unclamped average

30. Two Useful Properties of Trace
Golden-Thompson inequality:
For Hermitian matrices A and B

31. Finding lower bounds
Golden-Thompson inequality

32. Finding lower bounds Lower bound for log-likelihood
Golden-Thompson inequality
Lower bound for log-likelihood

33. Calculating the Gradients
Minimize the upper bound ?
Unclamped average

34. Visible qubits are clamped to their classical values given by the data
Clamped Hamiltonian
for
Infinite energy penalty for states different from v
Visible qubits are clamped to their classical values given by the data

35. We can now use sampling to estimate the steps
Estimating the Steps
Clamped average
Unclamped average
We can now use sampling to estimate the steps

36. Training the Transverse Field (Ga)
Minimizing the upper bound:
for all visible qubits, thus
cannot be estimated from measurements
Two problems:
Gn cannot be trained using the bound

37. Example: 10-Qubit QBM
Graph: fully connected (K10), fully visible

38. Example: 10-Qubit QBM Training set: M-modal distribution p = 0.9 M = 8
Random spin orientation
Single mode:
Hamming distance
p = 0.9
Multi-mode:
M = 8

39. Exact Diagonalization Results
KL-divergence:
Classical BM
Bound gradient
D=2
Exact gradient
(D is trained)
D final = 2.5

40. Sampling from D-Wave Probabilities cross at the anticrossing
Dickson et al., Nat. Commun. 4, 1903 (2013)
Probabilities cross at the anticrossing

41. Conclusions:
A quantum annealer can provide fast samples of quantum Boltzmann distribution
QBM can be trained by sampling
QBM may learn some distributions better than classical BM
See arXiv: