Quantum Boltzmann Machine Mohammad Amin DWave Systems Inc

Quantum Boltzmann Machine Mohammad Amin D-Wave Systems Inc. • 1 • Copyright© 2016, D-Wave Systems Inc.

Not the only use of QA Maybe not the best use of QA • 2 • Copyright© 2016, D-Wave Systems Inc.

Adiabatic Quantum Computation • 3 • s = t/tf gmin 0 s 1 Solution energy levels Initial state H(t) = (1 -s)HD + s. HP , tf ~ (1/gmin)2 Copyright© 2016, D-Wave Systems Inc.

Thermal Noise k. BT Bath Interaction energy levels System P 0 0 s 1 Dynamical freeze-out • 4 • Copyright© 2016, D-Wave Systems Inc.

Open quantum calculations of a 16 qubit random problem Classical energies • 5 • Copyright© 2016, D-Wave Systems Inc.

Equilibration Cause Correlation with simulated annealing Hen et al. , PRA 92, 042325 (2015) • 6 • Copyright© 2016, D-Wave Systems Inc.

Equilibration Cause Correlation with Quantum Monte Carlo Boixo et al. , Nature Phys. 10, 218 (2014) • 7 • Copyright© 2016, D-Wave Systems Inc.

Equilibration Cause Correlation with spin vector Monte Carlo Shin et al. , ar. Xiv: 1401. 7087 SVMC • 8 • SVMC Copyright© 2016, D-Wave Systems Inc.

Equilibration Can Mask Quantum Speedup Brooke et al. , Science 284, 779 (1999) Quantum advantage is expected to be dynamical • 9 • Copyright© 2016, D-Wave Systems Inc.

Equilibration Can Mask Quantum Speedup Ronnow et al. , Science 345, 420 (2014) Hen et al. , ar. Xiv: 1502. 01663 King et al. , ar. Xiv: 1502. 02098 Equilibrated probability!!! Computation time is independent of dynamics! • 10 • Copyright© 2016, D-Wave Systems Inc.

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 • 11 • Annealing time (ms) Copyright© 2016, D-Wave Systems Inc.

Residual Energy vs Annealing Time 50 random problems, 100 samples per problem per annealing time Frustrated loops (a=0. 25) • 12 • Annealing time (ms) Bimodal (J=-1, +1 , h=0) Annealing time (ms) Copyright© 2016, D-Wave Systems Inc.

Boltzmann sampling is #P harder than NP What can we do with a Quantum Boltzmann Distribution? • 13 • Copyright© 2016, D-Wave Systems Inc.

ar. Xiv: 1601. 02036 Evgeny Andriyash • 14 • Jason Rolfe Bohdan Kulchytskyy Roger Melko Copyright© 2016, D-Wave Systems Inc.

Machine Learning in our Daily Life • 15 • Copyright© 2016, D-Wave Systems Inc.

Introduction to Machine Learning Data Unseen data • 16 • Model 3 Copyright© 2016, D-Wave Systems Inc.

Probabilistic Models Data Probability distribution Model Variables Parameters q Training: Tune q such that • 17 • Copyright© 2016, D-Wave Systems Inc.

Boltzmann Machine Data Model Variables Parameters q Boltzmann distribution (b =1) • 18 • Copyright© 2016, D-Wave Systems Inc.

Boltzmann Machine Ising model: spins parameters • 19 • Copyright© 2016, D-Wave Systems Inc.

Adding Hidden Variables zi zn za = (zn , zi) visible hidden visible • 21 • hidden Copyright© 2016, D-Wave Systems Inc.

Training a BM Tune such that Maximize log-likelihood: We need an efficient way to calculate Or minimize: training rate gradient descent technique • 22 • Copyright© 2016, D-Wave Systems Inc.

Calculating the Gradient Average with clamped visibles • 23 • Unclamped average Copyright© 2016, D-Wave Systems Inc.

Training Ising Hamiltonian Parameters Clamped average Unclamped average Gradients can be estimated using sampling! • 24 • Copyright© 2016, D-Wave Systems Inc.

Question: Is it possible to train a quantum Boltzmann machine? Ising Hamiltonian • 25 • Transverse Ising Hamiltonian Copyright© 2016, D-Wave Systems Inc.

Transverse Ising Hamiltonian • 26 • Copyright© 2016, D-Wave Systems Inc.

Quantum Boltzmann Distribution Boltzmann probability distribution: Density matrix: • 27 • Projection operator Identity matrix Copyright© 2016, D-Wave Systems Inc.

Gradient Descent Classically: = = Clamped average • 28 • Unclamped average Copyright© 2016, D-Wave Systems Inc.

Calculating the Gradient cannot be estimated using sampling! ≠ ≠ Clamped average • 29 • Unclamped average Copyright© 2016, D-Wave Systems Inc.

Two Useful Properties of Trace Golden-Thompson inequality: For Hermitian matrices A and B • 30 • Copyright© 2016, D-Wave Systems Inc.

Finding lower bounds Golden-Thompson inequality • 31 • Copyright© 2016, D-Wave Systems Inc.

Finding lower bounds Golden-Thompson inequality Lower bound for log-likelihood • 32 • Copyright© 2016, D-Wave Systems Inc.

Calculating the Gradients Minimize the upper bound ? • 33 • Unclamped average Copyright© 2016, D-Wave Systems Inc.

Clamped Hamiltonian for Infinite energy penalty for states different from v Visible qubits are clamped to their classical values given by the data • 34 • Copyright© 2016, D-Wave Systems Inc.

Estimating the Steps Clamped average Unclamped average We can now use sampling to estimate the steps • 35 • Copyright© 2016, D-Wave Systems Inc.

Training the Transverse Field (Ga) Minimizing the upper bound: Two problems: cannot be estimated from measurements for all visible qubits, thus Gn cannot be trained using the bound • 36 • Copyright© 2016, D-Wave Systems Inc.

Example: 10 -Qubit QBM Graph: fully connected (K 10), fully visible • 37 • Copyright© 2016, D-Wave Systems Inc.

Example: 10 -Qubit QBM Training set: M-modal distribution Random spin orientation Single mode: Multi-mode: • 38 • Hamming distance p = 0. 9 M=8 Copyright© 2016, D-Wave Systems Inc.

Exact Diagonalization Results KL-divergence: Bound gradient D=2 Classical BM Exact gradient (D is trained) D final = 2. 5 • 39 • Copyright© 2016, D-Wave Systems Inc.

Sampling from D-Wave Dickson et al. , Nat. Commun. 4, 1903 (2013) Probabilities cross at the anticrossing • 40 • Copyright© 2016, D-Wave Systems Inc.

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 ar. Xiv: 1601. 02036 • 41 • Copyright© 2016, D-Wave Systems Inc.