LOCKHEED MARTIN SITE UPDATE 11 APRIL 2018 MUNICH, GERMANY Kristen Pudenz Senior Quantum Applications Engineer

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1 LOCKHEED MARTIN SITE UPDATE 11 APRIL 2018 MUNICH, GERMANY Kristen Pudenz Senior Quantum Applications Engineer

2 THE USC-LM QUANTUM COMPUTING CENTER Dr. Edward H. Ned Allen Chief Scientist and LM Senior Fellow Lockheed Martin Corporation Greg Tallant Program Manager and LM Fellow Lockheed Martin Aeronautics Skunk Works University of Southern California Information Sciences Institute Marina de Rey, California, USA

3 QUANTUM APPLICATIONS TEAM Kristen Pudenz, Ph.D. Quantum Applications Quantum Error Correction Steve Adachi, Ph.D. Quantum Machine Learning LM Fellow, Space Systems Chris Elliott, Ph.D. Quantum Enabled V&V 15+ Years of Flight Control Development Julia Kwok, M.E.E. Quantum Graph Coloring Classical Optimization Peter Stanfill 30+ Years of Flight Control Development

4 CENTER HISTORY AND TIMELINE DW-1 Rainier Early LM identified quantum information science as a high potential technology to address critical needs Nov Purchased access time and support for DW-1 Rainier quantum annealer from D-Wave Systems Mar Formed partnership with USC/ISI to host DW-1 and establish collaborative QC Center DW-2 Vesuvius Marina del Rey, CA Jan USC-LM QC Center Operational Mar 2013 Completed upgrade to 512-qubit DW-2 Vesuvius processor Feb 2016 Completed upgrade to qubit DW-2X Washington processor photo credit Fatemeh Kashfi.

5 MACHINE LEARNING: CONNECTION PRUNING EXPERIMENT Question: How much qubit connectivity is needed for Machine Learning? Restricted Boltzmann Machine (RBM) can in principle be a full bipartite graph We know we can achieve good Machine Learning performance using full bipartite RBMs D-Wave Chimera graph corresponds to a Locally Connected Boltzmann Machine (LCBM) Evidence suggests such LCBMs are too sparse for good ML performance Is there some intermediate range of sparsity that is realizable on QA hardware, yet still gives good ML performance? Characterizing Machine Learning application needs for qubit connectivity may help to inform QEO architecture design Start with an already-trained, full bipartite RBM Define a cutoff! and prune connections with " #$ <! Measure accuracy of the pruned RBM on the training/test set Keep increasing C and observe how accuracy deteriorates

6 EXAMPLE DEEP BELIEF NET Coarse-grained MNIST (CG-MNIST) Each training/test image consists of 32 superpixels Modeled using (32,32,32,10) Deep Belief Net Input layer with 32 nodes 2 hidden layers with 32 nodes each Output layer with 10 nodes RBM layers Layer 1 = 32x32 RBM Layer 2 = 32x32 RBM Layer 3 = 32x10 RBM original image coarse-grained image Adachi, S.H., Henderson, M.P. (2015) Application of Quantum Annealing to Training of Deep Neural Networks.

7 EFFECT OF PRUNING CONNECTIONS ON ACCURACY In this example, we can prune over half the layer 1 and 2 connections with minimal decrease in accuracy

8 WHAT DO THE PRUNED NETWORKS LOOK LIKE? Original Pruned

9 CONNECTIVITY EFFECTS Most nodes have degree less than or equal to 16

10 MOVING FORWARD Pruning after the fact is cheating Doesn t tell us before we train, which connections we can do without Suggests however that we can achieve good ML performance without full bipartite connectivity Next step is to try quantum-assisted training of networks that have sparser connectivity from the start Could see a slight tradeoff since sparser RBMs can be embedded on Chimera using shorter-length chains What kind of neural network could be embedded on Pegasus architecture? We need to be careful about the way in which quantum annealers approximate the Boltzmann distribution Temperature estimation Other methods of measuring and controlling distance between gathered samples and ideal distribution

11 GRAPH COLORING We are pursuing a greedy approach based on finding one color s vertex set at a time on the quantum annealer using maximum independent set (MIS) or vertex cover (VC) This allows us to color graphs by embedding an n variable problem Global graph coloring approach requires n*k variables Classical computing closes the loop for multiple MIS or VC cycles to find each color successively Find one independent set this is one color Remove independent set from graph Find next independent set and continue the loop until all vertices are colored We are examining the full algorithm on a test set of parameterized Erdos-Renyi graphs to facilitate comparisons to previous results using the global approach Embedding the graph to be colored takes most of the classical compute time We are investigating methods of adjusting the first embedding to make a compact yet easy to compute embedding for subsequent subgraphs as independent sets are removed

12 COLORING RESULTS For many graphs, we find a k-coloring that is smaller than the one planted at instance generation We do see some k- colorings with larger than planted k We are currently working on an improved algorithm that should have better results with slightly more classical compute time

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