Read the updated D-Wave 2000Q Technology Overview.

## Technology Information

### 2017

**D-Wave 2000Q Technology Overview**

**D-Wave Overview**

A brief introduction to D-Wave and quantum computing. Read the D-Wave Overview

### 2016

**IDC: Quantum Computing in the Real World**

IDC recently published a Technology Spotlight on quantum computing. Download the report here.

## D-Wave White Papers

### 2017

**Optimization with Clause Problems**

We introduce a new input class called clause problems, that can be used to study local constraint structures, which occur in inputs translated from general NP-hard problems to the D-Wave native topology. We describe a small family of clause problems that are contrived to create significant challenges for two classical competition solvers, simulated annealing (SA) and the Hamze–de Frietas–Selby solver (HFS). We identify key properties of these inputs that lead to poor performance by the classical solvers, and consider whether these properties might naturally arise in problems from real-world applications.

**Computational Power Consumption and Speedup**

Power consumption for computation is a serious and growing issue for the world. We rely more and more on computing in everything we do as we try to satisfy our ever-increasing thirst for mobile computing, automation, machine intelligence, cloud computing, and increasingly powerful supercomputers. Highly specialized coprocessors such as D-Wave’s quantum processing units (QPUs) show promise in significantly increasing the power effciency of computing. In a recent study, D-Wave’s 2000-qubit system was shown to be up to 100 times more energy effcient than highly specialized algorithms on state-of-the-art classical computing servers when considering pure computation time, suggesting immediate relevance to large-scale energy efficient computing.

**Quantum Annealing amid Local Ruggedness and Global Frustration**

We introduce a problem class with two attributes crucial to the evaluation of quantum annealing processors: local ruggedness (i.e., tall, thin energy barriers in the energy landscape) so that quantum tunneling can be harnessed as a useful resource, and global frustration so that the problems are combinatorially challenging and representative of real-world inputs. We evaluate the new 2000-qubit D-Wave quantum processing unit (QPU) on these inputs, comparing it to software solvers that include both GPU-based solvers and a CPU-based solver which is highly tailored to the D-Wave topology. The D-Wave QPU solidly outperforms the software solvers: when we consider pure annealing time, the D-Wave QPU is three to four orders of magnitude faster than software solvers in both optimization and sampling evaluations.

**Boosting integer factoring performance via quantum annealing offsets**

D-Wave quantum computing systems now allow a user to advance or delay the annealing path of individual qubits through the anneal offsets feature. Here we demonstrate the potential of this feature by using it in an integer factoring circuit. Offsets allow the user to homogenize dynamics of various computational elements in the circuit. This gives a remarkable improvement over baseline performance, in some cases making the computation more than 1000 times faster.

**Limits on Parallel Speedup for Classical Ising Model Solvers**

Why can’t we put together a million cores and make it run a million times faster? Parallel computing systems offer enormous potential for significant runtime speedups over computation by a single CPU core. However, many computational tasks cannot be effciently parallelized. We explore some practical limits to achieving parallel speedups, with reference to some classical optimization solvers that are competitors to D-Wave quantum computers.

**Partitioning Optimization Problems for Hybrid Classical/Quantum Execution**

In this white paper we introduce qbsolv, a tool that solves large quadratic unconstrained binary optimization (QUBO) problems by partitioning into subproblems targeted for execution on a D-Wave system. Using a classical subproblem solver rather than quantum annealing, qbsolv delivers state-of-the-art numerical results and executes almost twice as fast as the best previously known implementation. We have released qbsolv as open-source software to foster greater use and experimentation in such partitioning solvers and to establish the QUBO form as a target for higher-level optimization interfaces. The software can be acccessed on GitHub at github.com/dwavesystems/qbsolv.

### 2013

**Programming with D-Wave: Map Coloring Problem**

Quantum computing, as implemented in the D-Wave system, is described by a simple but largely unfamiliar programming model. Using a simple map coloring problem this white paper describes the entire set of transformations needed to nd solutions by executing a single quantum machine instruction (QMI) within this programming model. This “direct embedding” is one of several ways to program the D-Wave quantum computer.

## D-Wave Publications

### 2016

**Benchmarking Quantum Hardware for Training of Fully Visible Boltzmann Machines**

“Quantum annealing (QA) is a hardware-based heuristic optimization and sampling method applicable to discrete undirected graphical models. While similar to simulated annealing, QA relies on quantum, rather than thermal, effects to explore complex search spaces. For many classes of problems, QA is known to offer computational advantages over simulated annealing. Here we report on the ability of recent QA hardware to accelerate training of fully visible Boltzmann machines.”

(14 Nov 2016) https://arxiv.org/abs/1611.04528

**Discrete Variational Autoencoders**

“Probabilistic models with discrete latent variables naturally capture datasets composed of discrete classes. However, they are difficult to train efficiently, since back propagation through discrete variables is generally not possible. We introduce a novel class of probabilistic models, comprising an undirected discrete component and a directed hierarchical continuous component, that can be trained efficiently using the variational autoencoder framework. The discrete component captures the distribution over the disconnected smooth manifolds induced by the continuous component. As a result, this class of models efficiently learns both the class of objects in an image, and their specific realization in pixels, from unsupervised data; and outperforms state-of-the-art methods on the permutation-invariant MNIST, OMNIGLOT, and Caltech-101 Silhouettes datasets.”

(7 Sep 2016) http://arxiv.org/abs/1609.02200

**Global warming: Temperature estimation in annealers**

Jack Raymond, Sheir Yarkoni, Evgeny Andriyash

"Sampling from a Boltzmann distribution is NP-hard and so requires heuristic approaches. Quantum annealing is one promising candidate. The failure of annealing dynamics to equilibrate on practical time scales is a well understood limitation, but does not always prevent a heuristically useful distribution from being generated. In this paper we evaluate several methods for determining a useful operational temperature range for annealers.

(2 Jun 2016) Download PDF.

**Mapping constrained optimization problems to quantum annealing with application to fault diagnosis**

Zhengbing Bian, Fabian Chudak, Robert Israel, Brad Lackey, William G. Macready, Aidan Roy

Current quantum annealing (QA) hardware suffers from practical limitations such as finite temperature, sparse connectivity, small qubit numbers, and control error. We propose new algorithms for mapping boolean constraint satisfaction problems (CSPs) onto QA hardware mitigating these limitations. In particular we develop a new embedding algorithm for mapping a CSP onto a hardware Ising model with a fixed sparse set of interactions, and propose two new decomposition algorithms for solving problems too large to map directly into hardware.

(10 Mar 2016) http://arxiv.org/abs/1603.03111

**Quantum Boltzmann Machine**

Mohammad H. Amin, Evgeny Andriyash, Jason Rolfe, Bohdan Kulchytskyy, Roger Melko

Inspired by the success of Boltzmann Machines based on classical Boltzmann distribution, we propose a new machine learning approach based on quantum Boltzmann distribution of a transverse-field Ising Hamiltonian. Due to the non-commutative nature of quantum mechanics, the training process of the Quantum Boltzmann Machine (QBM) can become nontrivial. We circumvent the problem by introducing bounds on the quantum probabilities. This allows us to train the QBM efficiently by sampling. We show examples of QBM training with and without the bound, using exact diagonalization, and compare the results with classical Boltzmann training. We also discuss the possibility of using quantum annealing processors like D-Wave for QBM training and application.

(8 Jan 2016) http://arxiv.org/abs/1601.02036

### 2015

**Degeneracy, degree, and heavy tails in quantum annealing**

Andrew D. King, Emile Hoskinson, Trevor Lanting, Evgeny Andriyash, Mohammad H. Amin

"Both simulated quantum annealing and physical quantum annealing have shown the emergence of "heavy tails" in their performance as optimizers: The total time needed to solve a set of random input instances is dominated by a small number of very hard instances…On similar inputs designed to suppress local degeneracy, performance of a quantum annealing processor on hard instances improves by orders of magnitude at the 512-qubit scale, while classical performance remains relatively unchanged."

(23 Dec 2015) Link to PDF.

**Fast clique minor generation in Chimera qubit connectivity graphs**

Tomas Boothby, Andrew D. King, Aidan Roy

(27 Oct 2015) http://link.springer.com/article/10.1007/s11128-015-1150-6?wt_mc=internal.event.1.SEM.ArticleAuthorOnlineFirst

**Constructing SAT Filters with a Quantum Annealer**

Adam Douglass, Andrew D. King, Jack Raymond

"Presented here is a case study of SAT filter construction with a focus on constraint satisfaction problems based on MAX-CUT clauses (Not-all-equal 3-SAT, 2-in-4-SAT, etc.) and frustrated cycles in the Ising model... Solutions are sampled using a D-Wave quantum annealer, and results are measured against classical approaches."

( 27 Oct 2015) http://link.springer.com/chapter/10.1007%2F978-3-319-24318-4_9

**Performance of a quantum annealer on range-limited constraint satisfaction problems**

A.D.King, T.Lanting, and R.Harris

(3 Sep 2015) http://arxiv.org/pdf/1502.02098.pdf

**Benchmarking a quantum annealing processor with the time-to-target metric**

In this paper we introduce the Time to Target (TTT) metric and compare the performance of the D-Wave 2X on a host of native hardware problems against highly optimized and tuned solvers.

(20 Aug, 2015) Link to PDF

**Searching for quantum speedup in quasistatic quantum annealers**

Mohammad H. Amin

(13 Mar 2015) http://arxiv.org/pdf/1503.04216.pdf

**Computational Role of Collective Tunneling in a Quantum Annealer**

Sergio Boixo et al.

(19 Feb 2015) http://arxiv.org/pdf/1411.4036.pdf

### 2014

**Discrete optimization using quantum annealing on sparse Ising models**

Frontiers in Physics, (18 Sep 2014) http://journal.frontiersin.org/Journal/10.3389/fphy.2014.00056/abstract

**A practical heuristic for finding graph minors**

Jun Cai, Bill Macready, Aidan Roy

(12 Jun 2014) http://arxiv.org/pdf/1406.2741.pdf

**Entanglement in a quantum annealing processor**

T. Lanting et. al

In this paper we present experimental evidence that, during a critical portion of QA, qubits in the D-Wave processor become entangled and entanglement persists even as these systems reach equilibrium with a thermal environment. Our results provide an encouraging sign that quantum annealing is a viable technology for large-scale quantum computing.

Physical Review X (29 May 2014) https://journals.aps.org/prx/abstract/10.1103/PhysRevX.4.021041

**Architectural considerations in the design of a superconducting quantum annealing processor**

P. Bunyk et al.

Pre-print (21 Jan 2014) http://arxiv.org/pdf/1401.5504v1

### 2013

**Evidence for temperature dependent spin-diffusion as a mechanism of intrinsic flux noise in SQUIDs**

T. Lanting et al.

(23 Dec 2013) http://arxiv.org/pdf/1306.1512.pdf

**Thermally assisted quantum annealing of a 16-qubit problem**

N G Dickson et al.

"Efforts to develop useful quantum computers have been blocked primarily by environmental noiseHere we examine the environment’s effect on quantum annealing using 16 qubits of a superconducting quantum processor. For a problem instance with an isolated small-gap anticrossing between the lowest two energy levels, we experimentally demonstrate that, even with annealing times eight orders of magnitude longer than the predicted single-qubit decoherence time, the probabilities of performing a successful computation are similar to those expected for a fully coherent system. Moreover, for the problem studied, we show that quantum annealing can take advantage of a thermal environment to achieve a speedup factor of up to 1,000 over a closed system."

Nature Communications, 1903 (May 21 2013) http://www.nature.com/ncomms/journal/v4/n5/full/ncomms2920.html

**Adiabatic quantum optimization with qudits**

M.H. Amin et al.

Quant. Inf. Proc. 12, 1819-1829 (April 2013) doi:10.1007/s11128-012-0480-x/QuantInfProc12/

**Tunneling spectroscopy using a probe qubit**

A. J. Berkley et al.

Phys. Rev. B 87, 020502(R) (2013) doi:10.1103/PhysRevB.87.020502

### 2012

**Algorithmic approach to adiabatic quantum optimization**

N. G. Dickson et al.

Phys. Rev. A 85, 032303 (2012) doi:10.1103/PhysRevA.85.032303arXiv:1108.33031

**Approximate diagonalization method for large-scale Hamiltonians**

M. H. Amin et al.

Phys. Rev. A 86, 052314 (2012) doi:10.1103/PhysRevA.86.052314

### 2011

**Probing high-frequency noise with macroscopic resonant tunneling**

T. Lanting et al.

Physical Review B PhysRevB.83.180502 arXiv:1103.1931

**Quantum annealing with manufactured spins**

M. W. Johnson et al.

Many interesting but practically intractable problems can be reduced to that of finding the ground state of a system of interacting spins; however, finding such a ground state remains computationally difficult..Here we use quantum annealing to find the ground state of an artificial Ising spin system comprising an array of eight superconducting flux quantum bits with programmable spin–spin couplings. We observe a clear signature of quantum annealing, distinguishable from classical thermal annealing through the temperature dependence of the time at which the system dynamics freezes.

Nature 473, 194-198 (12 May 2011) http://www.nature.com/nature/journal/v473/n7346/full/nature10012.html

**Importance of Explicit Vectorization for CPU and GPU Software Performance**

N. Dickson et al.

Journal of Computational Physics arXiv:1004.0024

**The Ising model: teaching an old problem new tricks**

Z. Bian et al.

Link to PDF

**Investigating the Performance of an Adiabatic Quantum Optimization Processor**

K. Karimi et al.

Quantum Information Processing arXiv:1006.4147

**Does adiabatic quantum optimization fail for NP-complete problems?**

N. G. Dickson et al.

Phys. Rev. Lett. 106, Issue 5, 050502 arXiv:1010.0669

### 2010

**A scalable control system for a superconducting adiabatic quantum optimization processor**

M. W. Johnson et al.

Supercond. Sci. Technol. 23, 065004 arXiv:0907.3757

**Experimental Demonstration of a Robust and Scalable Flux Qubit**

R. Harris et al.

Physical Review B 81, 134510 (2010) arXiv:0909.4321

**High-Performance Physics Simulations Using Multi-Core CPUs and GPGPUs in a Volunteer Computing Context**

K. Karimi et al.

International Journal of High Performance Computing Applications, doi: 10.1177/1094342010372928arXiv:1004.0023

**Robust Parameter Selection for Parallel Tempering**

F. Hamze et al

International Journal of Modern Physics C, Volume 21, issue 5 (2010) arXiv:1004.2840

**Experimental Investigation of an Eight Qubit Unit Cell in a Superconducting Optimization Processor**

R. Harris et al.

Phys. Rev. B 82, 024511 (2010) arXiv:1004.1628

**Cotunneling in pairs of coupled flux qubits**

T. Lanting et al.

Phys. Rev. B 82, 060512(R) (2010) arXiv:1006.0028

**A scalable readout system for a superconducting adiabatic quantum optimization system**

A. J. Berkley et al.

Supercond. Sci. Technol. 23, 105014 (2010) arXiv:0905.0891

### 2009

**Non-Markovian incoherent quantum dynamics of a two-state system**

M. H. S. Amin et al.

Phys. Rev. B 80, 214302 (2009) arXiv:0907.4797

**Decoherence in adiabatic quantum computation**

Phys. Rev. A 79, 022107 (2009) arXiv:0708.0384

**A Compound Josephson Junction Coupler for Flux Qubits With Minimal Crosstalk**

Phys. Rev. B 80, 052506 (2009) arXiv:0904.3784

**Landau-Zener transitions in a superconducting flux qubit**

Phys. Rev. B 80, 012507 (2009) arXiv:0807.0797

**Geometrical dependence of the low-frequency noise in superconducting flux qubits**

**Consistency of the Adiabatic Theorem**

Phys. Rev. Lett. 102, 220401 (2009) arXiv:0810.4335

**Landau-Zener transitions in the presence of spin environment**

A. T. S. Wan et al.

Int. J. Quant. Inf. 7, 725 (2009) arXiv:cond-mat/0703085

**First Order Quantum Phase Transition in Adiabatic Quantum Computation**

M. H. S. Amin et al.

Phys. Rev. A 80, 062326 (2009) arXiv:0904.1387

**The Role of Single Qubit Decoherence Time in Adiabatic Quantum Computation**

Phys. Rev. A 80, 022303 (2009) arXiv:0803.1196

### 2008

**Minor-Embedding in Adiabatic Quantum Computation: I. The Parameter Setting Problem**

**Macroscopic Resonant Tunneling in the Presence of Low Frequency Noise**

Phys. Rev. Lett. 100, 197001 (2008) arXiv:0712.0845

**On the construction of model Hamiltonians for adiabatic quantum computing and its application to finding low energy conformations of lattice protein models**

Physical Review A 78, 012320 (2008) arXiv:0801.3625

**Effect of Local Minima on Adiabatic Quantum Optimization**

Phys. Rev. Lett. 100, 130503 (2008) arXiv:0709.0528

**Thermally Assisted Adiabatic Quantum Computation**

Phys. Rev. Lett. 100, 060503 (2008) arXiv:cond-mat/0609332

**Probing Noise in Flux Qubits via Macroscopic Resonant Tunneling**

Phys. Rev. Lett. 101, 117003 (2008) arXiv:0712.0838

**Realizable Hamiltonians for Universal Adiabatic Quantum Computers**

Phys. Rev. A 78, 012352 (2008) aXiv:0704.1287

### 2007

**Sign- and Magnitude-Tunable Coupler for Superconducting Flux Qubits**

Phys. Rev. Lett. 98, 177001 (2007) arXiv:cond-mat/0608253

**A Characterization of global entanglement**

Quant. Info. Proc. 6, 187 (2007) arXiv:quant-ph/0602143

### 2006

**Rabi oscillations in systems with small anharmonicity**

Low Temp. Phys. 32, 198 (2006) arXiv:cond-mat/0407080

**Four-Qubit Device with Mixed Couplings**

Phys. Rev. Lett. 96, 047006 (2006) arXiv:cond-mat/0509557

**Adiabatic quantum computation with flux qubits, first experimental results**

IEEE Trans. App. Supercond. 17, 113 (2006) arXiv:cond-mat/0702580

### 2005

**Simulated Quantum Computation of Molecular Energies**

Science 309 p. 1704, (2005) arXiv:quant-ph/0604193

**Hamiltonian for coupled flux qubits**

Phys. Rev. B, 71, 064503 (2005) arXiv:cond-mat/0310425

**Quantum nondemolition charge measurement of a Josephson qubit**

Phys. Rev. B 71, 140505 (2005) arXiv:cond-mat/0412286

**Silent phase qubit based on d -wave Josephson junctions**

Phys. Rev. B 71, 064516 (2005) arXiv:cond-mat/0310224

**Flux qubit in charge-phase regime**

Phys. Rev. B 71, 024504 (2005) arXiv:cond-mat/0311220

**Mediated tunable coupling of flux qubits**

New J. Phys. 7 230 (2005) arXiv:cond-mat/0501148

**Direct Josephson coupling between superconducting flux qubits**

Phys. Rev. B 72, (2005) 020503(R) arXiv:cond-mat/0501085

### 2004

**Evidence for Entangled States of Two Coupled Flux Qubits**

Phys. Rev. Lett. 93, 037003 (2004) arXiv:cond-mat/0312332

**Low-frequency measurement of the tunneling amplitude in a flux qubit**

Phys. Rev. B 69, 060501 (2004) arXiv:cond-mat/0303657

**Quasiparticle Decoherence in d-Wave Superconducting Qubits**

Phys. Rev. Lett. 92, 017001 (2004) arXiv:cond-mat/0304255

**Observation of macroscopic Landau-Zener tunneling in a superconducting device**

Euro. Phys. Lett. 65, 844, (2004) arXiv:cond-mat/0307506

**Wigner distribution function formalism for superconductors and collisionless dynamics of the superconducting order parameter**

Low Temp. Phys. 30, 661 (2004) arXiv:cond-mat/0404401

**Superconducting quantum storage and processing**

IEEE International Solid State Circuit Conference (ISSCC), Tech. Dig., p296(2004)

### 2003

**Anomalous current-phase relation as basis for HTS qubit**

Proceedings of the European Conference on Applied Superconductivity (EUCAS 2003)

**Nonequilibrium quasiclassical theory for Josephson structures**

Phys. Rev. B 68, 054505 (2003) arXiv:cond-mat/0207724

**Josephson-phase qubit without tunneling**

Phys. Rev. B 67, 100508 (2003) arXiv:cond-mat/0211638

**Dynamical Effects of an Unconventional Current-Phase Relation in YBCO dc SQUIDs**

Phys. Rev. Lett. 90, 117002 (2003) arXiv:cond-mat/0303144

**Quasiclassical Calculations of spontaneous current in restricted geometries**

"Towards the Controllable Quantum States" edited by H. Takayanagi and J. Nitta, World Scientific Publishing Co. (2003), arXiv:cond-mat/0207617

**Continuous Monitoring of Rabi Oscillations in a Josephson Flux Qubit**

Phys. Rev. Lett. 91, 097906 (2003) arXiv:cond-mat/0303433

**Theory of weak continuous measurements in a strongly driven quantum bit**

Phys. Rev. B 68, 134514 (2003) arXiv:cond-mat/0306004

**Tunable coupling of superconducting qubits**

Phys. Rev. Lett. 90, 127901 (2003) arXiv:cond-mat/0207112

### 2002

**Multi-Terminal Superconducting Phase Qubit**

Physica C 368, 310 (2002) arXiv:cond-mat/0109382

**High Temperature PI/2-SQUID**

IEEE Tran. Appl. Supercond. 12, 1877 (2002) arXiv:cond-mat/0107370

**Time reversal breaking states and spontaneous current pattern in Josephson junctions of d-wave superconductors**

"New Trends in Superconductivity", edited by J.F. Annett and S. Kruchinin, Kluwer, Academic Publishers (2002).

**Quasiclassical theory of spontaneous currents at surfaces and interfaces of d-Wave superconductors**

Physica B 318, 162 (2002) arXiv:cond-mat/0105486

**Low-frequency characterization of quantum tunneling in flux qubits**

Phys. Rev. B 66, 214525 (2002) arXiv:cond-mat/0208076

**d+is versus d+id time reversal symmetry breaking states in finite size systems**

Phys. Rev. B 66, 174515 (2002) arXiv:cond-mat/0205495

**DC-SQUID based on the mesoscopic multi-terminal Josephson junction**

Physica C 372-376P1, 184 (2002); Special issue: Proceeding of the 5th European Conference on Applied Superconductivity, Copenhagen, Denmark, (Sep. 2001) arXiv:cond-mat/0109384

### 2001

**Mesoscopic multi-terminal Josephson structures. I. effects of nonlocal weak coupling**

Low Temp Phys. 27, 616 (2001) arXiv:cond-mat/0109333

**Degenerate Ground State in a Mesoscopic YBa2Cu3O7-x Grain Boundary Josephson Junction**

Phys. Rev. Lett. 86, 5369 (2001) arXiv:cond-mat/0102404

**Mechanisms of spontaneous current generation in an inhomogeneous d-wave superconductor**

Phys. Rev. B 63, 212502 (2001) arXiv:cond-mat/0011416

## Third Party Publications

### 2017

**Graph Partitioning using Quantum Annealing on the D-Wave System**

Hayato Ushijima-Mwesigwa, Christian F. A. Negre, and Susan M. Mniszewski

"In this work, we explore graph partitioning (GP) using quantum annealing on the D-Wave 2X machine. Motivated by a recently proposed graph-based electronic structure theory applied to quantum molecular dynamics (QMD) simulations, graph partitioning is used for reducing the calculation of the density matrix into smaller subsystems rendering the calculation more computationally efficient...Results for graph partitioning using quantum and hybrid classical-quantum approaches are shown to equal or out-perform current “state of the art” methods. "

(4 May 2017) https://arxiv.org/abs/1705.03082v1

**Nonnegative/binary matrix factorization with a D-Wave quantum annealer**

Daniel O’Malley, Velimir V. Vesselinov, Boian S. Alexandrov, and Ludmil B. Alexandrov (Los Alamos National Laboratory)

"D-Wave quantum annealers represent a novel computational architecture and have attracted significant interest, but have been used for few real-world computations. Machine learning has been identified as an area where quantum annealing may be useful. Here, we show that the D-Wave 2X can be effectively used as part of an unsupervised machine learning method. This method can be used to analyze large datasets. The D-Wave only limits the number of features that can be extracted from the dataset. We apply this method to learn the features from a set of facial images."

(5 April 2017) https://arxiv.org/abs/1704.01605

**A Study of Complex Deep Learning Networks on High Performance, Neuromorphic, and Quantum Computers**

Oak Ridge National Laboratory: Thomas E. Potok, Catherine Schuman, Steven R. Young, Robert M. Patton; USC Information Sciences Institute: Federico Spedalieri, Jeremy Liu, Ke-Thia Yao; University of Tennessee: Garrett Rose, Gangotree Chakma

"Current Deep Learning approaches have been very successful using convolutional neural networks (CNN) trained on large graphical processing units (GPU)-based computers. Three limitations of this approach are: 1) they are based on a simple layered network topology, i.e., highly connected layers, without intra-layer connections; 2) the networks are manually configured to achieve optimal results, and 3) the implementation of neuron model is expensive in both cost and power. In this paper, we evaluate deep learning models using three different computing architectures to address these problems: quantum computing to train complex topologies, high performance computing (HPC) to automatically determine network topology, and neuromorphic computing for a low-power hardware implementation. We use the MNIST dataset for our experiment, due to input size limitations of current quantum computers. Our results show the feasibility of using the three architectures in tandem to address the above deep learning limitations. We show a quantum computer can find high quality values of intra-layer connections weights, in a tractable time as the complexity of the network increases; a high performance computer can find optimal layer-based topologies; and a neuromorphic computer can represent the complex topology and weights derived from the other architectures in low power memristive hardware."

(15 Mar 2017) https://arxiv.org/abs/1703.05364

### 2016

**Not Magic…Quantum**

**Los Alamos National Laboratory, 1663 Magazine, July 2016**

"Quantum computers have long been on the horizon as conventional computing technologies approach their physical limits. While general-purpose quantum computers remain on the horizon for the time being, a special kind of quantum computer already exists and could be a game changer for simulation and computing tools in support of Los Alamos National Laboratory’s mission of stockpile stewardship without nuclear testing. It may also enable a slew of broader national security and computer science applications. But first, it will undoubtedly draw a vibrant community of top creative thinkers in many scientific fields to Los Alamos."

**Spanning Tree Calculations on D-Wave 2 Machines**

M.A. Novotny, L. Hobl, J.S. Hall, and K. Michielsen (Department of Physics and Astronomy, and Center for Computational Sciences, Mississippi State University and Institute for Advanced Simulation, Ju ̈lich Supercomputing Centre)

"Calculations on D-Wave machines are presented, both for the 500-qubit and the 1000-qubit machines. Results are presented for spanning trees on the available K4,4 Chimera graphs of both machines. Comparing trees of approximately the same size, the frequency of finding the ground state for the 1000-qubit machine is significantly improved over the 500- qubit older generation machine."

(Feb 2016) Journal of Physics Conference Series http://iopscience.iop.org/article/10.1088/1742-6596/681/1/012005/meta

### 2015

**What is the Computational Value of Finite Range Tunneling?**

Vasil S. Denchev, Sergio Boixo, Sergei V. Isakov, Nan Ding, Ryan Babbush, Vadim Smelyanskiy, John Martinis, Hartmut Neven (Google scientists)

"Quantum annealing (QA) has been proposed as a quantum enhanced optimization heuristic exploiting tunneling. Here, we demonstrate how finite range tunneling can provide considerable computational advantage. For a crafted problem designed to have tall and narrow energy barriers separating local minima, the D-Wave 2X quantum annealer achieves significant runtime advantages relative to Simulated Annealing (SA). For instances with 945 variables this results in a time-to-99%-success-probability that is ∼10^{8} times faster than SA running on a single processor core. "

(30 Dec 2015) http://arxiv.org/abs/1512.02206

**Multiple Query Optimization on the D-Wave 2X Adiabatic Quantum Computer**

Immanuel Trummer and Christoph Koch, (E ́cole Polytechnique Federale de Lausanne scientists)

"In this paper, we tackle the problem of multiple query optimization (MQO)...While the problem sizes that can be treated are currently limited, we already find a class of problem instances where the quantum annealer is three orders of magnitude faster than other approaches."

(23 Oct 2015) http://arxiv.org/pdf/1510.06437v1.pdf

**Application of quantum annealing to training of deep neural networks**

Steven H. Adachi, Maxwell P. Henderson (Lockheed Martin scientists)

"We investigated an alternative approach [to Deep Learning] that estimates model expectations of Restricted Boltzmann Machines using samples from a D-Wave quantum annealing machine...In our tests we found that the quantum sampling-based training approach achieves comparable or better accuracy with significantly fewer iterations of generative training than conventional CD-based training."

(Oct 2015) Mathpubs: http://www.mathpubs.com/detail/1510.06356v1/Application-of-Quantum-Annea...

**Solving the Optimal Trading Trajectory Problem Using a Quantum Annealer**

Gili Rosenberg, Poya Haghnegahdar, Phil Goddard, Peter Carr, Kesheng Wu, Marcos López de Prado

“We solve a multi-period portfolio optimization problem using D-Wave Systems' quantum annealer. We derive a formulation of the problem, discuss several possible integer encoding schemes, and present numerical examples that show high success rates.”

(22 Aug 2015) http://arxiv.org/abs/1508.06182

**Guest Column: Adiabatic Quantum Computing Challenges**

ACM SIGACT News archive, Volume 46 Issue 1, March 2015, Pages 40-61 http://dl.acm.org/citation.cfm?id=2744459&dl=ACM&coll=DL&CFID=666501900&CFTOKEN=29005022

"The paper presents a brief introduction to quantum computing with focus on the adiabatic model which is illustrated with the commercial D-Wave computer. We also include new theory and experimental work done on the D-Wave computer. Finally we discuss a hybrid method of combining classical and quantum computing and a few open problems."

**Benchmarking Adiabatic Quantum Optimization for Complex Network Analysis**

Ojas Parekh, Jeremy Wendt, Luke Shulenburger, Andrew Landahl, Jonathan Moussa, John Aidun (Sandia National Laboratories scientists)

"We lay the foundation for a benchmarking methodology for assessing current and future quantum computers. We pose and begin addressing fundamental questions about how to fairly compare computational devices at vastly different stages of technological maturity. We critically evaluate and offer our own contributions to current quantum benchmarking efforts, in particular those involving adiabatic quantum computation and the Adiabatic Quantum Optimizers produced by D-Wave Systems, Inc. "

(Apr 2015) Link to PDF

**Computational Role of Multiqubit Tunneling in a Quantum Annealer**

Sergio Boixo, Vadim N. Smelyanskiy, Alireza Shabani, Sergei V. Isakov, Mark Dykman, Vasil S. Denchev, Mohammad Amin, Anatoly Smirnov, Masoud Mohseni, Hartmut Neven (Scientists from Google, NASA Ames, and D-Wave)

Quantum tunneling, a phenomenon in which a quantum state traverses energy barriers above the energy of the state itself, has been hypothesized as an advantageous physical resource for optimization. This paper demonstrates that multiqubit tunneling plays a computational role in the D-Wave processor.

(Feb 20 2015) Nature Communications http://www.nature.com/ncomms/2016/160107/ncomms10327/full/ncomms10327.html

### 2014

**First application of quantum annealing to IMRT beamlet intensity optimization**

Daryl P Nazareth and Jason D Spaans (Roswell Park Cancer Institute)

"Optimization methods are critical to radiation therapy. A new technology, quantum annealing (QA), employs novel hardware and software techniques to address various discrete optimization problems in many fields. We report on the first application of quantum annealing to the process of beamlet intensity optimization for IMRT...This initial experiment suggests that more research into QA-based heuristics may offer significant speedup over conventional clinical optimization methods, as quantum annealing hardware scales to larger sizes."

(1 May 2015) Institute of Physics and Engineering in Medicine http://iopscience.iop.org/article/10.1088/0031-9155/60/10/4137/pdf

**Reexamining classical and quantum models for the D-Wave One processor**

(12 Sep 2014) http://arxiv.org/abs/1409.3827

**Quantum annealing correction for random Ising problems**

(19 Aug 2014) http://arxiv.org/abs/1408.4382

**A Quantum Annealing Approach for Fault Detection and Diagnosis of Graph-Based Systems**

(30 Jun 2014) http://arxiv.org/abs/1406.7601

**Quantum Optimization of Fully-Connected Spin Glasses**

(29 Jun 2014) http://arxiv.org/abs/1406.7553

**Distinguishing Classical and Quantum Models for the D-Wave Device**

Walter Vinci, Tameem Albash, Anurag Mishra, Paul A. Warburton, Daniel A. Lidar

(17 Mar 2014) http://arxiv.org/abs/1403.4228

**Glassy Chimeras could be blind to quantum speedup: Designing better benchmarks for quantum annealing machines**

Helmut G. Katzgraber, Firas Hamze, Ruben S. Andrist

(12 Jan 2014) http://arxiv.org/pdf/1401.1546.pdf

### 2013

**Experimental determination of Ramsey numbers**

Z. Bian et al.

Phys. Rev. Lett. vol. 111, 130505 (2013) arXiv:1201.1842

**Error corrected quantum annealing with hundreds of qubits**

K.L. Pudenz et al.

(31 Jul 2013) arXiv:1307.8190

**Hearing the shape of Ising models: on the distinguishability power of Physics**

W. Vinci et al.

(3 Jul 2013) arXiv:1307.1114

**Experimental signature of programmable quantum annealing**

S Boxio et al.

Nature Communications, 2067 (28 June 2013) doi:10.1038/ncomms3067

**MAX 2-SAT with up to 108 qubits**

S. Santra

(12 Jul 2013) arXiv:1307.3931

**Quantum annealing with more than one hundred qubits**

S. Boxio et al.

(16 Apr 2013) arXiv:1304.4595

**How Fast Can Quantum Annealers Count?**

I. Hen

(21 Jan 2013) arXiv:1301.4956

**Experimental Evaluation of an Adiabatic Quantum System for Combinatorial Optimization**

C. C. McGeoch et al.

Download PDF

### 2012

**Construction of Energy Functions for Lattice Heteropolymer Models: A Case Study in Constraint Satisfaction Programming and Adiabatic Quantum Optimization**

R. Babbush et al.

(4 Nov 2012) arXiv:1211.3422

**Solving the Graph Isomorphism Problem with a Quantum Annealer**

I. Hen et al.

(6 Jul 2012) arXiv:1207.1712

**Robust Classification with Adiabatic Quantum Optimization**

V.S. Denchev et al.

(5 May 2012) arXiv:1205.1148

**Finding low-energy conformations of lattice protein models by quantum annealing**

A. Perdomo-Ortiz et al.

(24 Apr 2012) arXiv:1204.5485

**A Near-Term Quantum Computing Approach for Hard Computational Problems in Space Exploration**

V.N. Smelyanskiy et al.

(12 Apr 2012) arXiv:1204.2821

**Quantum Speedup by Quantum Annealing**

D. Nagaj et al.

Phys. Rev. Lett. 109, 050501 (2012) arXiv:1202.6257

### 2009

**Training a Large Scale Classifier with the Quantum Adiabatic Algorithm**

H. Neven, et al.

(4 Dec 2009) arXiv:0912.0779

### 2008

**Training a Binary Classifier with the Quantum Adiabatic Algorithm**

H. Neven et al.

(4 Nov 2008) arXiv:0811.0416

**Image recognition with an adiabatic quantum computer I. Mapping to quadratic unconstrained binary optimization**

H. Neven et al.

(28 Apr 2008) arXiv:0804.4457