ARNAB BANERJEEArnab Banerjee will join Purdue University, West Lafayette as an Assistant Professor in the Depart-ment of Physics and Astronomy, jointly associated with the Purdue Institute of Quantum Science and Engineering from January 2020. Since 2017, he has been a Scientist in the Quantum Condensed Matter Division of Oak Ridge National Laboratory

in the Neutron Spectroscopy Group, where he was also a post-doctoral fel-low from 2013. Prior to this, he received his Masters and PhD from the Uni-versity of Chicago in 2013, before which he got his Bachelors in Physics at the Indian Institute of Technology in 2006. Arnab Banerjee’s research mostly concentrates on the area of experimental exploration and understanding of quantum materials towards their functionalization. He has received the IOP Reviewer of the Year Award, MRS Postdoctoral Award, and the UT-Battelle Research Accomplishment Award, ORNL Director’s Award, and the Discovery Magazine Top 100 Story of the Year awards.

Quantum Annealing the Ising Shastry-Sutherland HamiltonianThe exactly-solved Ising Shastry-Sutherland Hamiltonian represent a gen-eral class of frustrated square-lattice Hamiltonians which are found in real materials. Variants of this model can produce a complex phase diagram with emergent fractional magnetization plateaus considered the hallmark of this model, and observed in real materials such as the rare-earth tetraborides. We present a novel embedding of the Shastry-Sutherland lattice into the Chimera layout of the D-Wave 2000Q processor. We show that a combina-tion of forward and reverse annealing quickly converges to the correct phase diagram and computes the fractional magnetization plateaus found in the classical limit of zero transverse fields, and lends access to its critical regimes. Furthermore, the results enable us to directly compute and compare the stat-ic structure factor to reciprocal space measurements in real Ising materials, such as the diffuse neutron scattering spectrum in HoB4. The results show that quantum annealing offers an efficient avenue to describe material prop-erties and neutron scattering data. Our work point to the unique prospect to combine quantum annealing with machine learning to seek the magnetic Hamiltonians in materials with naturally occurring defects, which has been an outstanding challenge in the condensed matter community.

Other authors: Paul Kairys, Kelly Boothby, Isil Ozfidan, Jack Raymond, Andrew D. King, Travis Humble; The work is supported by the Department of Energy, Of-fice of Science, Scientific User Facilities Division.

JACOB BIAMONTEDr. Jacob Biamonte is an American theoretical physicist, quantum computer scientist. He is As-sociate Professor and the Lead of Deep Quantum Labs at Skolkovo Institute of Science and Technol-ogy. Biamonte has made several contributions to the theory and implementation of quantum computers.

Biamonte was employed as one of the world’s first quantum software pro-grammers at D-Wave Systems Inc. in Vancouver B.C., Canada. From this work, he published several celebrated proofs establishing the computational uni-versality of specific quantum many-body ground states used in adiabatic quantum computing. He played a central role in developing aspects of con-temporary quantum computing theory, including research recognized as pi-oneering the emerging field that unites quantum information with complex network theory and machine learning.

Biamonte earned an award winning Doctorate at the University of Oxford. His collective research was recognized in 2013 with the Shapiro Award in Math-ematical Physics.

He has advised both government agencies and industry, has led several suc-cessful interdisciplinary research teams and projects, comprised of students and research staff with backgrounds spanning physics, mathematics and computer science. He supervised several extraordinarily talented postdocs, two of which are now Assistant Professors.

Prof. Biamonte serves on the editorial boards of several journals including NJP, he is an invited member of the Foundational Questions Institute, and the 2018 USERN Medal Laureate.

He has authored several celebrated works on the theory and implementation of quantum information processing, including research recognized as pio-neering the emerging field that unites quantum information with complex network theory and machine learning.

Universal Variational Quantum ComputationWe show that the variational approach to quantum enhanced algorithms ad-mits a universal model of quantum computation [1].

Variational quantum algorithms dominate contemporary gate-based quantum enhanced optimization [2], eigenvalue estimation [3] and machine learning [4]. Here we establish the quantum computational universality of variational quantum computation by developing two constructions which prepare states with high 2-norm overlap with the out- puts of quantum circuits. The fleeting resource is the number of expected values which must be iteratively minimized using a classical-to-quantum feed- back loop. The first approach is efficient in the number of expected values for n-qubit circuits containingO(polylnn) non-Clifford gates—the number of expected values has no dependence on

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Clifford gates appearing in the simulated circuit. The second approach adapts the Feynman-Kitaev clock construction yielding ∼ 4·n+9·(T +1) expected val-ues while introducing not more than T slack qubits, for a quantum circuit parti-tioned into T Hermitian blocks (gates). The variational model is hence universal and necessitates (i) state-preparation by a control sequence followed by (ii) measurements in one basis and (iii) gradient-free or gradient-based minimiza-tion of a polynomially bounded number of expected values.

Discussion

An interesting feature of the model of universal variational quantum com-putation is how many-body Hamiltonian terms are realized as part of the measurement process. This is in contrast with leading alternative models of universal quantum computation.

In the gate model, many-body interactions are simulated by sequences of two-body gates, where Hamiltonians are emulated using so called Trotter approximations, which can result in painfully long gate sequences. The adi-abatic model suffers from the large unrealistic gaps found when applying perturbative gadgets to approximate many-body interactions with two-body interactions [6]. The variational model of universal quantum computation does not suffer from either the Trotter overhead blowup nor does it require perturbative methods. Moreover, the coefficients weighting many-body terms need not be implemented into the hardware directly; this weight is compensated for in the classical-iteration process which in turns controls the quantum state being produced.

As the presented model is inherently agnostic to how the states are prepared, this enables experimentalists to now vary the accessible control parameters to minimize an external and iteratively calculated objective function. Though the absolute limitations of this approach in the absence of error correction are not known, a realizable method of error suppression could get us closer to implementation of the traditional textbook quantum algorithms such as Shor’s factorization algorithm.

References

[1] Biamonte Jacob. Universal Variational Quantum Computation. arXiv e-prints, arXiv:1903.04500, March 2019. [2] Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. A Quantum Approximate Optimiza- tion Algorithm. arXiv e-prints, arXiv:1411.4028, November 2014. [3] A. Peruzzo, J. McClean, P. Shad-bolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’Brien. A vari-ational eigenvalue solver on a photonic quantum processor. Nature Communi-cations, 5:4213, July 2014. [4] J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd. Quantum ma- chine learning. Nature, 549:195–202, September 2017. [5] Seth Lloyd. Quantum approximate optimization is com-putationally universal. arXiv e-prints, page arXiv:1812.11075, December 2018. [6] Jacob D. Biamonte and Peter J. Love. Realizable Hamiltonians for universal adiabatic quantum computers. Physical Review A, 78:012352, July 2008. [7] A. Yu. Kitaev, A. H. Shen, and M. N. Vyalyi.Classical and Quantum Computation. American Mathematical Society, Boston, MA, USA, 2002.

YUDONG CAODr. Yudong Cao is a scientist and Chief Technology Officer at Zapata Computing, a quantum comput-ing software company that spun out of Harvard University. Prior to founding Zapata, he was a postdoctoral researcher at Harvard. He obtained his Ph.D. from Purdue University in Computer Sci-ence, where his thesis received the 2016 Chorafas

award in Computer Science and Informatics.

Quantum Boltzmann machine using eigenstate thermalizationAbstract: Quantum Boltzmann machines are natural quantum generaliza-tions of Boltzmann machines that are expected to be more expressive than their classical counterparts, as evidenced both numerically for small systems and asymptotically under various complexity-theoretic assumptions. How-ever, training quantum Boltzmann machines using gradient-based methods requires sampling observables in quantum thermal distributions, a problem that is NP-hard. In this work, we find that the locality of the gradient ob-servables gives rise to an efficient sampling method based on the Eigenstate Thermalization Hypothesis, and thus through Hamiltonian simulation an efficient method for training quantum Boltzmann machines on near-term quantum devices. Furthermore, under realistic assumptions on the moments of the data distribution to be modeled, the distribution sampled using our algorithm is approximately the same as that of an ideal quantum Boltzmann machine. We demonstrate numerically that under the proposed training scheme, quantum Boltzmann machines capture multimodal Bernoulli distri-butions better than classical restricted Boltzmann machines with the same connectivity structure. We also provide numerical results on the robustness of our training scheme with respect to noise.

SUPRIYO DATTADr. Supriyo Datta is known for the work of his group on quantum transport which has been widely adopted in the field of nanoelectronics. It combines the non‐equilibrium Green function (NEGF) formalism of many-body physics with the Landauer formalism from mesoscopic physics, and is described in his books. He is also known for

innovative proposals that have inspired extensive experimental efforts in mo-lecular electronics, negative capacitance devices and spintronics.

p-Bits for Quantum-inspired AlgorithmsDigital computing is based on a deterministic bit with two values, 0 and 1. Quantum computing is based on a q-bit which is a delicate superposition of 0 and 1. This talk draws attention to something in between that could be viewed as a poor man’s q-bit, namely, a p-bit which is a robust classical en-tity fluctuating between 0 and 1, and can be built with existing technology to operate at room temperature1. Specifically I will present (1) experimental results demonstrating that MRAM technology with small modifications can be used to build p-circuits that implement quantum-inspired optimization algorithms2, and (2) SPICE simulations showing that p-circuits can be used to emulate a broad class of quantum Hamiltonians3.

[3] K.Y. Camsari et al. “Stochastic p-bits for Invertible Boolean Logic,” Phys. Rev. X, 3, 031014 (2017); [2] W.A. Borders et al. “Integer Factorization using Stochastic Magnetic Tunnel Junctions,” Nature (in press); [3] K.Y.Camsari et al. “Scalable Emulation of Stoquastic Hamiltonians with Room Temperature p-Bits,” arXiv preprint arXiv:1810.07144.

STEPHEN K. GRAYDr. Stephen K. Gray obtained a BSc. degree in Chemistry from Carleton University, Ottawa, and a Ph.D. degree in Chemistry from the University of California at Berkeley. After post-doctoral studies at Oxford University and the University of Chica-go he was an Assistant Professor of Chemistry at Northern Illinois University before joining the staff

at Argonne National Laboratory. He is currently Senior Scientist and Group Leader of the Theory and Modeling Group of Argonne’s Center for Nanoscale Materials. He has over 250 peer-reviewed papers and is a Fellow of the Amer-ican Physical Society. His current research interests include light-interactions with nanoscale materials and quantum information science.

Supervised Quantum Machine Learning with Photonic QuditsAn approach to supervised quantum machine learning based on qudits formed by frequency combs is discussed. In particular, we discuss how a quantum circuit learning model [Mitarai et al., PRA 98, 032309 (2018)] can be implemented with entangled biphoton states as input, electro-optic mod-ulators and pulse shapers as gate elements, and photon counts as output. We carry out simulations of several classification problems to validate our proposed implementation.

KATHLEEN HAMILTONDr. Kathleen Hamilton is a research staff mem-ber as part of the Quantum Information Science Group at Oak Ridge National Laboratory. She joined ORNL in 2016 and has developed methods and algorithms for emerging technology such quantum annealers, quantum computers and neuromorphic processors while exploring how

these platforms can be incorporated into scientific computing applications. She received her Ph.D. in Physics from the University of California at Riverside and a Master’s degree in Physics from the University of New Hampshire.

Training with gradient estimation on NISQ devices for quantum machine learning applicationsHybrid quantum-classical algorithms that train parameterized quantum cir-cuits are split between a quantum device that implements the circuit and a classical optimizer which is used to update the circuit parameters in order to minimize a given loss function. These approaches can be used to train quan-tum circuits for quantum machine learning applications in which a quantum circuit is trained as a discriminative or generative model. The gradient of a quantum circuit can be exactly evaluated, numerically computed or estimat-ed and this has allowed for the use of stochastic gradient descent and other gradient-based optimization methods in circuit training.

Efficient and scalable training of circuits deployed on current NISQ hardware remains an open question. There are several bottlenecks encountered in gra-dient-based training: accurate evaluation of circuit gradients in the presence of noise and high computational overhead when using large sets of training data. This talk will discuss these challenges in the context of quantum cir-cuit training for generative modeling and supervised learning applications and will present recent results for circuits trained on superconducting qubit platforms.

Funding acknowledgement: This work was jointly supported by the ASCR Testbed Pathfinder Program at Oak Ridge National Laboratory under FWP #ERKJ332 and the ASCR QCAT Program at Oak Ridge National Laboratory under FWP # ERKJ347.

NATHAN KILORANDr. Nathan Kiloran earned a Ph.D. in Physics from the Institute for Quantum Computing at the Univer-sity of Waterloo, and later served as a postdoc in Ma-chine Learning at the University of Toronto. For the past two years, he has worked at the Toronto-based quantum computing startup Xanadu, leading their software and machine learning teams.

PennyLane - Automatic differentiation and machine learning of quantum computationsPennyLane is a Python-based software framework for optimization and ma-chine learning of quantum and hybrid quantum-classical computations. It extends the automatic differentiation algorithms common in machine learn-ing to work with quantum and hybrid computations. The library provides a unified architecture for near-term quantum computing devices, integrating with leading software platforms such as Xanadu’s Strawberry Fields, IBM’s Qiskit, Rigetti’s Forest, Google’s TensorFlow, and Facebook’s PyTorch. Penny-Lane can be used for variational quantum eigensolvers, quantum approxi-mate optimization, quantum machine learning models, and many other applications.

KAROL KOWALSKIDr. Kowalski is one of the coauthors of NWChem, an open-source parallel software package for computational chemistry. He has also contributed to GAMESS quantum chemistry package. He has been working on many-body formulations for electronic structure simulations in the areas of strongly correlated molecular systems, linear re-

sponse, and Green’s function formulations. He is a co-PI on a number of DOE BES partnership projects. He also leads the BES Quantum Information Science effort at Pacific Northwest National Laboratory.

Coupled cluster downfolding techniques for quantum computing: dimensionality reduction of electronic Hamiltonians in studies of correlated molecular systemsMany aspects of the DOE-BES scientific portfolio cannot function without support from predictive many-body models that can provide valid informa-tion relevant to various types of chemical transformations, spectroscopies, and materials properties. Novel and predictive modeling tools capable of overcoming exponential computational barriers of conventional computing are needed to describe chemical transformations that involve quasi-degen-erate electronic states and corresponding potential energy surfaces that are relevant for comprehensive understanding of challenges in the areas of catalysis, actinide chemistry, and nuclear-waste storage, nitrogen fixation, and energy storage materials. In this talk we will outline the extension of recently introduced sub-system embedding sub-algebras coupled cluster (SES-CC) formalism to the unitary CC formalism. In analogy to the standard single-reference SES-CC formalism, In contrast to the standard single-refer-ence SES-CC theory, the unitary CC approach results in a Hermitian form of the effective Hamiltonian. Additionally, for the double unitary CC formalism (DUCC) the corresponding CAS eigenvalue problem provides a rigorous sepa-ration of external cluster amplitudes that describe dynamical correlation ef-fects - used to define the effective Hamiltonian - from those corresponding to the internal (inside the active space) excitations that define the components of eigenvectors associated with the energy of the entire system. The proposed formalism can be viewed as an efficient way of downfolding many-electron Hamiltonian to the low-energy model represented by a particular choice of CAS. The Hermitian character of low-dimensional effective Hamiltonians makes them an ideal target for quantum computing. To illustrate the perfor-mance and feasibility of the DUCC formalism and the level of dimensionality reduction provided by the DUCC approach, we will provide preliminary results obtained with the Quantum Phase Estimator (QPE) and Variational Quantum Eigensolver (VQE) for benchmark molecular systems described by Gaussian and plane-wave basis sets.

RICHARD LIDr. Richard Li is currently a senior computational scientist at Wuxi NextCode Genomics. He finished his Ph.D. in May 2019, under the supervision of Daniel Lidar at University of Southern California, studying applications of quantum annealing to machine learning. He is interested in understand-ing and applying new computational technologies.

Case studies in machine learning via quantum annealingRecently there has been significant interest in applying quantum computers to machine learning tasks by speeding up certain tasks or increasing the mod-eling power. Many of these approaches rely on circuit-model architectures of quantum computing, which currently are limited in terms of the number of qubits. On the other hand, physical devices that perform quantum annealing are approaching a size that can be used with real-world problems, but such devices can be noisy and their computational power is less well understood. In this talk, we present several cases studies using quantum annealers made by D-Wave, applying them to simplified, but real-world machine learning tasks in biology: the classification and ranking of transcription factor binding to DNA, and the classification of cancer based on genomic features. We iden-tify cases where quantum annealing has a relative advantage compared to standard classical approaches and make comparisons to classical annealing approaches.

ANTONIO MEZZACAPODr. Antonio Mezzacapo is currently a research staff member in the Quantum computing theory group at IBM T. J. Watson Research Center. His main fo-cuses are the development of new quantum algo-rithms targeting interacting many body systems, and modeling of quantum devices. He has worked on the design of the Qiskit Aqua and Qiskit chem-

istry software packages for quantum computing.

Quantum computing and artificial intelligenceQuantum computing and artificial intelligence are new angles of looking at the way we can compute. In this talk, I will discuss topics at the intersection of quantum computing and artificial intelligence, elucidating how these two areas of research can benefit each other and presenting some concrete examples.

MASOUD MOHSENIDr. Masoud Mohseni is a Senior Research Scientist at Google Quantum Artificial Intelligence Labora-tory, where he develops quantum machine learn-ing and physics-inspired optimization algorithms. A former research scientist and a principal investi-gator at the Research Laboratory of Electronics at MIT, Dr. Mohseni has been a scientific consultant

at leading industry initiatives on unconventional computing. He obtained his Ph.D in physics in 2007 from University of Toronto. He then moved to Harvard University for a postdoctoral program in quantum simulation. He has made seminal contributions to the theory of quantum transport and open quantum systems. Dr. Mohseni’s current research addresses some of the fundamental problems at the interface of artificial intelligence, quantum computing, and physics of complex systems.

Challenges and opportunities for hybrid quantum-classical machine learning and optimizationWe present an overview of our progress on quantum-assisted algorithms for optimization and machine learning at Quantum AI Lab at Google. We devel-op an end-to end discrete optimization platform that uses an interplay of thermal and quantum fluctuations to sample from inaccessible low-energy states of spin-glass systems that encode high-quality solutions of certain hard combinatorial optimization and probabilistic inference problems. We introduce several new techniques for quantum circuit learning on Noisy In-termediate-Scale Quantum (NISQ) processors. We show how one can learn to learn on parameterized quantum circuits via classical recurrent neural networks. We perform layer-wise learning on quantum neural networks to tackle the problem of vanishing gradients. We provide several applications of such quantum models for characterization of the dynamics of NISQ devices and classification of quantum data.

BARRY SANDERSDr. Barry Sanders is Director of the Institute for Quantum Science and Technology at the Univer-sity of Calgary, Lead Investigator of the Alberta Major Innovation Fund Project on Quantum Tech-nologies, a Thousand Talents Chair at the Univer-sity of Science and Technology China and a Vajra Visiting Faculty member of the Raman Research

Institute in India. He received his Bachelor of Science degree from the Uni-versity of Calgary in 1984 and a Diploma of Imperial College supervised by Professor Sir Thomas W. B. Kibble in 1985. He completed a Ph.D. in 1987 at Imperial College London supervised by Professor Sir Peter Knight. His post-doctoral research was at the Australian National University, the University of Queensland and the University of Waikato. Dr. Sanders was on the Macquarie University faculty from 1991 until moving to Calgary in 2003.

Dr. Sanders is especially well known for seminal contributions to theories of quantum-limited measurement, highly nonclassical light, practical quantum cryptography and optical implementations of quantum information tasks. His current research interests include quantum algorithms and implementa-tions of quantum information tasks.

Dr. Sanders is a Fellow of the Institute of Physics (U.K.), the Optical Society of America, the Australian Institute of Physics, the American Physical Society and the Royal Society of Canada, and a Senior Fellow of the Canadian Insti-tute for Advanced Research. He is a past President of the Australian Optical Society, past Founding Co-Chair of the Canadian Association of Physicists Division of Atomic, Molecular and Optical Physics and former Leader of the Optical Society of America Quantum Optical Science and Technology Techni-cal Group. In 2016, Sanders was awarded the Imperial College London Doctor of Science (DSc) degree.

Dr. Sanders is Editor-in-Chief of New Journal of Physics, a former Associate Editor of Physical Review A, a former Editor of Optics Communications and a former Editor of Mathematical Structures of Computer Science.

Machine learning for quantum controlWe develop a framework that connects reinforcement learning with classical and quantum control, and this framework yields adaptive quantum-control policies that beat the standard quantum limit, inspires new methods for improving quantum-gate design for quantum computing, and suggest new ways to apply classical and quantum machine learning to control.

ROLANDO SOMMADr. Rolando Somma is a scientist working at the forefront of quantum information at Los Alamos National Laboratory (LANL). Rolando received his Ph.D. from the Instituto Balseiro, Argentina, and was a postdoctoral researcher at LANL and Perim-eter Institute for Theoretical Physics, in Waterloo, Canada. Dr. Somma has coauthored over 50 pub-

lications in peer-reviewed journals that are highly cited, and has given over 60 invited seminars in top institutions and conferences. He is the recipient of several awards, including the 2006 LANL’s Postdoctoral Distinguished Perfor-mance Award and the 2015 Google Faculty Research Award. His pioneering work on quantum information has been reported in the media many times.

Quantum Algorithms for Systems of Linear EquationsHarrow, Hassidim, and Lloyd developed a quantum algorithm (HHL) to prepare a quantum state proportional to the solution of a system of linear equations. Their result is another demonstration of the power of quantum computing. In this talk, I will describe improved quantum algorithms for the linear system’s problem, with a significant reduction in complexity and other resources. One such quantum algorithm improves the complexity de-pendence of the HHL algorithm exponentially in the precision. The other quantum algorithm is based on quantum adiabatic evolutions and does not use complex subroutines like phase estimation or variable time amplitude amplification. It is quadratically faster than the HHL algorithm in terms of the condition number, it is only based on Hamiltonian simulation, and requires at most one or two ancillary qubits, making it attractive for near-term im-plementations.

KRISTAN TEMMEDr. Kristan Temme is a Research Staff Member in the Theory of Quantum Computation and Infor-mation group at IBM’s T.J. Watson Research Center. His research currently centers around quantum al-gorithms and noise in complex quantum systems with a focus on the applications of noisy interme-diate scale quantum devices.

Kristan has received a diploma in Physics from the University of Heidelberg in 2007. He completed his Ph.D in Physics at the University of Vienna in 2011 un-der the supervision of Frank Verstraete. During this time, he studied quantum Markov processes and complex quantum many-body systems. Between 2012 and 2014, he was a Erwin Schroedinger postdoctoral fellow at the Massachu-setts Institute of Technology in the group of Edward Farhi. Between March 2014 and September 2015, Kristan held the IQIM postdoctoral Fellowship of the Gordon and Betty Moore Foundation at the California Institute of Technol-ogy, in the group of John Preskill. He joined IBM research in the fall of 2015.

Quantum machine learning and the prospect of near-term applications on noisy devices.Prospective near-term applications of early quantum devices rely on accurate estimates of expectation values to become relevant. Decoherence and gate errors lead to wrong estimates. This problem was, at least in theory, remedied with the advent of quantum error correction. However, the overhead that is needed to implement a fully fault-tolerant gate set with current codes and current devices seems prohibitively large. In turn, steady progress is made in improving the quality of the quantum hardware, which leads to the believe that in the foreseeable future machines could be build that cannot be emu-lated by a conventional computer. In light of recent progress mitigating the effect of decoherence on expectation values, it becomes interesting to ask what these noisy devices can be used for. In this talk, we will present our ad-vances in finding quantum machine learning applications for noisy quantum computers.

NATHAN WIEBEDr. Nathan Wiebe is a researcher in quantum computing who focuses on quantum methods for machine learning and simulation of physical systems. His work has provided the first quan-tum algorithms for deep learning, least squares fitting, quantum simulations using linear-com-binations of unitaries, quantum Hamiltonian

learning, near-optimal simulation of time-dependent physical systems, efficient Bayesian phase estimation and also has pioneered the use of par-ticle filters for characterizing quantum devices, as well as many other con-tributions ranging from the foundations of thermodynamics to adiabatic quantum computing and quantum chemistry simulation. He received his Ph.D. in 2011 from the university of Calgary, studying quantum computing before accepting a post-doctoral fellowship at the University of Waterloo and then finally joining Microsoft Research in 2013. In 2019 he joined the faculty of the University of Washington and Pacific Northwest National Labs as a senior scientist.

Training Quantum Boltzmann MachinesWe provide a method for fully quantum generative training of quantum Boltzmann machines with both visible and hidden units while using quan-tum relative entropy as an objective. This is significant because prior methods were not able to do so due to mathematical challenges posed by the gradient evaluation. We present two novel methods for solving this problem. The first proposal addresses it, for a class of restricted quantum Boltzmann machines with mutually commuting Hamiltonians on the hidden units, by using a vari-ational upper bound on the quantum relative entropy. The second one uses high-order divided difference methods and linear-combinations of unitaries to approximate the exact gradient of the relative entropy for a generic quan-tum Boltzmann machine. Both methods are efficient under the assumption that Gibbs state preparation is efficient and that the Hamiltonian are given by a sparse row-computable matrix.

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