## A Quantum-Inspired Method for Three-Dimensional Ligand-Based Virtual Screening

By Maritza Hernandez, Guo Liang Gan, Kirby Linvill, Carl Dukatz, Jun Feng, & Govinda Bhisetti

Measuring similarity between molecules is an important part of virtual screening (VS) experiments deployed during the early stages of drug discovery. Most widely used methods for evaluating the similarity of molecules use molecular fingerprints to encode structural information. While similarity methods using fingerprint encodings are efficient, they do not consider all the relevant aspects of molecular structure. In this paper, we describe a quantum-inspired graph-based molecular similarity (GMS) method for ligand-based VS. The GMS method is formulated as a quadratic unconstrained binary optimization problem that can be solved using a quantum annealer, providing the opportunity to take advantage of this nascent and potentially groundbreaking technology. In this study, we consider various features relevant to ligand-based VS, such as pharmacophore features and three-dimensional atomic coordinates, and include them in the GMS method. We evaluate this approach on various datasets from the DUD_LIB_VS_1.0 library. Our results show that using three-dimensional atomic coordinates as features for comparison yields higher early enrichment values. In addition, we evaluate the performance of the GMS method against conventional fingerprint approaches. The results demonstrate that the GMS method outperforms fingerprint methods for most of the datasets, presenting a new alternative in ligand-based VS with the potential for future enhancement.

## Multi-Community Detection in Signed Graphs Using Quantum Hardware

Signed graphs serve as a primary tool for modelling social networks. They can represent relationships between individuals (i.e., nodes) with the use of signed edges. Finding communities in a signed graph is of great importance in many areas, for example, targeted advertisement. We propose an algorithm to detect multiple communities in a signed graph. Our method reduces the multi-community detection problem to a quadratic binary unconstrained optimization problem and uses state-of-the-art quantum or classical optimizers to find an optimal assignment of each individual to a specific community.

## A Variable Neighbourhood Descent Heuristic for Conformational Search Using a Quantum Annealer

By Dominic Marchand, Moslem Noori, Austin Roberts, Gili Rosenberg, Brad Woods, Ugur Yildiz, Marc Coons, David Devore, & Peter Margl

Discovering the low-energy conformations of a molecule is of great interest to computational chemists, with applications in in silico materials design and drug discovery. In this paper, we propose a variable neighbourhood search heuristic for the conformational search problem. Using the structure of a molecule, neighbourhoods are chosen to allow for efficient optimization using a binary quadratic optimizer. The method is flexible with respect to the choice of molecular force field and the number of discretization levels in the search space, and can be further generalized to take advantage of higher-order binary polynomial optimizers. It is well-suited for the use of devices such as quantum annealers. After carefully defining neighbourhoods, the method easily adapts to the size and topology of these devices, allowing for seamless scaling alongside their future improvements.

## VanQver: The Variational and Adiabatically Navigated Quantum Eigensolver

By Shunji Matsuura, Takeshi Yamazaki, Valentin Senicourt, & Arman Zaribafiyan

The accelerated progress in manufacturing noisy intermediate-scale quantum (NISQ) computing hardware has opened the possibility of exploring its application in transforming approaches to solving computationally challenging problems. The important limitations common among all NISQ computing technologies are the absence of error correction and the short coherence time, which limit the computational power of these systems. Shortening the required time of a single run of a quantum algorithm is essential for reducing environment-induced errors and for the efficiency of the computation. We have investigated the ability of a variational version of adiabatic quantum computation (AQC) to generate an accurate state more efficiently compared to existing adiabatic methods. The standard AQC method uses a time-dependent Hamiltonian, connecting the initial Hamiltonian with the final Hamiltonian. In the current approach, a navigator Hamiltonian is introduced which has a finite amplitude only in the middle of the annealing process. Both the initial and navigator Hamiltonians are determined using variational methods. The cluster operator of coupled-cluster theory, truncated to single and double excitations, is used as a navigator Hamiltonian. A comparative study of our variational algorithm (VanQver) with that of standard AQC, starting with a Hartree–Fock Hamiltonian, is presented. The results indicate that the introduction of the navigator Hamiltonian significantly improves the annealing time required to achieve chemical accuracy by two to three orders of magnitude. The efficiency of the method is demonstrated in the ground-state energy estimation of molecular systems, namely, H2, P4, and LiH.

## Smooth Structured Prediction Using Quantum and Classical Gibbs Samplers

By Behrooz Sepehry, Ehsan Iranmanesh, Michael P. Friedlander, & Pooya Ronagh

We introduce a quantum algorithm for solving structured prediction problems with a runtime that scales with the square root of the size of the label space, but scales in $$\widetilde O\left(\epsilon^{-3.5}\right)$$ with respect to the precision, $$\epsilon$$, of the solution. In doing so, we analyze a stochastic gradient algorithm for convex optimization in the presence of an additive error in the calculation of the gradients, and show that its convergence rate does not deteriorate if the additive errors are of the order $$O(\sqrt\epsilon)$$. Our algorithm uses quantum Gibbs sampling at temperature $$\Omega (\epsilon)$$ as a subroutine. Based on these theoretical observations, we propose a method for using quantum Gibbs samplers to combine feedforward neural networks with probabilistic graphical models for quantum machine learning. Numerical results using Monte Carlo simulations on an image tagging task demonstrate the benefit of the approach.

## Physics-Inspired Optimization for Quadratic Unconstrained Problems Using a Digital Annealer

By Maliheh Aramon, Gili Rosenberg, Elisabetta Valiante, Toshiyuki Miyazawa, Hirotaka Tamura, & Helmut G. Katzgraber

The Fujitsu Digital Annealer (DA) is designed to solve fully connected quadratic unconstrained binary optimization (QUBO) problems. It is implemented on application-specific CMOS hardware and currently solves problems of up to 1024 variables. The DA’s algorithm is currently based on simulated annealing; however, it differs from it in its utilization of an efficient parallel-trial scheme and a dynamic escape mechanism. In addition, the DA exploits the massive parallelization that custom application-specific CMOS hardware allows. We compare the performance of the DA to simulated annealing and parallel tempering with isoenergetic cluster moves on two-dimensional and fully connected spin-glass problems with bimodal and Gaussian couplings. These represent the respective limits of sparse versus dense problems, as well as high-degeneracy versus low-degeneracy problems. Our results show that the DA currently exhibits a time-to-solution speedup of roughly two orders of magnitude for fully connected spin-glass problems with bimodal or Gaussian couplings, over the single-core implementations of simulated annealing and parallel tempering Monte Carlo used in this study. The DA does not appear to exhibit a speedup for sparse two-dimensional spin-glass problems, which we explain on theoretical grounds. We also benchmarked an early implementation of the Parallel Tempering DA. Our results suggest an improved scaling over the other algorithms for fully connected problems of average difficulty with bimodal disorder. The next generation of the DA is expected to be able to solve fully connected problems up to 8192 variables in size. This would enable the study of fundamental physics problems and industrial applications that were previously inaccessible using standard computing hardware or special-purpose quantum annealing machines.

## Towards the Practical Application of Near-Term Quantum Computers in Quantum Chemistry Simulations: A Problem Decomposition Approach

By Takeshi Yamazaki, Shunji Matsuura, Ali Narimani, Anushervon Saidmuradov, & Arman Zaribafiyan

With the aim of establishing a framework to efficiently perform the practical application of quantum chemistry simulation on near-term quantum devices, we envision a hybrid quantum–classical framework for leveraging problem decomposition (PD) techniques in quantum chemistry. Specifically, we use PD techniques to decompose a target molecular system into smaller subsystems requiring fewer computational resources. In our framework, there are two levels of hybridization. At the first level, we use a classical algorithm to decompose a target molecule into subsystems, and utilize a quantum algorithm to simulate the quantum nature of the subsystems. The second level is in the quantum algorithm. We consider the quantum–classical variational algorithm that iterates between an expectation estimation using a quantum device and a parameter optimization using a classical device. We investigate three popular PD techniques for our hybrid approach: the fragment molecular-orbital (FMO) method, the divide-and-conquer (DC) technique, and the density matrix embedding theory (DMET). We examine the efficacy of these techniques in correctly differentiating conformations of simple alkane molecules. In particular, we consider the ratio between the number of qubits for PD and that of the full system; the mean absolute deviation; and the Pearson correlation coefficient and Spearman’s rank correlation coefficient. Sampling error is introduced when expectation values are measured on the quantum device. Therefore, we study how this error affects the predictive performance of PD techniques. The present study is our first step to opening up the possibility of using quantum chemistry simulations at a scale close to the size of molecules relevant to industry on near-term quantum hardware.

## Combinatorial Optimization on Gate Model Quantum Computers: A Survey

By Ehsan Zahedinejad & Arman Zaribafiyan

The advent of quantum computing processors with the possibility to scale beyond experimental capacities magnifies the importance of studying their applications. Combinatorial optimization problems can be one of the promising applications of these new devices. These problems are recurrent in industrial applications and they are in general difficult for classical computing hardware. In this work, we provide a survey of the approaches to solving different types of combinatorial optimization problems, in particular quadratic unconstrained binary optimization (QUBO) problems on a gate model quantum computer. We focus mainly on four different approaches including digitizing adiabatic quantum computing, global quantum optimization algorithms, quantum algorithms that approximate the ground state of a general QUBO problem, and quantum sampling. We also discuss the quantum algorithms that are custom designed to solve certain types of QUBO problems.

## Combinatorial Optimization by Decomposition on Hybrid CPU–non-CPU Solver Architectures

By Ali Narimani, Seyed Saeed Rezaei, & Arman Zaribafiyan

The advent of special-purpose hardware such as FPGA- or ASIC-based annealers and quantum processors has shown potential in solving certain families of complex combinatorial optimization problems more efficiently than conventional CPUs. We show that to address an industrial optimization problem, a hybrid architecture of CPUs and non-CPU devices is inevitable. In this paper, we propose problem decomposition as an effective method for designing a hybrid CPU–non-CPU optimization problem solver. We introduce the required algorithmic elements for making problem decomposition a viable approach in meeting real-world constraints such as communication time and the potential higher cost of using non-CPU hardware. We then turn to the well-known maximum clique problem, and propose a new decomposition method for this problem. Our method enables us to solve the maximum clique problem on very large graphs using non-CPU hardware that is considerably smaller in size than the graph. As an example, we demonstrate that the maximum clique problem on the com-Amazon graph, with 334,863 vertices and 925,872 edges, can be solved using a single problem embedding on fully connected hardware with at least 21 nodes, such as the D-Wave 2000Q. We also show that our proposed problem decomposition approach can improve the runtime of two of the best-known classical algorithms for large, sparse graphs, namely PMC and BBMCSP, by orders of magnitude. In the light of our study, we believe that even new non-CPU hardware that is small in size could become competitive with CPUs if it could be either mass produced and highly parallelized, or able to provide high-quality solutions to specific problems significantly faster than CPUs.

## Effective Optimization Using Sample Persistence: A Case Study on Quantum Annealers and Various Monte Carlo Optimization Methods

By Hamed Karimi, Gili Rosenberg, & Helmut G. Katzgraber

We present and apply a general-purpose, multi-start algorithm for improving the performance of low-energy samplers used for solving optimization problems. The algorithm iteratively fixes the value of a large portion of the variables to values that have a high probability of being optimal. The resulting problems are smaller and less connected, and samplers tend to give better low-energy samples for these problems. The algorithm is trivially parallelizable, since each start in the multi-start algorithm is independent, and could be applied to any heuristic solver that can be run multiple times to give a sample. We present results for several classes of hard problems solved using simulated annealing, path-integral quantum Monte Carlo, parallel tempering with isoenergetic cluster moves, and a quantum annealer, and show that the success metrics as well as the scaling are improved substantially. When combined with this algorithm, the quantum annealer’s scaling was substantially improved for native Chimera graph problems. In addition, with this algorithm the scaling of the time to solution of the quantum annealer is comparable to the Hamze–de Freitas–Selby algorithm on the weak-strong cluster problems introduced by Boixo et al. Parallel tempering with isoenergetic cluster moves was able to consistently solve 3D spin glass problems with 8000 variables when combined with our method, whereas without our method it could not solve any.

## Practical Integer-to-Binary Mapping for Quantum Annealers

By Sahar Karimi & Pooya Ronagh

Recent advancements in quantum annealing hardware and numerous studies in this area suggests that quantum annealers have the potential to be effective in solving unconstrained binary quadratic programming problems. Naturally, one may desire to expand the application domain of these machines to problems with general discrete variables. In this paper, we explore the possibility of employing quantum annealers to solve unconstrained quadratic programming problems over a bounded integer domain. We present an approach for encoding integer variables into binary ones, thereby representing unconstrained integer quadratic programming problems as unconstrained binary quadratic programming problems. To respect some of the limitations of the currently developed quantum annealers, we propose an integer encoding, named bounded-coefficient encoding, in which we limit the size of the coefficients that appear in the encoding. Furthermore, we propose an algorithm for finding the upper bound on the coefficients of the encoding using the precision of the machine and the coefficients of the original integer problem. Finally, we experimentally show that this approach is far more resilient to the noise of the quantum annealers compared to traditional approaches for the encoding of integers in base two.

## Free-Energy-based Reinforcement Learning Using a Quantum Processor

By Anna Levit, Daniel Crawford, Navid Ghadermarzy, Jaspreet S. Oberoi,

Recent theoretical and experimental results suggest the possibility of using current and near-future quantum hardware in challenging sampling tasks. In this paper, we introduce free-energy-based reinforcement learning (FERL) as an application of quantum hardware. We propose a method for processing a quantum annealer’s measured qubit spin configurations in approximating the free energy of a quantum Boltzmann machine (QBM). We then apply this method to perform reinforcement learning on the grid-world problem using the D-Wave 2000Q quantum annealer. The experimental results show that our technique is a promising method for harnessing the power of quantum sampling in reinforcement learning tasks.

## Reinforcement Learning Using Quantum Boltzmann Machines

By Daniel Crawford, Anna Levit, Navid Ghadermarzy, Jaspreet S. Oberoi, & Pooya Ronagh

We investigate whether quantum annealers with select chip layouts can outperform classical computers in reinforcement learning tasks. We associate a transverse fi eld Ising spin Hamiltonian with a layout of qubits similar to that of a deep Boltzmann machine (DBM) and use simulated quantum annealing (SQA) to numerically simulate quantum sampling from this system. We design a reinforcement learning algorithm in which the set of visible nodes representing the states and actions of an optimal policy are the fi rst and last layers of the deep network. In absence of a transverse field, our simulations show that DBMs train more effectively than restricted Boltzmann machines (RBM) with the same number of weights. Since sampling from Boltzmann distributions of a DBM is not classically feasible, this is evidence of advantage of a non-Turing sampling oracle. We then develop a framework for training the network as a quantum Boltzmann machine (QBM) in the presence of a signifi cant transverse field for reinforcement learning. This further improves the reinforcement learning method using DBMs.

## A Subgradient Approach for Constrained Binary Optimization via Quantum Adiabatic Evolution

By Sahar Karimi & Pooya Ronagh

An earlier work proposes a method for solving the Lagrangian dual of a constrained binary quadratic programming problem via quantum adiabatic evolution using an outer approximation method. This should be an efficient prescription for solving the Lagrangian dual problem in the presence of an ideally noise-free quantum adiabatic system. However, current implementations of quantum annealing systems demand methods that are efficient at handling possible sources of noise. In this paper, we consider a subgradient method for finding an optimal primal-dual pair for the Lagrangian dual of a constrained binary polynomial programming problem. We then study the quadratic stable set (QSS) problem as a case study. We see that this method applied to the QSS problem can be viewed as an instance-dependent penalty-term approach that avoids large penalty coefficients. Finally, we report our experimental results of using the D-Wave 2X quantum annealer and conclude that our approach helps this quantum processor to succeed more often in solving these problems compared to the usual penalty-term approaches.

## Enhancing Quantum Annealing Performance for the Molecular Similarity Problem

By Maritza Hernandez & Maliheh Aramon

Quantum annealing is a promising technique which leverages quantum mechanics to solve hard optimization problems. Considerable progress has been made in the development of a physical quantum annealer, motivating the study of methods to enhance the efficiency of such a solver. In this work, we present a quantum annealing approach to measure similarity among molecular structures. Implementing real-world problems on a quantum annealer is challenging due to hardware limitations such as sparse connectivity, intrinsic control error, and limited precision. In order to overcome the limited connectivity, a problem must be reformulated using minor-embedding techniques. Using a real data set, we investigate the performance of a quantum annealer in solving the molecular similarity problem. We provide experimental evidence that common practices for embedding can be replaced by new alternatives which mitigate some of the hardware limitations and enhance its performance. Common practices for embedding include minimizing either the number of qubits or the chain length, and determining the strength of ferromagnetic couplers empirically. We show that current criteria for selecting an embedding do not improve the hardware’s performance for the molecular similarity problem. Furthermore, we use a theoretical approach to determine the strength of ferromagnetic couplers. Such an approach removes the computational burden of the current empirical approaches, and also results in hardware solutions that can benefit from simple local classical improvement. Although our results are limited to the problems considered here, they can be generalized to guide future benchmarking studies.

## Boosting Quantum Annealer Performance via Quantum Persistence

By Hamed Karimi & Gili Rosenberg

We propose a novel method for reducing the number of variables in quadratic unconstrained binary optimization problems, using a quantum annealer to fix the value of a large portion of the variables to values that have a high probability of being optimal. The resulting problems are usually much easier for the quantum annealer to solve, due to their being smaller and consisting of disconnected components. This approach significantly increases the success rate and number of observations of the best known energy value in samples obtained from the quantum annealer, when compared with calling the quantum annealer without using it, even when using fewer annealing cycles. Use of the method results in a considerable improvement in success metrics even for problems with high-precision couplers and biases, which are more challenging for the quantum annealer to solve. The results are further enhanced by applying the method iteratively and combining it with classical pre-processing. We present results for both Chimera graph-structured problems and embedded problems from a real-world application.

## Prime Factorization Using Quantum Annealing and Algebraic Geometry

By Raouf Dridi & Hedayat Alghassi

We investigate prime factorization from two perspectives: quantum annealing and computational algebraic geometry, specifically Gröbner bases. We present a novel autonomous algorithm which combines the two approaches and leads to the factorization of all bi-primes up to just over 200 000, the largest number factored to date using a quantum processor. We also explain how Gröbner bases can be used to reduce the degree of Hamiltonians.

## Systematic and Deterministic Graph-Minor Embedding for Cartesian Products of Graphs

By Arman Zaribafiyan, Dominic Marchand, & S. Saeed C. Rezaei

The limited connectivity of current and next-generation quantum annealers motivates the need for efficient graph-minor embedding methods. These methods allow non-native problems to be adapted to the target annealer’s architecture. The overhead of the widely used heuristic techniques is quickly proving to be a significant bottleneck for solving real-world applications. To alleviate this difficulty, we propose a systematic and deterministic embedding method, exploiting the structures of both the input graph of the specific problem and the quantum annealer. We focus on the specific case of the Cartesian product of two complete graphs, a regular structure that occurs in many problems. We divide the embedding problem by first embedding one of the factors of the Cartesian product in a repeatable pattern. The resulting simplified problem consists of the placement and connecting together of these copies to reach a valid solution. Aside from the obvious advantage of a systematic and deterministic approach with respect to speed and efficiency, the embeddings produced are easily scaled for larger processors and show desirable properties for the number of qubits used and the chain length distribution. To conclude, we briefly address the problem of circumventing inoperable qubits by presenting possible extensions of the method.

## A Novel Graph-based Approach for Determining Molecular Similarity

By Maritza Hernandez, Arman Zaribafiyan, Maliheh Aramon, & Mohammad Naghibi

In this paper, we tackle the problem of measuring similarity among graphs that represent real objects with noisy data. To account for noise, we relax the definition of similarity using the maximum weighted co-k-plex relaxation method, which allows dissimilarities among graphs up to a predetermined level. We then formulate the problem as a novel quadratic unconstrained binary optimization problem that can be solved by a quantum annealer. The context of our study is molecular similarity where the presence of noise might be due to regular errors in measuring molecular features. We develop a similarity measure and use it to predict the mutagenicity of a molecule. Our results indicate that the relaxed similarity measure, designed to accommodate the regular errors, yields a higher prediction accuracy than the measure that ignores the noise.

## Homology Computation of Large Point Clouds Using Quantum Annealing

By Raouf Dridi & Hedayat Alghassi

In this paper we present a quantum annealing algorithm for computation of homology of large point clouds. It is designed to work on resizable cloud computing platforms. The algorithm takes as input a witness complex K approximating the given point cloud. It uses quantum annealing to compute a clique cover of the 1-skeleton of K and then uses this cover to blow up K into a Mayer-Vietoris complex. Finally, it computes the homology of the Mayer-Vietoris complex in parallel. The novelty here is the use of clique covering which, compared to graph partitioning, substantially reduces the computational load assigned to each slave machine in the computing cluster. We describe two different clique covers and their quantum annealing formulation in the form of quadratic unconstrained binary optimization (QUBO).

By Pooya Ronagh, Brad Woods, & Ehsan Iranmanesh

Quantum adiabatic evolution is perceived as useful for binary quadratic programming problems that are a priori unconstrained. For constrained problems, it is a common practice to relax linear equality constraints as penalty terms in the objective function. However, there has not yet been proposed a method for efficiently dealing with inequality constraints using the quantum adiabatic approach. In this paper, we give a method for solving the Lagrangian dual of a binary quadratic programming (BQP) problem in the presence of inequality constraints and employ this procedure within a branch-and-bound framework for constrained BQP (CBQP) problems.

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

By 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. The formulation incorporates transaction costs (including permanent and temporary market impact), and, significantly, the solution does not require the inversion of a covariance matrix. The discrete multi-period portfolio optimization problem we solve is significantly harder than the continuous variable problem. We present insight into how results may be improved using suitable software enhancements, and why current quantum annealing technology limits the size of problem that can be successfully solved today. The formulation presented is specifically designed to be scalable, with the expectation that as quantum annealing technology improves, larger problems will be solvable using the same techniques.

## Building an Iterative Heuristic Solver for a Quantum Annealer

A quantum annealer heuristically minimizes quadratic unconstrained binary optimization (QUBO) problems, but is limited by the physical hardware in the size and density of the problems it can handle. We have developed a meta-heuristic solver that utilizes D-Wave Systems’ quantum annealer (or any other QUBO problem optimizer) to solve larger or denser problems, by iteratively solving subproblems, while keeping the rest of the variables fixed. We present our algorithm, several variants, and the results for the optimization of standard QUBO problem instances from OR-Library of sizes 500 and 2500 as well as the Palubeckis instances of sizes 3000 to 7000. For practical use of the solver, we show the dependence of the time to best solution on the desired gap to the best known solution. In addition, we study the dependence of the gap and the time to best solution on the size of the problems solved by the underlying optimizer.

## Quantum Annealing Implementation of Job-Shop Scheduling

By Davide Venturelli, Dominic Marchand, & Galo Rojo

A quantum annealing solver for the renowned job-shop scheduling problem (JSP) is presented in detail. After formulating the problem as a time-indexed quadratic unconstrained binary optimization problem, several pre-processing and graph embedding strategies are employed to compile optimally parametrized families of the JSP for scheduling instances of up to six jobs and six machines on the D-Wave Systems Vesuvius processor. Problem simplifications and partitioning algorithms, including variable pruning and running strategies that consider tailored binary searches, are discussed and the results from the processor are compared against state-of-the-art global-optimum solvers.