Portfolio Optimisation Using the D-Wave Quantum Annealer
PPortfolio Optimisation Using the D-WaveQuantum Annealer
Frank Phillipson and Harshil Singh Bhatia TNO, the Netherlands Organisation for Applied ScientificResearch, The Hague, The Netherlands Department of Computer Science and Engineering, IndianInstitute of Technology, Jodhpur, IndiaDecember 3, 2020
Abstract
The first quantum computers are expected to perform well at quadraticoptimisation problems. In this paper a quadratic problem in finance istaken, the Portfolio Optimisation problem. Here, a set of assets is chosenfor investment, such that the total risk is minimised, a minimum return isrealised and a budget constraint is met. This problem is solved for severalinstances in two main indices, the Nikkei225 and the S&P500 index, usingthe state-of-the-art implementation of D-Wave’s quantum annealer and itshybrid solvers. The results are benchmarked against conventional, state-of-the-art, commercially available tooling. Results show that for problemsof the size of the used instances, the D-Wave solution, in its current, stilllimited size, comes already close to the performance of commercial solvers.
Keywords—
Quantum Portfolio Optimisation, Quadratic unconstrained binaryOptimisation, Quantum annealing, Genetic algorithm
Portfolio management is the problem of selecting assets (bonds, stocks, commodities)or projects in an optimal way. Classical portfolio management, as introduced byMarkowitz, focuses on efficient (expected) mean-variance combinations [1], and hasled to a broad spectrum of optimisation problems: single and multi-objective [2, 3],single and multi-period [4, 5], without and with [6, 5] transaction costs, deterministicor stochastic [7, 8] in all possible combinations. One of the basic problems is the singleobjective, maximising expected return, under budget and risk constraints. Risk isexpressed by the covariance matrix of all assets. In this paper we consider a variantminimising the risk, under budget and return constraints. This is applied in familytrust and pension funds, where a specific return is needed for future liabilities and alow risk is desirable. This leads to a quadratic optimisation problem with continuous a r X i v : . [ q -f i n . P M ] N ov r binary variables. The variables are continuous if a fraction of the budget is allocatedto an asset. Binary variables can be found when the choice is whether or not to investin a specific asset or project.Solving this kind of problems is not trivial. Integer quadratic programming (IQP)problems are NP-hard, the decision version of IQP is NP-complete [9]. Binary quadraticprogramming problems are also NP-hard in general [10], however, specific cases arepolynomially solvable [11]. Classical solvers such as CPLEX, Gurobi, Localsolver,are still getting better in solving these problems for bigger instances. Next to thesesolvers, heuristic approaches exist, based on meta-heuristics like Particle Swarms, Ge-netic Algorithms, Ant Colony and Simulated Annealing. A recent overview of theseapproaches for Portfolio Optimisation can be found in [12].Quadratic optimisation problems with binary decision variables are expected tobe a sweet-spot for near future quantum computing [13], using Quantum Annealing[14] or the Quantum Approximate Optimisation Algorithm (QAOA) on a gate modelquantum computer [15]. Quantum computing is the technique of using quantum me-chanical phenomena such as superposition, entanglement and interference for doingcomputational operations. The type of devices which are capable of doing such quan-tum operations are still being actively developed and are called quantum computers.We distinguish between two paradigms of quantum computing devices: gate-basedand quantum annealers. A practically usable quantum computer is expected to be de-veloped in the next few years. In less than ten years quantum computers are expectedto outperform everyday computers, leading to breakthroughs in artificial intelligence[16], the discovery of new pharmaceuticals and beyond [17, 18]. Currently, variousparties, such as Google, IBM, Intel, Rigetti, QuTech, D-Wave and IonQ, are devel-oping quantum chips, which are the basis of the quantum computer [19]. The size ofthese computers is limited, with the state-of-the-art being around 70 qubits for gate-based quantum computers and 5000 qubits for quantum annealers. In the meantime,progress is being made on algorithms that can be executed on those quantum com-puters and on the software (stack) to enable the execution of quantum algorithms onquantum hardware [20].This also means that Portfolio Optimisation is one of the promising applicationsof quantum computing in finance [21]. The work of [22] presents an implementationof Markowitz’s portfolio selection on D-Wave, where the expected return is maximisedwhilst minimising the covariance (risk) of the portfolio under a budget constraint.They formulate this as an Ising problem and solve it on the D-Wave One, having 128qubits, for 63 potential investments within 20 µ s on the quantum processor. Theyindicate that the solution depends on the weights added to each of the objectivesand constraints. The same is done in [23], where reverse quantum annealing is usedto optimise the risk-adjusted returns by the use of the metrics of Sharpe ratio. In[24] the stock returns, variances and co-variances are modelled in the graph-theoreticmaximum independent set (MIS) and weighted maximum independent set (WMIS)structures under combinatorial optimisation. These structures are mapped into theIsing physics model representation of the underlying D-Wave One system. This isbenchmarked against the MATLAB standard function quadprog.A newer version of the D-Wave hardware is used in [25]. They also perform astock selection out of a universe of U.S. listed, liquid equities based on the Markowitzformulation and the Sharpe ratio. They approach this first classically, then by anapproach also using the D-Wave 2000Q. The results show that practitioners can usea D-Wave system to select attractive portfolios out of 40 U.S. liquid equities. Theresearch has been extended to 60 U.S liquid equities [26]. lgorithms have also been created for the gated quantum computer. In [27] analgorithm is given for Portfolio Optimisation that runs in poly( log ( N )), where N isthe size of the historical return data set. The number of required qubits here isalso N . An alternative method for a gated quantum computer is given by [28]. In[29] a quantum-enhanced simulation algorithm is used to approximate a computation-ally expensive objective function. Quantum Amplitude Estimation (QAE) providesa quadratic speed-up over classical Monte Carlo simulation. Combining QAE withquantum optimisation can be used for discrete optimisation problems like PortfolioOptimisation.With the introduction of the 5000 qubit Advantage quantum system by D-Waveand the new hybrid solver tooling a new heuristic approach is operational for in-production quantum computing applications . In this paper we use the new function-ality of D-Wave for the portfolio management problem and compare its performancewith other state-of-the-art, commercially available tooling.In the remainder of this paper first the Portfolio Optimisation problem is formu-lated. This is a quadratic, constrained, binary optimisation problem that can in prin-ciple be solved by commercial solvers. The solvers we use in this paper as benchmarkfor the quantum approach are presented in Section 3. In Section 4 the implementationof the problem on the D-Wave quantum annealer is shown. The results of solvinginstances of two main stock indices, the Nikkei225 and the S&P500, on the quantumannealer and the comparison with the benchmark approaches, will be presented inSection 5. We end with some conclusions and recommendations. We look at a Portfolio Optimisation problem, where we have N assets to invest in, P , ..., P N . The expected return of assets i equals µ i and the risk of the asset, denotedby the standard deviation, equals σ i . The returns of the assets are correlated, expressedby the correlation ρ ij for the correlation between assets i and j . We have now thereturn vector µ = { µ i } and the risk matrix Σ = { σ ij } where σ ij = σ i if i = j and σ ij = ρ ij σ i σ j if i (cid:54) = j .Assume that we have a budget to select n assets out of N . We want the returnto be higher than a certain value R ∗ and are searching for that set of n assets thatrealise the target return against minimal risk. We therefore define x i = 1 if asset i isselected and x i = 0 otherwise. This gives the following optimisation problem:min x T Σ x, (1)s.t. N (cid:88) i =1 x i = n, (2) µ T x ≥ R ∗ , (3)which is a quadratic, constrained, binary optimisation problem. Benchmarks
We want to compare the performance of the D-Wave hardware with other state of theart, commercially available tooling. For this we selected two solvers and two meta-heuristics.The first solver is LocalSolver [30], which is a black-box local-search solver of 0-1programming with non-linear constraints and objectives. A local-search heuristic isdesigned according to the following three layers: Search Strategy, Moves and EvolutionMachinery. LocalSolver performs structured Moves tending to maintain the feasibilityof solutions at each iteration whose evaluation is accelerated by exploiting invariantsinduced by the structure of the model. Unlike other math optimisation software,LocalSolver hybridises different optimisation techniques dynamically. We used anacademic licensed local version.The second solver is Gurobi [31], from Gurobi Optimisation, Inc., which is a power-ful optimiser designed to run in multi-core with capability of running in parallel mode.Gurobi uses the Branch and Bound Algorithm to solve Mixed-Integer Programming(MIP) models. It is based on four basic principles: pre-solve, cutting planes, heuris-tics, and parallelism. Each node in the Branch and Bound search tree is a new MIP.For our current MIQP (Mixed Integer Quadratic Programming), a Simplex Algorithmis used to solve the root node. We used an academic licensed local version of Gurobi.The third benchmark is a standard MATLAB implementation of Genetic Algo-rithms, which is a method for solving both constrained and unconstrained optimisationproblems based on a natural selection process that mimics biological evolution. Thealgorithm repeatedly modifies a population of individual solutions. It is a stochastic,population-based algorithm that searches randomly by mutation and crossover amongpopulation members.The last benchmark is the Simulated Annealing approach as implemented by D-Wave in their Ocean environment. Their sampler implements the simulated annealingalgorithm, based on the technique of cooling metal from a high temperature to im-prove its structure (annealing). This algorithm often finds good solutions to hardoptimisation problems.
In this paper we use the newly introduced (2020) functionality of D-Wave for theportfolio management problem and compare its performance with the commercialsolvers described in Section 3. In this section we describe how the problem can beimplemented on the D-Wave hardware. For this, the QUBO representation will bederived, a method to find the parameters is given and D-Wave’s quantum and hybridalgorithms are explained.
The D-Wave hardware solves Ising or QUBO problems. The QUBO [32] is expressedby the optimisation problem:QUBO: min/max y = x t Qx, (4) here x ∈ { , } n are the decision variables and Q is a n × n coefficient matrix.Another formulation of the problem, often used, equalsQUBO: min/max H = x t q + x t Qx, (5)or a combination of multiple of these termsQUBO: min/max H = λ · H + λ · H + · · · , (6)where λ , λ , . . . are weights that can used to tune the problem and include constraintsinto the QUBO. For already a large number of combinatorial optimisation problemsthe QUBO representation is known [32, 33]. Many constrained binary programmingproblems can be transformed easily to a QUBO representation. Assume that we havethe problem min y = c t x, subject to Ax = b, (7)then we can bring the constraints to the objective value, using a penalty factor λ fora quadratic penalty: min y = c t x + λ ( Ax − b ) t ( Ax − b ) . (8)Using P = Ic , the matrix with the values of vector c on its diagonal, we getmin y = x t P x + λ ( Ax − b ) t ( Ax − b ) = x t P x + x t Rx + d = x t Qx, (9)where matrix R and the constant d follow from the matrix multiplication and theconstant d can be neglected, as it does not influence the optimisation problem.A QUBO problem can be easily translated into a corresponding Ising problem of N variables s i ( i = 1 , .., N ) with s i ∈ { − , } given by :min y = N (cid:88) i =1 h i s i + N (cid:88) i =1 N (cid:88) j = i +1 J ij s i s j (10)The Ising model and QUBO model are related by s i = 2 x i − We now create a QUBO formulation for the problem given in Equation (1)-(3). Incase Equation (3) is an equality, this problem can be translated easily to a QUBOformulation: min (cid:0) λ x T Σ x + λ ( N (cid:88) i =1 x i − n ) + λ ( µ T x − R ∗ ) (cid:1) . (11)In the case of inequalities in Equation (3) we have to add additional K slackvariables y k ( k = 1 , . . . , K ), where K = (cid:98) log ( (cid:80) Ni =1 ( µ i )) (cid:99) . Scaling the µ values (inthousands, millions, etc.) will help reduce the number of variables. This leads tomin (cid:0) λ x T Σ x + λ ( N (cid:88) i =1 x i − n ) + λ ( µ T x − R ∗ − K (cid:88) k =1 k y k ) (cid:1) . (12) λ . Open dots are violating theoriginal constraints, while closed dots are valid solutions. It is known that the performance of the D-Wave hardware is depending strongly onthe choice of the penalty coefficient λ . When determining the penalty coefficients, wecan set λ = 1 and look for good values for λ and λ . In Figure 1 it is shown thatthe choice of λ greatly influences the best found solution. Values for λ lower that500 gives an invalid solution, i.e. the solutions do not meet the constraints given inthe original problem. Values for λ higher than 500 give allowed solutions. A valueof 500 or just above would be optimal. Rule of thumb is that the gain of violating aconstraint must be lower than the costs. For λ this means that violating the associatedconstraint in the optimal solution x ∗ for stock i gives a benefit of (cid:80) Nj =1 σ ij x ∗ j . Aroundthe optimal value, the biggest benefit would be for stock i for which the sum of thesmallest n values of σ ij is the largest, meaning (cid:99) λ = max i n (cid:88) j =1 σ i { j } where σ i { j } represents the j -th smallest covariance value for asset i .This is more complicated for λ . Again, we have to look at the optimal solution x ∗ and exchange a zero and one in this solution, such that n stocks are chosen, leadingto x (cid:48) . This is done such that ( x ∗ − x (cid:48) ) T Σ( x ∗ − x (cid:48) ) /µ T ( x ∗ − x (cid:48) ) is maximised. Theprocedure we used is as follows:1. A n sums S i = (cid:80) nj =1 σ i { j } ,2. A µ i between these n stocks,3. (cid:99) λ = A /A .4 Solving QUBO The most recent hardware, D-Wave Advantage, features a qubit connectivity basedon Pegasus topology. The QUBO problem has to be transformed to this structure.Due to the current limitation of the chip size, a compact formulation of the QUBOand an efficient mapping to the graph is required. This problem is known as MinorEmbedding. Minor Embedding is NP-Hard and can be handled by D-Wave’s Systemautomatically, hence we do not attempt to optimise it. While some qubits in the chipare connected using external couplers, the D-Wave QPU (Quantum Processing Unit)is not fully connected. Hence a problem variable has to be duplicated to multipleconnected qubits. Those qubits should have the same value, meaning the weight oftheir connection should be such that it holds in the optimisation process. All thesequbits representing the same variables are part of a so-called chain, and their edgeweights is called the chain strength ( γ ) , which is an important value in the optimisationprocess. In [34], it has been indicated that if γ is large enough, the optimal solutionswill match γ ≥ Σ ij | Q ij | . However the goal is to find the smallest value to avoidrescaling the problem. The problem of finding the smallest γ is NP-Hard. On a higherlevel, D-Wave offers hybrid solvers. These solvers implement state of the art classicalalgorithms with intelligent allocation of the QPU to parts of the problem where itbenefits most. These solvers are designed to accommodate even very large problems.This means that also most parameters of the embedding are set automatically. Bydefault, samples are iterated over four parallel solvers. The top branch implementsa classical Tabu Search that runs on the entire problem until interrupted by anotherbranch completing. The other three branches use different decomposers to sample outa part of the current sample set to different samplers. The D-Wave’s System sampleruses the energy impact as the criteria for selection of variables. We solved the Portfolio Optimisation problem for two indices, the Nikkei225 andthe S&P500. We solved several instances for both indices on the D-Wave annealer,using the implementation described in Section 4, and all four benchmark approachesdescribed in Section 3. The instances were run on a Intel(R) Core(TM) [email protected] 8GB RAM personal computer. We used the newest version of the D-Wave Leap environment solvers, Hybrid Binary Quadratic Model Version 2, for binaryproblems. The solver is, due to the underlying QPU, a stochastic solver. For this, weran the hybrid solver five times for each problem instance and performed a parametergrid search for λ and λ for each problem instance.We used the implementation first to select a number of stocks of the Nikkei225index. The Nikkei225 is a stock market index for the Tokyo Stock Exchange. It hasbeen calculated daily by the Nihon Keizai Shimbun (The Nikkei) newspaper since1950. It is a price-weighted index, operating in the Japanese Yen, and its componentsare reviewed once a year. The Nikkei measures the performance of 225 large, publiclyowned companies in Japan from a wide array of industry sectors. We took quarterlydata of the Nikkei225 index of the last five year. The used return ( µ ) of each stock isthe five year return, and the covariance σ ij the calculated covariance over all quartersof the last 5 years. From this index, we took a subset N of which n are selected tominimise the risk under a budget and return ( R ∗ ) constraint. Note that ( R ∗ ) and n are related here: R ∗ = 3000 and n = 20 means an average 5-year return of the n R ∗ Size( Q ) (cid:99) λ (cid:99) λ λ λ Solution Best50 10 0 60 76 0 70 0 256 256 ∗
50 25 0 60 320 0 365 0 3,035 3 , ∗
50 10 1,200 60 76 0.04 76 0.1 321 321 ∗
50 25 1,200 60 320 0.10 365 0.1 3,058 3 , ∗
100 20 0 111 137 0 100 0 809 809 ∗
100 50 0 111 607 0 700 0 10,999 10 , ∗
100 20 2,500 111 137 0.05 100 0.1 1,126 1 , ∗
100 50 2,500 111 607 0.09 700 0.1 11,065 11 , ∗
150 20 0 161 116 0 100 0 628 628 ∗
150 50 0 161 502 0 500 0 7,993 7 , ∗
150 20 3,000 161 116 0.04 100 0.1 1,271 1 , ∗
150 50 3,000 161 502 0.10 500 0.1 8,315 8 , ∗
200 20 0 211 100 0 100 0 530 529200 50 0 211 428 0 500 0 6,000 5,950200 20 3,250 211 100 0.04 100 0.1 1,308 1,262200 50 3,250 211 428 0.08 500 0.1 6,859 6,500225 20 0 236 88 0 88 0 484 482225 50 0 236 385 0 385 0 5,468 5,379225 20 3,500 236 88 0.05 100 0.1 1,504 1,455225 50 3,500 236 385 0.07 385 0.1 6,610 6,131Table 1: 20 instances for the Nikkei225 index with their parameters and HQPUsolutions. portfolio of 3000 /
20 = 150 percent.In Table 1 we show the 20 test instances we created on the Nikkei225, each hav-ing a different combination of (
N, n, R ∗ ). For each instance we show the size of thecoefficient matrix of the QUBO (Size( Q )), the calculated ( λ ) and optimal (cid:99) λ value ofthe parameters and the resulting value of the objective function of the best solutionfound by the hybrid solver. Also the best known solution is shown.In Table 2 the results of all benchmark tooling per test instance. In the table areshown the results of the hybrid solver (HQPU), Simulated Annealing (SA), GeneticAlgorithm (GA), the Gurobi solver (GB) and LocalSolver (LS). We see that Local-Solver finds for all instances the best solution. Gurobi also finds this solution, however,it runs out of memory (locally) for instances bigger than N = 150. For the instancesit solves, it proves optimality by closing the optimality gap to 0%. LocalSolver isnot able to close this optimality gap within reasonable time for most larger instances.The HQPU gives reasonable results, optimal for the smaller instances and well within5% of the optimal solution for the larger cases, with exception for the last instance.The Simulated Annealing stays close the HQPU solution and the Genetic Algorithmimplementation underperforms in comparison with the other solvers.The performance is also depending on the computation time. In Table 3 thecalculation times are listed. Here only the time the solver requires are mentioned, thetime to build the problem is out of scope. Best solving times are realised by Gurobi.LocalSolver performs well regarding to finding a good solution. However, for mostcases the optimality gap is not closed to 0% which means the solver keeps running n R ∗ HQPU SA GA GB LS Best Solution50 10 0 256 256 256 256 256 256 ∗
50 25 0 3,035 3,035 3,236 3,035 3,035 † , ∗
50 10 1,200 321 321 321 321 321 321 ∗
50 25 1,200 3,058 3,058 3,236 3,058 3,058 3 , ∗
100 20 0 809 809 1,102 809 809 † ∗
100 50 0 10,999 10,999 12,318 10,999 10,999 10 , ∗
100 20 2,500 1,126 1,135 1,450 1,126 1,126 1 , ∗
100 50 2,500 11,065 11,065 12,397 11,065 11,065 11 , ∗
150 20 0 628 628 1,047 628 628 † ∗
150 50 0 7,993 8,321 10,561 7,993 7,993 † , ∗
150 20 3,000 1,271 1,260 2,937 1,256 1,256 † , ∗
150 50 3,000 8,315 8,270 12,659 8,270 8,270 † , ∗
200 20 0 530 534 1,151 - 529 † † † † † † † † † for LocalSolver are not proven optimal, solver stops beforeGAP=0. 9 n R ∗ HQPU SA GA GB LS50 10 0 1.0 8 58 < < > < < < > < < < < > < > < > < > > > > > > > > > until the maximum admitted time is achieved. Simulated Annealing and Geneticalgorithms have increasing calculation times for the instances. The HQPU is quitefast and independent from the instance size here. The times displayed in the tablecorrespond to one call to the hybrid solver, where we used 5 calls per instance. D-Waveallows the user to set a time limit. When the user does not specify this, a minimumtime limit is calculated and used. For all instances here we did not adjust the timelimit.The second exercise we perform on the bigger S&P500. The S&P500 is a stockmarket index that measures the stock performance of 500 large companies listed onstock exchanges in the United States. It is one of the most commonly followed equityindices. The S&P500 index is a capitalisation-weighted index and the 10 largest com-panies in the index account for 26% of the market capitalisation of the index: AppleInc., Microsoft, Amazon.com, Alphabet Inc., Facebook, Johnson & Johnson, Berk-shire Hathaway, Visa Inc., Procter & Gamble and JPMorgan Chase. For this indexwe created five instances, which are depicted with the results of the solvers in Table 4and Table 5.Again we see that Gurobi is not able to run instances bigger than N = 150.LocalSolver finds the best solutions but is not able to close the optimality gap. Inthe three largest instances the gap stayed 100% within the allowed 600s. Genetic n R ∗ Size( Q ) HQPU SA GA GB LS Best100 50 3500 112 5518 5518 6959 5518 5518 † ∗
200 50 3500 213 3121 3141 5896 - 3121 † † † † † for LocalSolver are not proven optimal, solver stops beforeGAP=0. N n R ∗ HQPU SA GA GB LS100 50 3500 1.0 8 61 < > > > > > algorithms performed poorly again. The hybrid approach and Simulated Annealingwere able to stay close to the LocalSolver solutions, within 3% except for the lastinstance. Although we gave the HQPU more time to solve, we adjusted the timelimit by hand, it stayed on 18% above the best solution found by LocalSolver, as theSimulated Annealing solver did. The solving times of the HQPU stay quite reasonableas compared to the other solvers. Quantum computing is still in its early stage. However, the moment of quantumcomputers outperforming classical computers is coming closer. In this phase, hybridsolvers, using classical methods combined with quantum computing, are promising.The quantum paradigm that is already maturing is quantum annealing combined withthe Hybrid Solvers that D-Wave is offering. In this paper we showed the performance ofthis hybrid approach, applied to a quadratic optimisation problem occurring in finance,Portfolio Optimisation. Here a portfolio of assets is selected, minimised the total riskof the portfolio. This portfolio has to meet some return and budget constraints.We suggested an implementation of this problem on the newest D-Wave quantumannealer using the hybrid solver. The performance on a problem where 50 stocks needsto be selected from an existing index, the Nikkei225 or the S&P500, was already rea-sonable, in comparison with the performance of other commercially available solvers.Of these solvers, LocalSolver performed the best, finding the optimal solution in allcases very quickly. However, proving optimality, and thus finishing the optimisation ask, was not realised within reasonable time. The hybrid quantum solver was ableto find a solution within 3% of the optimal solution for the S&P500 index up to 400stocks. Further improvement of the hybrid solvers and the enlargement of the QPUin the coming years lead to the expectation that this computing paradigm is close toreal business applications.For further research we recommend to dive into the hybrid approach and lookfor improvements. The used method is a standard method, where D-Wave offers theopportunity to tailor the methodology in the Leap environment. Also other quantumoptimisation algorithms could be used, like the QAOA algorithm on a gated quantumcomputer. Acknowledgments
The authors want to thank Hans Groenewegen for providing the financial data under-lying this research and Irina Chiscop for her valuable review and comments.
References [1] Harry Markowitz.
Harry Markowitz: selected works , volume 1. World Scientific,2009.[2] Marius Radulescu and Constanta Zoie Radulescu. A multi-objective approach tomulti-period: Portfolio optimization with transaction costs. In
Financial DecisionAid Using Multiple Criteria , pages 93–112. Springer, 2018.[3] Panos Xidonas, George Mavrotas, Christis Hassapis, and Constantin Zopouni-dis. Robust multiobjective portfolio optimization: A minimax regret approach.
European Journal of Operational Research , 262(1):299–305, 2017.[4] Jo¨elle Skaf and Stephen Boyd. Multi-period portfolio optimization with con-straints and transaction costs. In
Working Manuscript . Citeseer, 2009.[5] Konstantinos Liagkouras and K Metaxiotis. Multi-period mean–variance fuzzyportfolio optimization model with transaction costs.
Engineering applications ofartificial intelligence , 67:260–269, 2018.[6] Cristinca Fulga and Bogdana Stanojevi´c. Single period portfolio optimizationwith fuzzy transaction costs. In . Vilnius Gediminas Technical University, 2008.[7] Tao Pang and Azmat Hussain. A stochastic portfolio optimization model withcomplete memory.
Stochastic Analysis and Applications , 35(4):742–766, 2017.[8] Tao Pang and Katherine Varga. Portfolio optimization for assets with stochasticyields and stochastic volatility.
Journal of Optimization Theory and Applications ,182(2):691–729, 2019.[9] Alberto Del Pia, Santanu S Dey, and Marco Molinaro. Mixed-integer quadraticprogramming is in np.
Mathematical Programming , 162(1-2):225–240, 2017.[10] Michael R Garey and David S Johnson.
Computers and intractability , volume174. freeman San Francisco, 1979.
11] Duan Li, Xiaoling Sun, Shenshen Gu, Jianjun Gao, and Chunli Liu. Polynomi-ally solvable cases of binary quadratic programs. In
Optimization and OptimalControl , pages 199–225. Springer, 2010.[12] Majid Zanjirdar. Overview of portfolio optimization models.
Advances in Math-ematical Finance and Applications , 5(4):1–16, 2020.[13] Pooya Ronagh, Brad Woods, and Ehsan Iranmanesh. Solving constrainedquadratic binary problems via quantum adiabatic evolution.
Quantum Infor-mation & Computation , 16(11-12):1029–1047, 2016.[14] Catherine C McGeoch. Adiabatic quantum computation and quantum annealing:Theory and practice.
Synthesis Lectures on Quantum Computing , 5(2):1–93, 2014.[15] Edward Farhi and Aram W Harrow. Quantum supremacy through the quantumapproximate optimization algorithm. arXiv preprint arXiv:1602.07674 , 2016.[16] Niels Neumann, Frank Phillipson, and Richard Versluis. Machine learning in thequantum era.
Digitale Welt , 3(2):24–29, 2019.[17] Matthias M¨oller and Cornelis Vuik. On the impact of quantum computing tech-nology on future developments in high-performance scientific computing.
Ethicsand Information Technology , 19(4):253–269, 2017.[18] Felix G¨unther Gemeinhardt.
Quantum Computing: A Foresight on Applications,Impacts and Opportunities of Strategic Relevance . PhD thesis, Universit¨at Linz,2020.[19] Salonik Resch and Ulya R Karpuzcu. Quantum computing: an overview acrossthe system stack. arXiv preprint arXiv:1905.07240 , 2019.[20] Mario Piattini, Guido Peterssen, Ricardo P´erez-Castillo, Jose Luis Hevia,Manuel A Serrano, Guillermo Hern´andez, Ignacio Garc´ıa Rodr´ıguez de Guzm´an,Claudio Andr´es Paradela, Macario Polo, Ezequiel Murina, et al. The talaveramanifesto for quantum software engineering and programming. In
QANSWER ,pages 1–5, 2020.[21] Roman Orus, Samuel Mugel, and Enrique Lizaso. Quantum computing for fi-nance: overview and prospects.
Reviews in Physics , 4:100028, 2019.[22] Nada Elsokkary, Faisal Shah Khan, Davide La Torre, Travis S Humble, and JoelGottlieb. Financial portfolio management using d-wave quantum optimizer: Thecase of abu dhabi securities exchange. Technical report, Oak Ridge NationalLab.(ORNL), Oak Ridge, TN (United States), 2017.[23] Davide Venturelli and Alexei Kondratyev. Reverse quantum annealing approachto portfolio optimization problems.
Quantum Machine Intelligence , 1(1-2):17–30,2019.[24] Michael Marzec. Portfolio optimization: Applications in quantum computing.
Handbook of High-Frequency Trading and Modeling in Finance , pages 73–106,2016.[25] Jeffrey Cohen, Alex Khan, and Clark Alexander. Portfolio optimization of 40stocks using the dwave quantum annealer. arXiv preprint arXiv:2007.01430 , 2020.[26] Jeffrey Cohen, Alex Khan, and Clark Alexander. Portfolio optimization of 60stocks using classical and quantum algorithms. arXiv preprint arXiv:2007.08669 ,2020.
27] Patrick Rebentrost and Seth Lloyd. Quantum computational finance: quantumalgorithm for portfolio optimization. arXiv preprint arXiv:1811.03975 , 2018.[28] Iordanis Kerenidis, Anupam Prakash, and D´aniel Szil´agyi. Quantum algorithmsfor portfolio optimization. In
Proceedings of the 1st ACM Conference on Advancesin Financial Technologies , pages 147–155, 2019.[29] Julien Gacon, Christa Zoufal, and Stefan Woerner. Quantum-enhancedsimulation-based optimization. arXiv preprint arXiv:2005.10780 , 2020.[30] Thierry Benoist, Bertrand Estellon, Fr´ed´eric Gardi, Romain Megel, and KarimNouioua. Localsolver 1.x: A black-box local-search solver for 0-1 programming. , pages 299–316, 2011.[31] LLC Gurobi Optimization. Gurobi optimizer reference manual, 2020.[32] Fred Glover, Gary Kochenberger, and Yu Du. A tutorial on formulating and usingqubo models. arXiv preprint arXiv:1811.11538 , 2018.[33] Andrew Lucas. Ising formulations of many np problems.
Frontiers in Physics ,2:5, 2014.[34] Carleton Coffrin. Challenges with chains: Testing the limits of a d-wave quantumannealer for discrete optimization. 2019.,2:5, 2014.[34] Carleton Coffrin. Challenges with chains: Testing the limits of a d-wave quantumannealer for discrete optimization. 2019.