# Strategic Portfolio Optimization Using Simulated, Digital, and Quantum Annealing
###### Abstract  
In this work, we introduce a new workflow to solve portfolio optimization problems on annealing platforms. We combine a classical preprocessing step with a modified unconstrained binary optimization (QUBO) model and evaluate it using simulated annealing (classical computer), digital annealing (Fujitsu's Digital Annealing Unit), and quantum annealing (D-Wave Advantage). Starting from [PERSON]'s theory on portfolio optimization, our classical preprocessing step finds the most promising assets within a set of possible assets to choose from. We then modify existing QUBO models for portfolio optimization, such that there are no limitations on the number of assets that can be invested in. Furthermore, our QUBO model enables an investor to also place an arbitrary amount of money into each asset. We apply this modified QUBO to the set of promising asset candidates we generated previously via classical preprocessing. A solution to our QUBO model contains information about what percentage of the whole available capital should be invested into which asset. For the evaluation, we have used publicly available real-world data sets of stocks of the New York Stock Exchange as well as common ETFs. Finally, we have compared the respective annealing results with randomly generated portfolios by using the return, variance, and diversification of the created portfolios as measures. The results show that our QUBO formulation is capable of creating well-diversified portfolios that respect certain criteria given by an investor, such as maximizing return, minimizing risk, or sticking to a certain budget. q 202220220220220220220220220220220220220220220220220220222022022022022022022202202220220222022022202220220222022022202202202202202220220220  
computers promise to solve currently intractable problems, such as the factoring of numbers (see [PERSON]'s algorithm [1]), in polynomial time. However, currently available quantum computers suffer from noise, missing error correction, low numbers of qubits, and more, and are thus far from reaching the theoretically possible speedups. Current research efforts can be split into two main classes. The first class is concerned with the investigation of possible near-term advantages using currently available quantum and classical hardware. The second class is concerned with the creation of quantum algorithms that may exhibit speedups over current classically available algorithms, but require so-called fault-tolerant quantum hardware. Commercially available quantum computers can also be split into two main classes: quantum annealers and quantum-gate computers. The former, manufactured for example by D-Wave Systems, are based upon the adiabatic theorem to perform calculations. Many of the (decision) problems currently thought to be intractable have an optimization version that can easily be mapped onto the native input format of quantum annealers, namely the Ising model (or the equivalent QUBO model). As of now, these models have become a quasi-standard for examining NP-complete and NP-hard problems in the realm of quantum computing. There is a plethora of known transformations for NP-complete/-hard problems to the QUBO model (see e.g., [2]), as well as guidelines on how to formulate QUBO models in general (see e.g., [3]). The performance of various QUBO formulations for certain problems has been subject to intensive investigations in the past couple of years (see e.g., [4; 5; 6; 7; 8; 9; 10; 11]) and is still an ongoing effort. In this paper, we focus on the applicability of annealing techniques to the NP-hard problem of portfolio optimization [5], a well-known topic for investment funds and individual investors. The pioneering work by [PERSON] [12] in 1952 can be considered the foundation of portfolio optimization. The goal here is to distribute an investor's capital on assets such that a given objective, such as maximizing the return or minimizing the risk, is optimized.