Quantum Bridge Analytics II: QUBO-Plus, network optimization and combinatorial chaining for asset exchange

Abstract

Quantum Bridge Analytics relates to methods and systems for hybrid classical-quantum computing, and is devoted to developing tools for bridging classical and quantum computing to gain the benefits of their alliance in the present and enable enhanced practical application of quantum computing in the future. This is the second of a two-part tutorial that surveys key elements of Quantum Bridge Analytics and its applications. Part I focused on the Quadratic Unconstrained Binary Optimization (QUBO) model which is presently the most widely applied optimization model in the quantum computing area, and which unifies a rich variety of combinatorial optimization problems. Part II (the present paper) introduces the domain of QUBO-Plus models that enables a larger range of problems to be handled effectively. After illustrating the scope of these QUBO-Plus models with examples, we give special attention to an important instance of these models called the Asset Exchange Problem (AEP). Solutions to the AEP enable market players to identify exchanges of assets that benefit all participants. Such exchanges are generated by a combination of two optimization technologies for this class of QUBO-Plus models, one grounded in network optimization and one based on a new metaheuristic optimization approach called combinatorial chaining. This combination opens the door to expanding the links to quantum computing applications established by QUBO models through the Quantum Bridge Analytics perspective. We show how the modeling and solution capability for the AEP instance of QUBO-Plus models provides a framework for solving a broad range of problems arising in financial, industrial, scientific, and social settings.

Appendices

Appendix 1: Blockchains and decentralized market making

Decentralized market making is an intriguing concept that would require a detailed exploration, as it will likely emerge as critical factor for enabling scalable liquidity. But there are many questions to be answered. For example, what is the value of contributions by the decentralized market makers? Also, could these small investments held by the market—provided to equalize values in an exchange—be aggregated into baskets, and could those baskets be traded? How do we accurately assess the risks of items in baskets, to flow them up to the basket, to avoid “toxic assets” being included?

Finally, it should be noted that a computational system or agent that learns what a user wants to buy or sell, or might be willing to trade, would be quite valuable as an e-commerce tool because it provides a means to unveil the deeper purchase intentions of users. AI based agents could assist not only in the process of helping the user to determine what they might be willing to trade for or buy but could even help the user discover new purchase intentions that might lead to greater personal satisfaction. In other words, instead of just contributing to the accumulation of more useless stuff in their lives, such a system could explore more complex human values, as opposed to those reflecting desires and whims stimulated by media and advertising.

For example, if an AI held a model that understood the OCEAN Big Five personality traits, which was used so effectively by Cambridge Analytica in 2016, it could predict that the user has a high degree in a single trait, openness to new experiences. By balancing knowledge around both investment planning and personality traits, the advisor could provide more balanced advice to the user that would lead to greater personal satisfaction and fulfillment. A strictly financial based AI advisor would simply recommend one asset class over other, or the diversification into additional classes. But an AI advisor that used both financial optimization as well as heuristics about human personality and psyche, understanding the complex needs of the investor, might suggest to keep 95% of the portfolio within financial instruments, but propose that 5% could be invested in experiential learning for the user, in other words, investing in him or herself. This could include travel to learn a new language or a workshop to learn a new skill, possibly with permission to tap into the user’s online “bucket list”—the list of things you'd like to do before you “kick the bucket.”

To put this into the context of the AEP problem and combinatorial chaining, consider a situation with User A who has inherited a somewhat odd abstract painting from a distant relative in France, that doesn't have much value on the resale market in America. However, on a combinatorial exchange market, there may be a chance of trading it for something not only less objectionable but desirable for all parties. Her asking price is a value of $3000. Now, because her interaction with the exchange is managed by a user agent with access to her private “bucket list,” the trusted agent can now look for something that matches items on his list. It turns out that she has always wanted to take a class at the Cordon Bleu cooking school and to learn some French. So our agent can scan against other agents and listings, to find User B who wants to trade a $3000 workshop pass at Cordon Bleu for ten day stay in a beachfront Airbnb on some nice tropical island. The combinatorial chain holds that in place while finding a third or fourth transaction to make the combinatorial exchange pareto-optimal for all users. Fortunately, it finds User C who has a modest bungalow on a beach in the Marquesas, which doesn't get much Airbnb interest because it is too remote. However, that person looks at the painting, and realizes it was painted by the singer Jacques Brel, who was a great singer but lousy painter, and actually has quite a bit of value in the Marquesas because Jacques Brel spent his last days on the island of Hiva Oa, following the footsteps of Paul Gauguin and learning how to paint untamed landscapes that were so bad they looked abstract. So his agent offers a 3 week stay for that painting!

In this way, an AI-based financial advisor would advise in a more human and humane way. Thus, metaheuristic optimization via asset exchange technology could be applied directly to the issues of happiness, life goals and meaning. For user A, the lifelong goal of learning how to master the art of French cooking. For User B, a desperately needed vacation he couldn't afford otherwise. And for User C, the lifelong goal of appearing on Antique Roadshow, to show off a barn find of a lifetime. We thus can ascend from cold process of optimizing utility functions to optimizing the human condition.

Appendix 2: Illustration of the network structure of NetAEP0

The structure of the network NetAEP0 created in Sect. 4 is illustrated in the following diagram, where the i nodes are represented in their duplicated form i[R] and i[S], giving rise to the arc i[R] → i[S], for a network with N = {1, …, 6}. The assets α are represented by the letters A, B, C, D and E, giving rise to asset nodes of the form (α, i[S]) and (α, j[R]) which are joined by arcs (α, i[S]) → (α, j[R]) (called α-linking arcs in Sect. 4), where i and j may vary but the asset α (= A, B, …, etc.) must be the same in each such arc. It should be noted that these linking arcs do not have limiting bounds on their flows other than an implicit lower bound of 0.

The arcs of the network can be represented by a succession of columns of R-labeled nodes and S-labeled nodes, in a pattern that begins with the R-labeled i nodes i[R], followed by the S-labeled i nodes i[S], followed in turn by the S-labeled asset nodes (α, i[S]), then followed by the R-labeled asset nodes (α, j[R]) and finally followed by the R-labeled i nodes i[R] to repeat the pattern. A further interesting pattern seen in the diagram is that all S-labeled nodes have exactly 1 arc entering but may have multiple arcs leaving, while all R-labeled nodes have exactly 1 arc leaving but may have multiple arcs entering. The i nodes are enclosed in circles in the diagram and the asset nodes are enclosed in rectangles.

Since the asset arcs (linking arcs) do not have bounds on their flows, the foregoing pattern implies that an asset arc whose S-labeled node has a single arc out can be collapsed to be represented only by the R-labeled node, and an asset arc whose R-labeled node has a single arc in can be collapsed to be represented only by the S-labeled node. It should be emphasized that the staged structure shown in the diagram above is slightly misleading, since cycles typically vary in length and, in addition, duplicated i nodes may be encountered at various stages without implying they form a cycle that can be traced back to a previous instance of a duplicated node. The i indexes and the assets in the diagram have been ordered to show the patterns produced by arranging the nodes in columns. By contrast, the algorithm given in Sect. 4 for generating the network applies for any ordering of the indexes i in N and is independent of any ordering of the assets, which shows that such orderings are irrelevant in the general case.

Appendix 3: Illustration for reverse scanning

Appendix 4: Additional applications of the Asset Exchange Problem in public health and pandemic containment

The AEP arises in a variety of contexts, and recently, a new class of problems has become quite pertinent: the application of optimization processes to nonpharmaceutical interventions used in public health and pandemic containment. The basis of our approach on a form of cooperative optimization, where multiple parties with complex criteria collaborate as well as compete for resources, becomes additionally relevant in such situations where the objective to be optimized can be expressed in terms of the number of lives saved.

This is an area that is still in its infancy, and we make general observations about its nature and importance that should be considered speculative at the current stage. In the form of cooperative group optimization treated here, our approach generalizes processes that seek exchanges of pandemic resources or classes of citizens to enable combinatorial optimization. For example, a simple instance of such a system would be in the setting of elementary schools, in the formation of classroom bubbles to limit community spread of a highly infectious virus. In this example, a school wishes to allocate students and teachers to classroom bubbles to control the spread of the virus that currently poses a threat to health.

At present, schools assign students to bubbles in an ad hoc fashion. It becomes important to consider how this can be improved if additional information were available to assess the risk of infection for individual students, or infection via their home bubbles due to family members in high-risk occupations or the incidence of vaccination. We then pose the question: how would the formation of bubbles be modified to reduce the overall risk for the community? Each bubble is like a portfolio, and the assets in those portfolios contain their own risk levels. Key considerations are summarized by posing the further questions: Is it preferable to assign low risk students to low-risk bubbles, or to spread the risk around? If the school intended to achieve a kind of incremental herd immunity, then it might be supposed that it would be better to assign a low number of higher risk students from anti-vax families, with a larger number of low-risk students that belong to highly vaccinated families. However, when the wishes of individual vaccinated parents enter into the equation, these parents may not wish to allow their children to mix with children from unvaccinated households. Is there some science that could help to settle this issue?

The AEP framework makes it possible to address such issues using combinatorial swaps—moving interchangeable students around between bubbles, to seek more optimal configurations. Again, by enabling all potential complex exchanges, splits and aggregations to compose bubbles optimally, the school could decrease its overall risk of community spread. However, the computational complexity of this type of complex combinatorial exchange trading produces an NP-hard problem. This is further complicated by the inclusion of constraints—whether some students have special needs, certain students tend to be disruptive, and so forth—which increase the complexity of the problem.

The foregoing example is only one of many other possible areas that can be explored with both QUBO and AEP approaches, ranging from the allocation of scarce testing kits, to the early detection of asymptomatic spreaders, to the implementation of anomaly detection applications for improving the odds of early detection of variants of interest and concern. However, the most interesting is the vaccine allocation problem.

In the midst of a pandemic such as the one we are facing today, public health officials must make rapid vaccine allocation decisions—choosing who will receive the vaccines, where and when. Currently, there are two approaches to distributing vaccines to the general population: pull-based (in which requests from healthcare providers for specific quantities are filled) and push-based allocation (which is focused on distributing quantities to achieve equity among priority groups). The current preferred solution is to distribute the majority of vaccine doses via pull-based requests, while a smaller percentage of vaccine doses are held back by public health officials who must make decisions on how to allocate these doses to address and correct the inequities that may arise in a strictly pull-based process.

These decisions are extremely complex, and intersect with both logistical and ethical perspectives, especially when dealing with priority groups such as healthcare personnel, the elderly and people with certain health risks. The solutions must be fair, simple, effective (such as getting the right type of vaccines to specific locations), cognizant of cold chain constraints and quantity management. It is simply not possible for humans to make optimal decisions in this case. This is an ideal problem for mathematical optimization, that could help humans in making more optimal decisions.

Researchers have already begun using mixed integer programming to address this problem. This tool has proven to be valuable in deciding what percentage of available vaccines to keep in reserve for push-based allocation. As demonstrated in studies reported in Part 1 of this tutorial (Glover et al., 2021a, 2022), the use of algorithms specifically designed for QUBO and QUBOPlus models yield better solutions than MIP algorithms for these problem classes when the MIP algorithms are allowed to run one to three orders of magnitude longer. Moreover, the best algorithms for QUBO-related applications can obtain high quality solutions to problems much larger than the MIP approaches are capable of handling. In the QUBO-Plus category, the AEP model and its specialized algorithm afford a way to enlarge the strategies available for identifying a more efficient and equitable way to distribute vaccines, considering the factors highlighted here. Furthermore, with the significant risk for wastage of ultracold doses, faster optimization technologies could be valuable for optimizing real-time local distribution situations.

Finally, it may be possible to use these QUBO-related optimization technologies in the realm of behavioral economics to address situations where people often do not act in their own best interest. Hundreds of behavioral biases affect our daily decision-making. Many of those biases are accentuated in situations of stress, fatigue, propaganda, or fear. During a pandemic, both during lockdown and as vaccines have been released, factors like optimism bias and vaccine hesitancy impact the calculations people make on the probability of getting sick and dying. Compounding this situation, the lifting of social distancing measures can produce a false sense of security that worsens some of these biases and causes people to take fewer precautions. Through the relevance of the AEP model and its solution approach for cryptocurrency applications, it may be possible to use the AEP technology, in concert with cryptocurrency-based behavioral incentive tokens or nudge-based interventions, to address these challenges in behavioral economics as well. [See: https://www.zs.com/insights/how-behavioral-science-can-solve-vaccine-hesitancy and https://mgmt.wharton.upenn.edu/wp-content/uploads/2021/03/Covid19_Vaccine-Uptake-Behavioral-Science_Task-Force_Report_Feb-23-2021.pdf].

In sum, in the fight against a virus such as COVID-19, using technology that enjoys two orders of magnitude faster optimization for significantly larger variable spaces would allow for faster and better predictions about the spread of the virus, the ability to cluster higher risk cases more quickly and accurately for scarce testing supplies, and perhaps also, demonstrate that AI can be ethical.