# A Quantum Walk Through Actuarial Science

Quantum computing concepts can be applied to measure and manage entangled risks February 2021*Photo: iStock.com/Akiyoko*

Why might actuaries want to use quantum computing? This article will offer an intriguing—although not mathematically rigorous—explanation that is hopefully conceptually tantalizing!

A quantum computer uses quantum mechanics to drive calculations instead of classical physics like today’s computers. Hopefully this will become a new programming paradigm for actuaries in the future—because although one programming paradigm can make a problem difficult to solve, changing the paradigm may give the problem a new perspective that can greatly simplify it.

This article will not go directly from quantum computing to actuarial science—that is a bridge too far. Instead, it will take a detour through quantum economics and quantum finance, which are emerging areas of study that use mathematical tools established in quantum mechanics to supplement current economics and finance models to gain new insights and solve new problems.

The quantum mechanics topics that will be addressed in this article are superposition, duality and entanglement. These concepts then will be viewed through the lens of quantum economics and quantum finance and applied to actuarial science topics to create new areas of interest that I call quantum insurance and quantum reinsurance. Lastly, Q# will be introduced, which is a programming language used to develop code for a quantum computing simulator.

### The Qubit

Let us take some of the mystique out of quantum mechanics and quantum computing. If you have read anything at all on these topics, you probably have heard of the bit in our current computers versus a qubit in quantum computing. These are smallest units of calculation within both types of computing systems. A bit can be a zero or one, but not both, whereas a qubit can be zero or one at the same time.

This makes the qubit sound very mystical, but it is not. If you think back to your linear algebra days, a qubit is a modeled linear combination of orthonormal basis vectors. In layperson terms, think of the orthonormal basis vectors as a two-dimensional coordinate system where the axes are at right angles of each other, just like the x and y axes of graphs in our elementary algebra class. The linear combination part of the description simply is stating that the qubit can be anywhere within the coordinate system. Hence, saying a qubit can be both a zero and one at the same time is not more magical than saying I can be zero on the x axis and one and the y axis, or vice versa. While the description may not be 100 percent correct, it should paint a simple mental picture to keep in mind. While the qubit is represented by this coordinate system, it is said to be in superposition in quantum mechanics. The basis vectors in this coordinate system are also called state-vectors. The state-vectors do not actually exist—they are simply a mathematical tool.

There is another way to look at the bit versus the qubit. The bit is based off modeling concepts that are mutually exclusive and distinct, making the world binary. A concept is or is not a realization of something. A coin is always distinctly a head or tail. Conversely, the qubit is modeling that says the world is a combination or superposition of two opposing states. It is not completely one thing or the other. The coin is neither heads nor tails when it is standing on its edge—its true outcome is not realized until it falls on one of its sides. In other words, do you like your sets to model as distinct or fuzzy?

### Duality

In the physical world, the properties of the qubit are photons of light. While the photons are in superposition, they display the characteristics of being waves as displayed through the double slit experiment. But once a scientist tries to determine, by measurement, which slit the light went through, the light suddenly takes on the behavior of particles. This is called particle-wave duality.^{1} This apparent transition from properties of a wave to a particle, due to measurement, is said to collapse the wave to a single observable attribute.

In classical mechanics, there is no real difference between states and measurements; however, in quantum mechanics, the difference is profound.^{2} A lot of issues are made about the collapse of the state due to quantum measurement. But the collapse also occurs in classic probability; it is known as updating the probability distribution when given new information.^{3}

### Evolution of States

The qubit evolves with certainty over time through operators that put the qubit into a new state. Knowing the states of a qubit does not mean you know the result of the measurement with certainty. In quantum mechanics, the point is not to measure the states, but to measure the observables such as the spin of an electron particle. The classical evolution of states determines the ability to predict the outcome of the experiment. The quantum evolution of states permits us to compute the probabilities of the outcomes of later experiments.^{4}

### Money Versus Value

Quantum mechanics was developed when it was realized that energy comes in discrete amounts, called quanta, instead of continuously. The same is true for money—it comes in discrete packets, such as a paycheck or premium for an insurance policy.^{5}

In classical economics, money is used as a simple tool to represent value and is seen as an easy way to broker trade. In quantum economics, money and value are two different concepts. Value is a fuzzy concept represented by the state of the qubit. Before a transaction, value is in a superposition of trade and no trade. It is only at the time of transaction that money is used as the measurement device to determine the actual observed worth of an item.^{6}

This is especially true for illiquid instruments. What is the value of an illiquid asset, a reinsurance treaty or a block of variable annuities? The instruments have a deterministic evolution through their contracts, which allows a probability to be put on their value. The money required to purchase the instrument is indeterministic until a transaction is completed.

### Entanglement

Entanglement is related to combining multiple qubits or subsystems into a composite system. It details the amount of information and observables that can be known about the composite system, the subsystems and the relationships among them. An excerpt from the 2014 book *Quantum Mechanics: The Theoretical Minimum* explains it well:

*“Entanglement is the quantum mechanical extension of correlation … Entanglement is not an all-or-nothing proposition. Some [qubit] states are more entangled than others … What is it about maximally entangled states that is so fascinating? [It] can be summed up in two statements.*

*An entangled state is a complete description of the combined system. No more can be known about it.**In a maximally entangled state, nothing is known about the individual subsystems.*

*[Number 2 implies that] there is no certainty in the result when measuring each of the subsystems … [Versus] in classical physics, the use of probability is always associated with an incompleteness of knowledge relative to all that could be known. There is more that could be known about the system.”*

The important point about entanglement is when a system is combined, there are different levels of entanglement—and the more entangled the systems are, the less is known about the individual subsystems. Therefore, entanglement has a physical component along with an informational component, and the measurement of one qubit has an impact on the other qubit. This is what Einstein called “spooky action at a distance.”^{7}

### Financial Entanglement

In quantum finance, the relationship between money and credit is a form of social entanglement, explicitly encoded using contracts. This entanglement, when aggregated, can affect the entire financial system. It is analogous to entangled spin of two electrons.^{8}

Consider a loan contract between a debtor and creditor with each party represented as a qubit. If we look at the state of each party, then they are both in a superposition of default and no default. If the debtor changes state and defaults, then this action will have an immediate impact on the state of the creditor even though the creditor was not directly manipulated or notified of the default. The debt/credit relationships throughout the economy create an intricate web of entanglement, which was on full display during the 2008 housing crisis.^{9}

### Quantum Reinsurance

Reinsurance can be viewed anecdotally to the debt and credit of the banking system, where the creditor is the reinsurance organization and the debtor is the ceding company. The ceding company during the life of the treaty can be in a superposition of solvency and insolvency. This creates the intricate web of entanglement between the reinsurer and the ceding company.

Even with information about both companies, the more entangled the companies, the less information there is about the individual independent entities. For example, let’s say a reinsurer has a 95 percent quota share on the entire business of a ceding company. It would not make sense to model the ceding company as an entity independent from the reinsurer, because the organizations are so intertwined that the results would not tie back to reality. Modeling both companies requires information about the entire system. The information about the states of the individual organizations is essentially gone.

By the design of regulation, to get reinsurance credit, there must be a transfer of six major risks:

- Mortality
- Morbidity
- Lapse
- Credit quality of assets
- Reinvestments
- Disintermediation

If any of these risks are deemed significant, they must be transferred to the reinsurer.^{10} These risks could be modeled as the degree of entanglement between the two organizations. The greater the risk transfer, the more entangled they are. Modeling like this gives the regulators a mathematical way to measure the amount of risk transfer.

For instance, organizing the risks of a complex global reinsurer into mutually exclusive buckets underestimates the complexity of a reinsurance operation. This complexity only can be appreciated when the degree to which the risks are ensnared and how new treaties impact the entire organization are considered. This complexity becomes more chaotic when considering that ceding companies can be in many different regulatory jurisdictions. The insurance contracts within the treaties are little more than exchanging multiple options and guarantees between the insurance organization and its policyholders. Beyond this, the reinsurance treaty is an exchange of multiple options between the reinsurer and its ceding companies.

### Quantum Insurance

Representing insurance through entanglement can be taken a step further. What if the risks of the ceding company were modeled as individual qubits in superposition? For example, mortality is in a superposition of dead and alive, just like Schrodinger’s cat. Morbidity, lapse, assets, reinvestment and disintermediation are in a superpositions of sick and not sick, pay and not pay, default and not default, above rate of return and below rate of return, and surrender and not surrender, respectively. This means an insurance organization is an entangled system of risks. Actuaries who have done corporate risk management know what I mean!

It is exceedingly difficult to segregate risks into independent buckets because they are so intertwined. Using a quantum approach to risk management may make the exercise easier and more intuitive because risks appear to be fuzzy instead of mutually exclusive. This would make the modeling of reinsurance boil down to modeling the merging of two entangled insurance organizations, which sounds conceptually alluring to me.

### Building a Model

If modeling insurance and reinsurance in this manner, there is no reason to wait for quantum computers to be invented. Microsoft realized the technology industry cannot build a quantum computer and then get everyone up to speed on how to use it—it saw the need to start training people now, so everyone is ready to take advantage of quantum computing when it comes available. For this reason, Microsoft created a Quantum Developers Kit (QDK), which is completely free and available for download. It features a new language called Q#. With the QDK, you get Q# libraries, a quantum simulator and extensions for other .NET languages and Python. Integrated development applications that support it are Visual Studio, Visual Studio Code, and Jupyter Notebooks.^{11}

The full state simulator can simulate about 30 qubits on a local computer. If the application grows beyond this point, the application can be moved to Microsoft Azure (see a complete online training and step-by-step tutorial on Microsoft Learn.)

### The Power of Expression

Technology not only gives us the ability to do a task; it gives us a power of expression. There is power in being able to express ideas in a natural way. Quantum computing may provide a natural way to express new or unresolved topics in actuarial science. It may give us faster ways to do projection or gain insights. At the very least, it is an interesting thought experiment to stretch our current understanding of risk and all its complexity.

### Why?

This article is written for *The Actuary’s* February 2021 theme of “Advancements in Technology and the Role of the Actuary.” As a profession, we must be willing to constantly learn new ways of modeling risk and how to reframe problems. This means as technology advances, our role is to force ourselves to think about old problems in new ways, continually learn new topics and challenge ourselves. We need to actively try new ideas, even if they may seem radical and a little unconventional at the time. This is always a worthwhile exercise, regardless if it produces tangible results today. You never know when that knowledge might be useful. It is important to not fear being wrong in the pursuit of knowledge.

### In Closing

This article was a journey through quantum concepts as applied to economics, finance and actuarial science. It started by looking at the fundamental features of quantum mechanics, such as superposition, duality, the evolution of state and entanglement. These principles then were related to the topics of quantum economics and finance, such as the duality of value and money and the entanglement between debtor and creditor in banking. Then, all this information was adapted to insurance risk and reinsurance to display the possible quantum applications, such as in the entanglement of risks.

Framing actuarial science in quantum terms is not a mainstream idea, especially modeling using quantum computing. But this does not mean that it is not worth pursuing. There is such a thing as productive and profitable play. This play is important to advance our personal careers, and it is also important for the viability of our profession.

**Bryon Robidoux, FSA, CERA,**is actuary for The Standard. He is also a contributing editor for

*The Actuary*.

### References:

- 1. Baggott, J. E. 2013.
*The Quantum Story: a History in 40 Moments*. Oxford University Press. ↩ - 2. Susskind, Leonard, and Art Friedman. 2014.
*Quantum Mechanics: The Theoretical Minimum*. New York: Basic Books. ↩ - 3. Rieffel, Eleanor, and Wolfgang Polak. 2014.
*Quantum Computing: A Gentle Introduction.*Cambridge, Massachusetts: The MIT Press. ↩ - 4. Supra note 2. ↩
- 5. Orrell, David. 2020.
*Quantum Economics and Finance: An Applied Mathematics Introduction.*Panda Ohana Publishing. ↩ - 6. Ibid. ↩
- 7. Supra note 5. ↩
- 8. Supra note 5. ↩
- 9. Supra note 5. ↩
- 10. Tiller, John E., and Denise Fagerberg Tiller. 2005.
*Life, Health & Annuity Reinsurance.*3rd ed., ACTEX Publications, Inc. ↩ - 11. Microsoft. What Are the Q# Programming Language and QDK?
*Microsoft*, May 5, 2019. ↩

Copyright © 2021 by the Society of Actuaries, Chicago, Illinois.