QEF12 – A quantum oscillator model of stock markets

We’ve seen how the quantum model of supply and demand can be used to model transactions and here we’re going to look at how we can use it to simulate the stock market.

With the quantum model supply and demand we have propensity functions for the buyer and the seller and the joint propensity function is equal to the product of those. A market maker would maximize profit if the spread between the buyer and the seller price, i.e. the distance between the two peaks, is equal to the standard deviation of the joint propensity.

This system is described by a linear entropic force but it does not behave like a classical oscillator because we know that the price is indeterminate and of course prices don’t actually oscillate. We therefore go to the quantum harmonic oscillator which is the quantum version of a spring system. The ground state is a normal distribution where the mass scales with the inverse variance. We’re going to define the frequency for the stock market in terms of turnover, so if the turnover is once per year then that will be the frequency.

One way that you can use the quantum harmonic oscillator to simulate the statistics of stock markets is to look at higher energy levels. This plot shows a tracing of the oscillations for the case where we have three energy levels. Most of the energy is in the ground state but there is also some energy in the next two energy levels.

We’re going to look at a slightly simpler version here but first of all we can compare with the actual sort of data that you get from a stock market. This plot is an order book for Apple stock over a single hour and so we’ve got the buyers putting up orders at lower prices and sellers at higher prices. These are not the same as for propensity functions because they don’t reveal all the preferences and of course orders in the middle region are going to clear so will disappear. However these orders do give an idea of the propensity functions which are shown by the shaded areas in the background.

Now, suppose that someone comes and makes a very large order and you want to know how much the price is going to change in response to that – it’s going to go up but by how much? According to the square-root law, which is an empirically derived relation from finance, the price change is given by the square-root of the size of the trade divided by the daily traded volume, all multiplied by the daily volatility, and then multiplied by a numerical constant Y of order unity.

In the oscillator model this makes sense because the restoring force is linear so the energy required to perturb the system is going to vary with the square of the displacement, or equivalently the displacement varies with the square-root of the energy. And of course the energy in a quantum harmonic oscillator is just a multiple of the frequency. When you have a large purchase that’s like boosting the frequency over that time period, which boosts the energy. Comparing these two formulas show that they’re in agreement if we have Y being of the order the square root of two. That sounds about right because in the oscillator model we’re assuming all the energy goes into lifting the price, so the actual price change might be a bit less than that.

This gives us a picture for how we could model prices in general. We can assume that we have an oscillator in its ground state, but then it’s displaced by perturbations which shift it from side to side. It is then going to oscillate in a coherent state, which means it stays in a normal distribution but it’s moving from side to side. The probability of being in a particular energy level varies with a Poisson distribution. We will assume that the spread between the bid and the ask price is equal to the standard deviation in order to maximize the profit.

So to summarise we have a normal distribution, which represents the joint propensity function, bouncing back and forth between the dashed lines in the figure. However there’s another participant in the market which is the market maker. They only transact at the particular ask/bid prices, shown by the vertical bars. We can therefore compute the probability of the ask or bid prices being selected.

If you look at the propensity for the ask price being obtained it follows a sinusoidal of plot. The dashed line is the actual result, the gray line in the background is the sinusoidal plot of the sort that you would obtain with a with a simple two-state quantum system for example. Similarly the propensity function for the bid would just be the opposite of that, so the propensity adds to one (we are assuming here that a transaction takes place at one of the prices).

Notice that for this particular setting the propensity varies from 0.25 to 0.75, so that’s plus or minus 25 percent which is consistent with the kind of shift that you might expect from the quarter law from quantum decision theory.

We can use this model to simulate stock prices. Here the squares are the ask price, circles are the bid price, and we are assuming that we have random perturbations, so unlike the previous plot the oscillations are not allowed to continue because random noise is being applied. We’re also going to have some noise added to the spread, i.e. the difference between the bid and the ask.

The noise in the spread turns out to be important because it’s a main contributor to the fat tails which you see when you look at the overall probability density for a long simulation shown here. This compares with the dash line which is the statistics for the Dow Jones Industrial Average.

To summarize, in neoclassical theory it is assumed that supply and demand cancel out in equilibrium, so there’s no concept of dynamics, force, mass, energy and so on. The quantum model in contrast is obtained by quantizing a linear entropic force. This linear restoring force is what explains the empirically derived square-root law of price impact, and the oscillator model can also be used to simulate the dynamics of stock markets such as price change distributions.

Further reading:

Orrell D (2022) Quantum oscillations in the stock market. Wilmott (forthcoming).

Previous: QEF11 – The money bomb

Playlist: Quantum Economics and Finance

Quantum economics – the story so far

This piece gives a brief summary of my work to date (2016-2021) in quantum economics.

The idea that the financial system could best be represented as a quantum system came to me (dawned on me? evolved?) while working on The Evolution of Money (Columbia University Press, 2016). “Money objects bind the virtual to the real, and abstract number to the fuzzy idea of value, in a way similar to the particle/wave duality in quantum physics,” I offered. “Money serves as a means to quantify value, in the sense of reducing it to a mathematical quantity – but as in quantum measurement, the process is approximate.” Price is best seen as an emergent feature of the financial system. I summarised this theory in two papers for the journal Economic Thought: “A Quantum Theory of Money and Value” and “A Quantum Theory of Money and Value, Part 2: The Uncertainty Principle“.

While I had some background in quantum physics – I studied the topic in undergraduate university, taught a course on mathematical physics one year at UCL, and encountered quantum phenomena first-hand while working on the design of particle accelerators in my early career – my aim in the book (co-authored with Roman Chlupaty) was not to impose quantum ideas onto the economy. My primary research interest was in computational biology and forecasting and I had not touched quantum mechanics in many years. The dual real/virtual nature of money just had an obvious similarity to the dual nature of quantum entities, and in fact I was surprised that I appeared to have been the first to make this connection in a serious way and come up with a quantum theory of money.

I was aware that a number of researchers were working in applying quantum models to cognition and psychology, but it was only after finishing the book that I learned about the area of quantum finance (I also discovered a separate paper on “Quantum economics” by the physicist Asghar Qadir from 1978, which argued that the quantum formalism was well suited to modelling things like economic preferences). The reason I hadn’t come across these works in my research about money was because just like in neoclassical economics there was no discussion of that topic. Nor was there much discussion of quantum phenomena such as entanglement or interference. Instead the emphasis in quantum finance (as this paper notes) was on using quantum techniques to solve classical problems such as the Black-Scholes option-pricing algorithm, or portfolio optimisation.

My motivation was completely different. In books such as Economyths, and The Money Formula (with Paul Wilmott), I had investigated the drawbacks and limitations of these traditional models – so rather than invent more efficient ways of solving them, I wanted to replace them with something more realistic. Money was the the thing which linked finance and psychology, so a quantum theory of money could be a first step in developing a new approach to economics.

I sketched out the basic idea as an Economic Thought paper “Quantum economics” which served as a blueprint for my 2018 book of the same name. It tied together the quantum theory of money, with ideas from quantum finance, quantum cognition, quantum game theory, and the broader field of quantum social science. The ideas were also summarised in a piece for Aeon magazine – which was when I found out why no one had probably bothered to develop a quantum theory of money. The article was not well received, by economists but especially it seemed by physicists, some of whom went out of their way to trash the idea.

I was not new to having my work come under criticism. Indeed, much of my career has focused on pointing out the drawbacks and limitations of mathematical models, which has frequently brought me into conflict with people who don’t see it that way, starting with my D.Phil. thesis on model error in weather forecasting (see Apollo’s Arrow). My book Economyths also drew howls of outrage from some economists. However quantum economics felt different, and seemed to touch on a range of taboos, in particular from physicists who have long resisted the adoption of quantum ideas by other fields. But quantum mathematics is not owned by physicists, it is simply an alternative version of probability which was first used to model subatomic particles, but also can be used to describe phenomena such as uncertainty, entanglement, and interference which affect mental systems including the economy.

While writing the book I developed in parallel an online mathematical appendix which presented some key results from quantum cognition, finance, and game theory (an early version was translated into Russian). Because my aim was to develop a theory of quantum economics, I also started applying quantum methods to some key economic problems, including supply and demand, option pricing, stock market behaviour, and the debt relationship which underlies the creation of money. This online appendix later grew into my technical book Quantum Economics and Finance: An Applied Mathematics Introduction, first published in 2020 and now in its second edition.

For supply and demand, my idea was to model the buyer and seller in terms of a propensity function, which describes a probabilistic propensity to transact as a function of price. A simple choice is to describe the propensity function as a normal distribution. The joint propensity function is the product of the buyer and seller functions. The next step is to use the concept of entropic force to derive an expression for the forces which describes the tendency for each party to move the price closer to their preferred price point. The joint force is just the sum of the forces for the buyer and seller. However there is a contradiction because the probability distribution does not match that produced by an oscillation. To resolve this, we quantize the force to obtain a quantum harmonic oscillator whose ground state matches the joint propensity function. This model, which sounds elaborate but is actually quite minimal in terms of parameters, applies to economic transactions in general, so has numerous applications, including the stock market. The paper “A quantum model of supply and demand” was published in the journal Physica A in 2020.

Typical propensity functions for buyer (to the left), seller (right), and joint (shaded).

The question of how to price options is one of the oldest problems in finance. The modern method dates back to a 1900 thesis by Bachelier and is based on the concept of a random walk. For the quantum version, the logical place to start was with the quantum version of this which is a quantum walk. Instead of assuming that the log price will follow a normal distribution with a standard deviation that grows with the square-root of time, the model has two peaks which speed away from each other linearly in time. It therefore captures the psychological stance of an investor who has a bullish or bearish view on the asset (e.g. price might grow by 10 percent each year), but balances that with the possibility that the opposite might happen in order to obtain a fair price for the option. When coupled with the quantum model of supply and demand, the algorithm can be used to predict option price and volume. “A quantum walk model of financial options” was published in Wilmott magazine in 2021, and the theory was reported on the same year by the Economist in an article “A quantum walk down Wall Street“.

Probability distribution for a quantum walk (solid) versus random walk (dashed).

Finally a main question in quantum economics is the interaction between mind and money which underlies the debt relationship, and also the creation of money objects in the first place. Both of these topics are traditionally neglected in mainstream economics. In quantum economics it is easy to show that the debt relationship can be modelled as a simple circuit with two qubits, representing the debtor and creditor, entangled by a C-NOT gate which represents the loan contract. Interestingly, it turns out that the same circuit can be used to represent the decision-making process within the mind of a single person, where there is an interplay between a subjective context and the final decision. In quantum cognition, this is usually modelled as a two-stage process; however it can also be modelled using two entangled qubits, in which the context and the decision are separated out, as in the debt model. This result was published in a 2021 Frontiers in Artificial Intelligence paper, co-authored with Monireh Houshmand, called “Quantum propensity in economics“. A related paper published in Quantum Reports, that discusses applications including mortgage default, is “The color of money: threshold effects in quantum economics“. 

Two-qubit entanglement circuit for debt contract (A is debtor, B is creditor), or quantum cognition (A is context, B is decision).

For a full list of my research in quantum economics and finance, including links to these and other papers, please see the page Quantum Economics Resources. These findings and others are also presented in my technical book Quantum Economics and Finance: An Applied Mathematics Introduction, and for a general audience in Money, Magic, and How to Dismantle a Financial Bomb: Quantum Economics for the Real World (available 02/2022). The work continues! – if readers are interested in getting involved, please drop me a line here or through LinkedIn.

Ten reasons to (not) be quantum

While the use of quantum models is becoming more popular in the social sciences including economics, it is still the case that when many people, especially those with a training in physics, hear the expressions “quantum economics” or “quantum finance” they immediately reach for some off-the-shelf arguments about why it must be nonsense (or some smelling salts). Here is a compilation of the usual ones, along with responses.

1. Quantum mechanics was developed for subatomic particles, so it should not be applied to human systems. As one website claimed, “It’s only when you look at the tiniest quantum particles – atoms, electrons, photons and the like – that you see intriguing things like superposition and entanglement.” An article wonders why we “see the common-sense [classical] states but not the imponderable superpositions?”

Response: Bohr’s idea of superposition and complementarity was borrowed from psychology, as when we hold (or ponder) conflicting ideas in our heads at the same time, and the concepts of mental interference or entanglement are not so obscure. Also, many ideas from quantum mechanics such as the Hilbert space were invented independently by mathematicians. And calculus was developed for tracking the motion of celestial bodies but we don’t ban its application to other things.

2. Quantum is too hard for non-physicists to understand. According to the physicist Sean Carroll, “No theory in the history of science has been more misused and abused by cranks and charlatans – and misunderstood by people struggling in good faith with difficult ideas – than quantum mechanics.”

Response: There is often a confusion between quantum probability, which is a mathematical tool, and quantum physics, which is about subatomic particles. Yes, the physics of subatomic particles is complicated – so are things like classical fluid dynamics. But quantum probability is just the next-simplest type of probability after the classical one. And the misuse of mathematical models which has created the most societal problems is the classical methods used in economics. As a side note, most people involved in quantum economics and quantum finance are physicists or (like me) mathematicians. Which brings us to:

3. Quantum economics is physics envy, or an attempt to “appropriate the high prestige of physics” as one physicist put it.

Response: Mainstream economics is directly inspired by, and based on, concepts from classical mechanistic science. There is nothing inherently wrong with using the same mathematical tools for different areas, what is strange is when the tools used don’t change or adapt. As John Cleese said: “people like psychologists and biologists have still got physics envy but it’s envy of Newtonian physics and they haven’t really noticed what’s been happening the last 115 years.”

4. Quantum is flaky, pretentious, pseudoscientific hype or woo. Sample usage: “As a quantum physicist, I’ve developed a reflexive eyeroll upon hearing the word applied to anything outside of physics. It’s used to describe homeopathy, dishwasher detergents and deodorant.” Cue nerd jokes about “quantum healing” or “quantum astrology”. In his description of what he called the Intellectual-Yet-Idiot, Nassim Taleb included anyone who “Has mentioned quantum mechanics at least twice in the past five years in conversations that had nothing to do with physics.”

Response: Quantum is a mathematical toolbox – it might come across as flaky or pretentious for a person to talk about it in the wrong context, but not to use it in their work.

5. Entanglement is unique to special physical systems which can maintain quantum coherence. One science journalist told me that “Dollars don’t become quantum mechanically entangled. If they did, we’d be building quantum computers out of money.” A paper takes it as given that “one could not possibly idealize traders and investors as quantum objects exhibiting non-classical properties, like superposition or long-distance entanglement.” Physicists often conflate entanglement with Bell tests: one explained that “you can never violate a Bell inequality using systems like dice, dollars, or bank accounts. There is simply no way, and certainly no experiment has ever done so. (Maybe one or two ‘crackpot’ people claim otherwise, but they are not to be trusted.)” Another writes: “It is a common mistake to confuse classical statistical correlations with the correlations implied by quantum entanglement. The difference was made explicit by John Bell. There is no chance that correlations in statistical economics will violate a Bell inequality … unless you fiddle the data.” In other words, using entanglement in an economics model is a sign of either gullibility, or scientific fraud.

Response: In mathematical terms, entanglement is a straightforward property of Hilbert spaces, and we can use it to model social and financial systems, including traders’ decisions or the behaviour of money. The Bell test is not a definition of entanglement, it is a way of teasing out a particular form of entanglement for subatomic particles. It is true that we can’t build quantum computers out of money, but nor can we build classical computers – does that mean money is not classical?

6. Quantum is too complicated and mathematical – we need simpler models and less math. Variants: The economy cannot be reduced to equations, people are not subatomic particles. May quote Lin Yutang.

Response: The need for simple models is a theme of many of my books, however what counts is things like the number of parameters in a model. Quantum probability is more complicated than classical probability, but it is the simplest way to capture phenomena such as superposition, interference, and entanglement, which characterise many key mental and financial processes (for example, the quantum walk model for pricing options or the two-qubit model for quantum decisions are not complicated). People are not subatomic particles, but nor are they classical particles, which doesn’t stop economists from using classical models, or talking about physics-like forces of supply and demand (they are just assumed to be at equilibrium, so cancel out). And while it is true that human behaviour cannot be reduced to equations of any sort, we use equations all the time to simulate the economy. Again, many of my books, such as Apollo’s Arrow, or Truth or Beauty, have criticised the overreach of mathematical models, but that is a separate issue and applies as much to classical models.

7. Quantum is a forced analogy or a metaphor. As economist Paul Samuelson once wrote, “There is really nothing more pathetic than to have an economist or a retired engineer try to force analogies between the concepts of physics and the concepts of economics … and when an economist makes reference to a Heisenberg Principle of [quantum] indeterminacy in the social world, at best this must be regarded as a figure of speech or a play on words, rather than a valid application of the relations of quantum mechanics.”

Response: Quantum probability is a mathematical tool, which is not the same as an analogy or metaphor. The purpose of a metaphor is usually to describe something which is abstract and complicated in terms of something that is more concrete, so it would make more sense to go the other way and use human behaviour as a metaphor to help describe subatomic behaviour.

8. The brain has not been shown to rely directly on quantum processes.

Response: Quantum effects appear to be exploited by biological systems in a number of processes (see quantum biology) but whether they are used in the brain or not makes no difference to economics. The argument is not that the economy inherits quantum properties from subatomic interactions in the brain, but that it can be modelled as a quantum system in its own right. For example, a debt contract can be expressed using a quantum circuit in a way which captures effects such as uncertainty, subjective context, power relationships, and so on.

9. Markets are not quantum because there is no uncertainty. For example, something like a bank account, or an order book for a stock market, has clearly posted amounts and prices. One person compared her bank account to Schrödinger’s cat: “I am a PhD physicist, so for me the word quantum that gets thrown around is a bit ridiculous … So think about your bank account, it might be empty until you open it, so are you telling me that this is uh quantum finance or quantum economics okay you can have a million in your account or you can have zero we don’t know?”

Response: While it may be true that bank accounts are not like Schrödinger’s cat, I will let The Economist answer that one, from an article called “Schrödinger’s markets” in the print edition: “on a closer look finance bears a striking resemblance to the quantum world. A beam of light might seem continuous, but is in fact a stream of discrete packets of energy called photons. Cash flows come in similarly distinct chunks. Like the position of a particle, the true price of an asset is unknowable without making a measurement – a transaction – that in turn changes it. In both fields uncertainty, or risk, is best understood not as a peripheral source of error, but as the fundamental feature of the system.”

As computer scientist Scott Aaronson notes, quantum methods are adapted to handle “information and probabilities and observables, and how they relate to each other.” Since the financial system seems a pretty good example of information, probabilities, and observables (in this case through transactions) it seems like a suitable approach. Much of the confusion comes down to the fact that quantum economics is not quantum physics applied to the economy, but rather quantum mathematics applied to the economy (see figure below). Physicists often struggle with this because they tend to mistake their elegant models for reality, however quantum mathematics should be viewed as a mathematical toolbox that can be applied to either physical or social systems where appropriate.

The idea of quantum economics is not that physics can be directly applied to social behaviour as shown here …

… but instead that quantum mathematics can be applied to both physical and social systems. Figures from: Quantum Economics and Finance: An Applied Mathematics Introduction

Physicists tend to see entanglement for example as a special property of subatomic particles – or “a surprising feature of the world” as one emphasized – and object that the two parties in a loan contract are not entangled in the same way. To understand the entanglement, it is necessary to lift the level of analysis from physical people, to mental constructs – which is entirely appropriate, given that money and value are mental constructs. From the perspective of the debt contract, if the debtor decides to default, then the state of the loan also changes immediately for both parties. And debt contracts are a feature of the world too (even if they are less remarkable or surprising than quarks or whatever).

The above nine reasons for rejecting a quantum approach, which are the ones most commonly produced, are very superficial and are easily dismissed with a little reflection. (Skeptics sometimes prefer to say that they don’t understand or are “not convinced” without giving a specific reason, but my aim is not to convince people of anything, it is to lay out the facts as I see them and let others do their own research and come to their own conclusions.) Also, arguing against these reasons, as I have done above, will in my experience have absolutely no effect. One reason is that getting the quantum approach seems to involve something of an aha moment where it suddenly clicks into place. The other reason though is that they are not the real reason. So why is it that no one even tried to apply quantum methods to the economy until about a century after they were invented? This points to:

    10. Quantum economics touches on a range of taboo topics.

For the full picture read Money, Magic, and How to Dismantle a Financial Bomb: Quantum Economics for the Real World. Finally, given the numerous reasons to not take a quantum approach, I should point out that there also many reasons why the opposite is true, and the economy is amenable to a quantum treatment! In particular, quantum is the best framework for expressing in mathematical terms the complex interactions between mind and money. To see why, the best place to start is again with the books, or see this brief summary. For a mathematical treatment, see Quantum Economics and Finance: An Applied Mathematics Introduction.

[Update] My work in economics has seen me called a number of things including a conspiracy theorist, and the intellectual equivalent of a climate-change denier. More recently an academic physicist read this piece and wrote, in a now-deleted tweet, that I was a charlatan who was ducking and weaving in order to avoid any criticism. I replied that he may have read the post, but he hadn’t understood it. He said “I judge you are not a crank. I judge you are a charlatan.” Then he thought about it (references to names redacted):

Any physicist worth their salt should agree with him that the only test is whether quantum math proves useful in modelling and prediction.

Quantum economics and finance – video series

This series of short videos introduces the key ideas of quantum economics and finance. The only background assumed is basic linear algebra. The material is based on the book Quantum Economics and Finance: An Applied Mathematics Introduction.

Notes:

QEF01 – Introduction to Quantum Economics and Finance

QEF02 – Quantum Probability and Logic

QEF03 – Basics of Quantum Computing

QEF04 – Quantum Cognition

QEF05 – The Quantum Walk

QEF06 – The Penny Flip Game

QEF07 – The Prisoner’s Dilemma

QEF08 – Quantizing Propensity

QEF09 – Threshold Effects in Quantum Economics

QEF10 – A Quantum Option Pricing Model

QEF11 – The Money Bomb

QEF12 – A Quantum Oscillator Model of Stock Markets

QEF01 – Introduction to Quantum Economics and Finance

These are the notes to the first of a series of presentations on quantum economics and finance. For the video version see here.

Quantum economics and finance uses quantum mathematics to model phenomena including cognition financial transactions and the dynamics of money and credit. In these talks we’ll be talking about topics including: why quantum in the first place; quantum probability and logic; basics of quantum computing; quantum cognition ; quantum walk; quantum game theory; quantum supply and demand; threshold effects; option pricing; and the money bomb. 

So why quantum in the first place?  The quantum revolution in physics was born when physicists found that at the subatomic level energy was always exchanged in terms of discrete parcels which they called quanta, from the Latin for “how much”. In economics the equivalent is financial transactions, like when you buy an ice cream in Italy and you say “quanto costa” which makes the quantum connection a little clearer. So money behaves in some ways like an object but not a classical one. It shows signature properties of quantum systems such as discreteness, indeterminacy, entanglement, duality, interference and so on.

Perhaps the most obvious such property is the way that money jumps. In physics Erwin Schrodinger said “If we have to go on with these damned quantum jumps then I’m sorry that I ever got involved” but with financial transactions of course the same thing happens all the time. For example when you tap your card at a store, the money doesn’t flow out continuously, it just jumps.

In physics the position of a particle is fundamentally indeterminate and is in a sense constructed by the measurement procedure. It’s the same thing in markets. If you put your house up for sale you will have a fuzzy idea of the price but the actual monetary value is only determined at the moment of the sale.

And this is money’s job: it’s a way to collapse value down to number. Theorists often talk about money but one thing that isn’t often emphasized is the most basic feature which is is its connection with numbers. If you look at a US dollar bill for example you see it has a numerical 1 and a word “one” in each corner, so it’s got quite a few ones, and then it’s got “one dollar” down at the bottom, and a big “one” in the middle, and there’s a lot more ones on the other side, so they’re really emphasizing the connection with one, and that is money’s most basic property – that it combines the properties of a real owned thing with a virtual number.

These dual real/virtual properties are reflected in the two main historical theories of money which are bullionism – money is gold and nothing else as JP Morgan said – and chartalism, which is the idea that credit alone is money as Alfred Mitchell-Innes said the next year. But then Bitcoin comes along and on the one hand it seems to be completely virtual, but on the other hand it’s also real as you’ll notice if you happen to lose the hard drive in which your bitcoins are located.

The duality of money is therefore similar to the duality of light. Wave-particle complementarity has been reflected in theories of light that go back millennia – Aristotle thought light was a wave, and Newton thought it was particles, and this bounced back and forth until finally the quantum theory came along and showed that it has properties of both at the same time. It’s the same with money.

In economics we’re used to treating preferences as something like fixed and known objects, adjusted for some cognitive biases, but often our preferences are made up in response to questions which act like a kind of a measurement event. So thoughts and ideas behave in some ways like objects but they’re not classical objects. In physics, Bohr’s theory of wave particle complementarity was actually inspired by the observation from psychologists that we can hold opposite ideas in the mind at the same time in superposition, and in fact it’s these interference terms which play a very important role in quantum cognition as we’ll see.

In physics, particles can mysteriously become entangled so they act as a single system. In the financial system there is a much more direct form of entanglement where financial assets and virtual liabilities have these quantum characteristics of entanglement.

In economics there’s this idea of rational economic man, who is like a kind of robot. The picture which is emerging from quantum social science is a quantum economic person who’s entangled, indeterminate, dynamic, paradoxical and alive. As the philosopher Slavoy Zizek said, a fact rarely noticed is that quantum physics appears to defy our common sense view of material reality, but it seems to apply somewhat better to human reality where the human spirit encounters itself outside itself.

Further reading:

Orrell D and Chlupatý R (2016) The Evolution of Money. New York: Columbia University Press.

Orrell D (2016) A quantum theory of money and value. Economic Thought 5(2): 19-36.

Orrell D (2020) The value of value: A quantum approach to economics, security and international relations. Security Dialogue 51(5): 482–498.

Next: QEF02 – Quantum Probability and Logic

Playlist: Quantum Economics and Finance

QEF02 – Quantum Probability and Logic

What does it mean when we say we want to quantize economics? Well, it doesn’t mean that we’re emulating (or somehow abusing) quantum physics. Quantum mathematics is about information and probabilities and observables and how they relate to each other, as Scott Aaronson said, and money is a form of information that does not behave like a classical object.

Now when we try to apply quantum ideas to areas outside of physics we encounter a lot of obstacles. One is that we’ve constantly been told by famous physicists and mathematicians that quantum mechanics is fundamentally incomprehensible, so we’ll never really understand it. We have these weird phenomena like Schrodinger’s cat which can be alive or dead at the same time. We’re often told that concepts such as superposition and entanglement only apply to the tiniest quantum particles and you’ll never encounter them in your everyday life. And then finally there’s this impenetrable mathematics. But let’s just forget physics for the time being, and think of something much simpler like a coin toss.

If we wanted to model the state of a coin toss where we don’t know the outcome, there are two possible outcomes heads or tails, so we would want a two-dimensional space. And we could represent the state of the coin, if it was a balanced coin, with a diagonal line (see screenshot of video below).

That balance between the two states is really the idea of quantum probability. We can represent heads as an up arrow in Dirac notation, which is used in quantum mechanics, or as a vector (1,0)^T. Tails can be a down arrow or a vector (0,1)^T.  Our superposed state is a mix of these two, a balanced combination of up and down.

In order to get the probabilities we’ll just take the projections. The projection on the horizontal axis, when squared, is going to give the probability of heads, and the probability of tails is going to be the square of the projection onto the vertical axis. The reason we are squaring them is because we want the projections to always add to one, and by the Pythagorean theorem that will be the case.

So the basic difference between classical and quantum probability is that classical probability uses what’s called the 1-norm, so the options are heads or tails, while quantum uses this 2-norm which involves the square. Making this change from a 1-norm to a 2-norm leads to all these different quantum concepts such as superposition, negative probability, interference, and entanglement.

So what do we mean by negative probability. Well if we just flip our state over, mirroring it around the vertical axis, then now we’ve got a negative probability for heads. The size of the probability when we take the norm, because we’re squaring it, is obviously positive again so nothing has really changed, except that when we add probabilities together we can get a plus and a minus canceling out in what are called interference effects.

We can also think of more complicated systems. Imagine for example that we had two coins, so now there are four different things that we need to keep track of: there’s heads and tails for one coin, and heads and tells for the other coin. That’s four dimensions which is obviously not easy to draw. But quantum coins can be entangled so for example you might have the only possibilities as heads-heads or tails-tails. So what this means is that just by making this switch from a one norm to a two norm we’re allowing all of these quantum phenomena such as superposition, interference, entanglement and so on.

Further reading:

Aaronson S (2013) Quantum Computing Since Democritus. Cambridge: Cambridge University Press.

Previous: QEF01 – Introduction to Quantum Economics and Finance

Next: QEF03 – Basics of Quantum Computing

Playlist: Quantum Economics and Finance

QEF03 – Basics of Quantum Computing

The approach which we’ll be following is based more on quantum computing than on quantum mechanics per se, so let’s have a look at how these quantum computers work.

Quantum computers perform their operations using qubits which are in a superposed state, so one big difference between them and classical computers is that they’re inherently probabilistic – you have to run a simulation many times and then sample the output .

The basic mathematical tool we’ll be using is the Hilbert space. This can be viewed as a generalization of Euclidean space, with the difference that there are complex coefficients. Quantum probability involves negative probabilities (in the sense of projections) so we need complex numbers for example when we take the square root of a probability.

The dual state is the complex conjugate of the transpose of , also called the Hermitian conjugate, which is written as .

The inner product between two elements and is denoted , and is analogous to the dot product in a normal vector space, with the difference that the result can again be complex. The norm of is given by which is a non-negative real number.

The outer product is denoted , and is like multiplying a column vector by a row vector.

A unitary matrix is one whose inverse is its conjugate transpose. Unitary matrices preserve probabilities and play a big role in quantum computing, where logic gates are represented by unitary matrices that act on qubits. An example of a gate is the NOT gate which has the effect of flipping a qubit, so if the input is then the output will be . There’s the Hadamard gate which we’ve already seen. It takes a an input of say and puts it into a superposed state. There’s a rotation gate which simply rotates the qubit by an angle , so if the input is then the output will be a superposed state which has the probability of being measured in and of being measured in . The symbol for measurement is the gauge symbol.

Multiple qubits are denoted as a tensor product , so two qubits gives you a column vector with four elements.

A state in a Hilbert space is entangled if it does not factor as a tensor product of the form where and . For example , which has an equal probability of being observed in the state or , is not entangled because it can be written as a tensor product; however the state , which has an equal probability of being observed in the state or , cannot be similarly decomposed, so is entangled.

Gates can act on multiple qubits, an example is the C-NOT gate which here flips the state of the lower qubit depending on the state of the upper qubit: , , , .

C-NOT gate

The Toffoli gate on the left is similar but acts on three qubits: it flips the state of the first qubit depending on the state of the two control qubits. The figure below shows a quantum circuit with three qubits. The first gate is a Toffoli gate, the middle gate is a C-NOT gate which acts on two qubits, and the last gate on the right is a NOT gate which flips the bottom qubit.

A quantum circuit to increment a counter

If we analyse the effect of this circuit, we find that , , and so on. The circuit therefore has the effect of incrementing a binary counter.

Another circuit that we’re going to be using quite a lot is the two-qubit figure one shown in the video screenshot below. We have a unitary matrix acting on the first qubit which might for example rotate that qubit around in a particular direction, and the same thing for the lower qubit. Then the top qubit is going to be acting as a control on the second one. As we’ll see this circuit can be used in quantum cognition, where the top qubit can represent a context for a decision, or also in quantum finance where we’re going to be using this to model the debt relationship.

Previous: QEF02 – Quantum Probability and Logic

Next: QEF04 – Quantum Cognition

Playlist: Quantum Economics and Finance

QEF04 – Quantum Cognition

Now that we’ve got most of the mathematical and computational background out of the way, let’s get on to some applications in quantum cognition. Quantum methods were first adopted here because there are problems in behavioral economics which are not easily handled using classical logic, and it’s it’s easier to address them using quantum methods because you can use features such as interference and entanglement.

In quantum cognition we’re going to be modeling mental states using what amount of qubits. So imagine if we started off with an initialized qubit , and then we’re going to act on it by a gate which puts it into a certain state, and then we measure the result which will be a or . This could represent for example different outcomes or decisions.

One of the first applications of quantum cognition was to the order effect. This refers to the phenomenon seen with surveys where the response to questions depends very much on the order in which the questions are asked. One example was a survey done back in the 90s of whether Clinton and Gore were trustworthy and it turned out that the answer was sensitive to the order of the questions. This order effect can be modeled using the quantum formalism as a sequence of projections.

The order effect

Here the main horizontal and vertical axes correspond to the frame for addressing the Clinton question, and then the the dashed lines represent the Gore axes which are at an angle to that. The initial state shown by the grey line is at angle of about 40 degrees so it’s roughly equally balanced in the Clinton axis for saying Yes or No. So let’s say that the response to the question is Yes, Clinton is trustworthy. Then that is then used as the starting point for the next question about Gore (dotted line) which gives one end point. But if the order of the question is reversed then you project first on to the Gore axis, and then onto the Clinton axis, and you get a different result. The reason is that there’s a kind of interference caused by the shift in the mental frame.

If we go through the exercise of calculating all the different probabilities for the case where the order is Clinton and then Gore we find the table of probabilities looks like this:

Results with Clinton question followed by Gore question

We can get exactly the same result if we use the circuit below, which we have already seen. Here we’ve got a rotation gate which in this case rotates by which is preparing our state, and then we’ve got a second qubit which is similarly rotated by the angle which represents this relative shift in mental frame.

More generally this same circuit can be used to simulate any decision B which is influenced by a context A. Some examples include preference reversal where we change our mind depending on the context, the endowment effect where we value something more if we own it than if we don’t own it, and the disjunction effect.

The disjunction effect goes back to a 1992 experiment from Tversky and Shafir. They asked students to imagine that they have a tough exam coming up, and they have an opportunity to buy a vacation to Hawaii at a very good price. Would they take the offer?

In one version of the test they were told the result of the exam. If the result was a pass then 54 percent chose to buy, if the result was fail 57 percent chose to buy. So in each case more than half. But then there was another version in which they were told they will not know the result, and in this case only 32 percent chose to buy. This is odd because the outcomes can only be pass or fail, so you’d expect it to be close to the average or about 55 and a half percent, but no only 32 percent chose to buy. So this is an example of some kind of mental interference effect, where the reasons effectively cancel out.

One way to model this using the quantum method is to do something similar to the dual slit experiment in physics (see video image below), where light from a source gets split into two channels and forms an interference pattern when it recombines. In this case we take the test A which we can either pass (A+) or fail (A-). Then we have the decision to buy a vacation and again it can either split to B+ or B-. The setup is therefore similar to the order effect, where A plays the role of a context (the first question) and B represents the final decision, and again it can be represented using the same simple two-qubit circuit above.

In general, if we suppose that A represents a subjective context, and B represents an objective term such as a numerical payoff, then if we rate the overall attractiveness on a scale 0 to 1, and assume a uniform prior for the various probability terms, it is easily seen that the interference between the subjective and objective factors has an expected value equal to a quarter. Yukalov and Sornette (2015) call this result the quarter law. In the case of the disjunction effect, the interference is negative – if the person knows the outcome, more than half buy the vacation (average 55 percent), but if they are uncertain, this reduces to 32% (a reduction of 23% or about a quarter).

Another application of quantum cognition is to the question of debt. The state of a debt depends on whether or not the debtor is going to default. If the debtor is going to default the debt is worth nothing, if the debtor is not going to default for sure then that debt is worth its face value. An early example of a debt-based form of money was the tallies which were used in England in the middle ages. Suppose that the sovereign wanted to collect a tax debt. A tally stick would then be marked with the the value of the debt, and split down the middle. The sovereign would keep the longer version of the stick which was called the stock and they would hand the debtor the shorter piece of wood which was the foil. When the debt was repaid in the form of produce or whatever then the two sides of the stick were matched and destroyed to extinguish the debt.

We can model this using a version of the same two-qubit circuit. A NOT gate flips the first qubit to symbolize the creation of a debt. For the the debtor we can use the Hadamard gate for simplicity, which puts the qubit into a superposed state . The C-NOT gate has the debtor acting as a control. The outcomes are which is entangled. The debt is either in the state or which means one of these people is going to have the money, the debtor or the creditor, with a 50-50 chance.

Because the tallies represented a claim on a debt, that meant that they had monetary value and could circulate as money objects. So what we think of as a cognitive phenomenon – the decision on whether to default or not – is ultimately what creates the value for money. The sovereign’s job is to convince the debtor that they must not default on the debt, and as we’ll discuss later that involves a certain kind of a work or kind of energy which is really what forms the basis for money.

Further reading:

Busemeyer J and Bruza P (2012) Quantum Models of Cognition and Decision. Cambridge: Cambridge University Press.

Orrell D (2020) Quantum Economics and Finance: An Applied Mathematics Introduction. UK: Panda Ohana.

Wang Z, Solloway T, Shiffrin RS and Busemeyer JR (2014) Context effects produced by question orders reveal quantum nature of human judgments. Proceedings of the National Academy of Sciences 111(26): 9431–6.

Wendt A (2015) Quantum Mind and Social Science: Unifying Physical and Social Ontology. Cambridge: Cambridge University Press.

Yukalov VI and Sornette D (2015) Preference reversal in quantum decision theory. Frontiers in Psychology 6: 1–7.

For an online app which demonstrates the order effect, see here.

Previous: QEF03 – Basics of Quantum Computing

Next: QEF05 – The Quantum Walk

Playlist: Quantum Economics and Finance

QEF05 – The Quantum Walk

One algorithm which is used a lot in quantum computing and also in quantum cognition is the quantum walk, which is a quantum version of the classical random walk model.

The classical random walk was described by the statistician Karl Pearson in a 1905 paper using the example of a drunken man, who takes a step in one direction, another step in a different direction, and so on. The expected distance traveled is seen to grow with the square root of time but “the most probable place to find a drunken man who is at all capable of keeping on his feet is somewhere near his starting point.”

The same idea had actually already been used by Louis Bachelier in his 1900 thesis on option pricing to argue that an investor’s expected profit or loss was zero because prices move randomly up or down, but the best forecast for an assets future price is its current price. The classical random walk can be modeled using a coin toss, so you toss the coin at each time step and move to the left if the coin is tails and move to the right if it’s heads up. If you repeat this many times, what you find is that many paths end up near the center, but the only way to get to one of the extremes is to get the same result at each coin toss (all heads or all tails) which is very unlikely. So the final probability distribution for position will be concentrated near the middle, and converges to a normal distribution.

The random walk was first used by Bachelier to price options, then it was used in nuclear physics, and then it fed back from there into finance again, and now it’s ubiquitous in finance and it forms the core of all the models used to calculate things like the price of options and financial derivatives in general.

There is a quantum circuit which does the quantum version of a random walk. It consists of a gate we have already seen which increments a counter (interpreted as a step to the right), another which decrements a counter (a step to the left), and in addition there is a Hadamard gate which puts the bottom control qubit into a superposition state. The empty control circle means the control is active when the qubit is in the state rather than .

The quantum walk circuit

The part of this lower qubit which is in the state is going to increment the counter, the part which is in is going to decrement it. The result, when repeated in a loop, is a wave function which is evolving over time. The only random part comes at the end when we collapse the wave function down during measurement. The quantum walk explores numerous different paths (see screenshot from video at end of post below) and when these paths meet somewhere they don’t have to add together, instead they can subtract from one another because of interference. The resulting distribution is quite interesting, and it’s very different from the the classical one as seen in the figure below.

Quantum walk versus normal distribution

The classical random walk (dashed line) converges as mentioned to the normal distribution, while the quantum walk (solid line) has these strange peaks on either side and is actually quite low in the middle because of the interference. This is interesting because in traditional finance theory we’re always thinking about everything being normally distributed, so prices stay close to their starting point. But while this might be objectively fairly accurate, when you think about the mental frame of investors it’s quite different. Most people buying options on a stock aren’t thinking about that stock staying at almost the same price as it is now, instead they’re going to be thinking about it either increasing by a certain amount per year or decreasing by a certain amount per year kind. They might be biased towards one picture or the other but they’re also going to be holding the opposite picture in their head at the same time.

Another difference between the classical and quantum walks is that the latter grows much more quickly, linearly in time rather than with the square-root of time. This feature of the quantum walk is one reason why it’s used a lot in quantum computing for things like search algorithms. Now we can make the the quantum walk act in a more classical fashion if we want by adding decoherence, as occurs when you get some random noise. Decoherence is the enemy of quantum computers because it makes the quantum computers behave in a more classical fashion. A big part of the challenge of quantum computing is to isolate the qubits from their environment so that they don’t get this decoherence. If we want, we can add some decoherence to a quantum walk model quite simply in the algorithm just by adding some random noise at each time step, and if enough is added then the quantum walk converges to the normal distribution. But is this really what we want or do we want to be able to exploit these quantum properties in some way?

In quantum cognition the quantum walk model has been used to model various cognitive effects including signal detection, how people assign ratings to stimuli, and general decision making. In terms of neural processes you can think of it as modeling a kind of a parallel cognitive architecture that involves both cooperative and competitive interactions, which results in interference effects. So as a model of how we think about the future it seems like an improvement over the classical random walk model, which is really more a model of something like a dust particle which doesn’t think about the future. Later on we’re going to be using the quantum walk to model subjective beliefs about future asset price changes as opposed to the actual observed changes which again are not the same thing.

Further reading:

Kempe J (2003) Quantum random walk – an introductory overview. Contemporary Physics 44: 307.

Kvam PD, Pleskac TJ, Yu S and Busemeyer JR (2015) Quantum interference in evidence accumulation. Proceedings of the National Academy of Sciences 112 (34): 10645-10650.

Previous: QEF04 – Quantum Cognition

Next: QEF06 – The Penny Flip Game

Playlist: Quantum Economics and Finance

QEF06 – The Penny Flip Game

Game theory is very important in economics and it’s interesting to ask how games change when they’re played using quantum logic instead of classical logic.

Quantum game theory seems to have started back in 1999 when quantum computing was still in its infancy. There are two games that we’re going to talk about, one is the prisoner’s dilemma which will be the subject of the next segment, and the first one is the penny flip game. This is an extremely simple game where we have two players A and B. Player A starts by positioning a coin in the up state and player B can choose to flip the coin or not without seeing. Then player A can choose to flip the coin or not without B seeing. Player B then chooses to flip the coin or not and if the coin ends heads up then B wins, otherwise A wins.

We’ll denote the up/down states of the coin as usual , so up will be or (1,0)^T and down will be or (0,1)^T. The choices to flip or not flip a coin can then be represented by the NOT gate which is flip and the identity I which keeps things the same. A quantum circuit for this game would look like this where we’ve got the various different moves, and measure the final outcome to see whether it’s heads up or tails up.

Quantum circuit for the penny flip game

When you play the game with random moves you find that half the time the result is going to be up, so B wins, and half the time the result is going to be down, so A wins. Each player should therefore win 50 percent of the time. Suppose though that after playing a number of games, player B wins every time.

The situation is a bit like a trick performed by magician Darren Brown, where he takes somebody from the audience onto the stage, and that person holds a a coin behind their back and then holds both hands out in front, and Darren Brown has to guess which hand is holding the coin, and he does this several times in a row. What is going on?

In his case of course the answer is magic, but for our case it’s that player B is cheating by applying the Hadamard transformation (see video screenshot below). This puts the coin in a superposed state of up and down. Whether the coin is then flipped or not flipped by A has no effect on the superposed date. Player B then gets to apply the Hadamard transformation again at the end because they get the last move, and that has the effect of always putting the coin back in the up state. The answer is therefore always up and B wins.

A classical analogy of this would be that player B turns a normal coin by 90 degrees so it’s on its edge. If player A flips the coin or not it is still going to remain on its edge, and then player B turns it by 90 degrees again so it is face up and and B wins the game. Of course in the classical version a coin on its edge has a 50-50 chance of falling either way but the quantum coin can exist in a superposed state. As David Meyer, who invented this game back in 1999, pointed out quantum strategies can be more successful than classical ones and the reason quantum computers promise to have vastly stronger computational power than classical computers is because they can they can play these quantum tricks, and do moves which are simply not possible using classical computers. It is also exactly these moves that seem to play such a key role in human cognition.

Further reading:

Meyer DA (1999) Quantum Strategies. Physical Review Letters 82, 1052.

Previous: QEF05 – The Quantum Walk

Next: QEF07 – The Prisoner’s Dilemma

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