Author: David Orrell, Author and Economist

March 16, 2021

In 2019, Google announced that they had achieved ‘quantum supremacy’ by showing they could run a particular task much faster on their quantum device than on any classical computer. Research teams around the world are competing to find the first real-world applications and finance is at the very top of this list.

However, quantum computing may do more than change the way that quantitative analysts run their algorithms. It may also profoundly alter our perception of the financial system, and the economy in general. The reason for this is that classical and quantum computers handle probability in a different way.

**The quantum coin**

In classical probability, a statement can be either true or false, but not both at the same time. In mathematics-speak, the rule for determining the size of some quantity is called the norm. In classical probability, the norm, denoted the 1-norm, is just the magnitude. If the probability is 0.5, then that is the size.

The next-simplest norm, known as the 2-norm, works for a pair of numbers, and is the square root of the sum of squares. The 2-norm therefore corresponds to the distance between two points on a 2-dimensional plane, instead of a 1-dimensional line, hence the name. Since mathematicians love to extend a theory, a natural question to ask is what rules for probability would look like if they were based on this 2-norm.

It is only in the final step, when we take the magnitude into account, that negative probabilities are forced to become positive

For one thing, we could denote the state of something like a coin toss by a 2-D diagonal ray of length 1. The probability of heads is given by the square of the horizontal extent, while the probability of tails is given by the square of the vertical extent. By the Pythagorean theorem, the sum of these two numbers equals 1, as expected for a probability. If the coin is perfectly balanced, then the line should be at 45 degrees, so the chances of getting a heads or tails are identical. When we toss the coin and observe the outcome, the ambiguous state “collapses” to either heads or tails.

Because the norm of a quantum probability depends on the square, one could also imagine cases where the probabilities were negative. In classical probability, negative probabilities don’t make sense: if a forecaster announced a negative 30 percent chance of rain tomorrow, we would think they were crazy. However, in a 2-norm, there is nothing to prevent negative probabilities occurring. It is only in the final step, when we take the magnitude into account, that negative probabilities are forced to become positive. If we’re going to allow negative numbers, then for mathematical consistency we should also permit complex numbers, which involve the square root of negative one. Now it’s possible we’ll end up with a complex number for a probability; however the 2-norm of a complex number is a positive number (or zero). To summarise, classical probability is the simplest kind of probability, which is based on the 1-norm and involves positive numbers. The next-simplest kind of probability uses the 2-norm, and includes complex numbers. This kind of probability is called quantum probability.

**Quantum logic**

In a classical computer, a bit can take the value of 0 or 1. In a quantum computer, the state is represented by a qubit, which in mathematical terms describes a ray of length 1. Only when the qubit is measured does it give a 0 or 1. But prior to measurement, a quantum computer can work in the superposed state, which is what makes them so powerful.

So what does this have to do with finance? Well, it turns out that quantum algorithms behave in a very different way from their classical counterparts. For example, many of the algorithms used by quantitative analysts are based on the concept of a random walk. This assumes that the price of an asset such as a stock varies in a random way, taking a random step up or down at each time step. It turns out that the magnitude of the expected change increases with the square-root of time.

Quantum computing has its own version of the random walk, which is known as the quantum walk. One difference is the expected magnitude of change, which grows much faster (linearly with time). This feature matches the way that most people think about financial markets. After all, if we think a stock will go up by eight percent in a year then we will probably extend that into the future as well, so the next year it will grow by another eight percent. We don’t think in square-roots.

This is just one way in which quantum models seem a better fit to human thought processes than classical ones. The field of quantum cognition shows that many of what behavioural economists call ‘paradoxes’ of human decision-making actually make perfect sense when we switch to quantum probability. Once quantum computers become established in finance, expect quantum algorithms to get more attention, not for their ability to improve processing times, but because they are a better match for human behaviour.

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