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 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.

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.

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