What is a quantum random walk?

The previous note showed that the random walk hypothesis for prices holds up well for direction and fails for the distribution of the size of moves. This note turns to one of the candidates being studied as an alternative: the quantum random walk. Despite the name, there is no need to start with quantum computers. The aim is to see what changes when a single element of the classical random walk is replaced.

The classical random walk

A walker stands on a number line. At each step a coin is tossed: heads, one step right; tails, one step left. After t steps, the walker’s position follows a binomial distribution, which approaches a normal distribution as t grows.

Two points matter here.

• Each outcome is described by a probability, and probabilities are added together.
• The spread of the distribution (its standard deviation) grows in proportion to √t. After 100 steps the walker is typically only about 10 steps from the start.

This is the skeleton of the random walk model of prices discussed in the previous note.

The quantum random walk: what is replaced

The quantum random walk was proposed by Aharonov, Davidovich and Zagury in 1993 (1); its present form, with no measurement along the way, was developed in later work (2). The version described here is the most widely studied one, the discrete-time quantum random walk.

The walker’s state has two parts: a position and a coin. The coin has two states, heads and tails, but in quantum mechanics it can also be in a superposition of the two. The state is described not by probabilities but by complex-valued probability amplitudes.

Each step consists of two operations.

1. The coin operation. A fixed unitary matrix is applied to the coin state. The standard choice is the Hadamard matrix, which takes heads to an equal superposition of heads and tails, and tails to a superposition of the two with opposite signs.
2. The conditional shift. The heads component moves one step right and the tails component one step left.

The difference from the classical walk is that the coin is never looked at along the way. The outcome of each step is not fixed; the walk proceeds as amplitudes. Only after t steps is the position measured, and the squared modulus of the amplitude at each position gives the probability of finding the walker there.

Interference changes the shape of the distribution

Because nothing is measured along the way, the amplitudes of the different paths leading to a given position are added together. Amplitudes can be positive or negative (and in general complex), so some positions see paths reinforce one another and others see them cancel. This is interference.

As a result, the distribution of a quantum random walk with the Hadamard coin looks entirely different from the classical one.

• Thin in the middle, peaked at the edges. Cancellation is common near the origin, and the probability concentrates around ±t/√2 from the starting point.
• Spread grows with t. The standard deviation grows in proportion to t rather than √t. This kind of spreading is called ballistic, as opposed to diffusive.
• The support is finite. After t steps the walker can only be between −t and t; the probability outside that range is exactly zero. Konno further showed that, as t grows, the distribution of the position divided by t approaches a definite distribution on the interval from −1/√2 to 1/√2 (3, 4). For large t, then, almost all the probability lies within ±t/√2. From a symmetric initial state, the density of this limiting distribution grows without bound towards the two ends of the interval (while the total probability near the ends remains finite). The two peaks are this limiting shape showing through.
• The initial state sets the bias. Depending on the coin’s initial state, the distribution can lean to the left or the right; starting from a suitable superposition makes it symmetric.

These properties are examined in detail in the 2001 analysis by Ambainis and colleagues (2) and in Kempe’s 2003 introductory review (5).

Without interference, the walk turns classical

Every distinctive property of the quantum random walk comes from interference. In 2004 Romanelli and colleagues (6) split the time evolution of the quantum random walk into a term corresponding to a classical Markov process and a term representing interference. The variance grows as t² because of the interference term; when interference is lost (decoherence) — for instance because a measurement is made at every step — the variance returns to growing in proportion to t, as in a classical random walk.

Every distinctive property of the quantum random walk can therefore be traced to a single added ingredient: interference.

The continuous-time quantum random walk

There is also a continuous-time quantum random walk with no coin. Farhi and Gutmann formulated it in 1998 (7) as a quantum system evolving in continuous time on a graph. Its mathematical construction differs from the discrete-time version, but it shares many properties, including ballistic spreading (5). Both are used in research on quantum algorithms.

No quantum computer required

The quantum random walk grew up within research on quantum computation. But studying a quantum random walk on a line does not require a quantum computer. The state after t steps is just a list of two complex numbers per position — the amplitudes for heads and tails — and an ordinary computer can calculate it exactly.

This will matter again when a later note sets out what “quantum-inspired” means. Neither of the two studies introduced below shows a speed-up on a quantum computer; both centre on using the mathematical formalism of quantum mechanics as a model of probability distributions. That the models can run directly on a quantum computer is listed by Orrell as one of their advantages and is also mentioned by De Backer and colleagues, but both models can be computed on an ordinary computer.

Research applying it to finance

There is not yet much research applying quantum random walks to finance. Two studies taking different directions are outlined here.

A model of option prices (Orrell, 2021)

David Orrell (8) proposed replacing the binomial model — an option-pricing model built on the classical random walk — with a quantum random walk. Investors’ subjective expectations about future prices are represented as a quantum random walk before measurement, and the prices actually observed in the market are read as the state fixed by measurement. The interpretation borrows from “quantum cognition”, an approach studied in the psychology of decision-making.

In Orrell’s model, adding decoherence brings the results arbitrarily close to the classical model, and decoherence is also read as representing the role of external information.

A model of return distributions (De Backer et al., 2025)

De Backer and colleagues (9) examined the distributions produced by quantum random walks as models of return distributions, comparing them with classical models: geometric Brownian motion, Lévy stable distributions and power-law distributions.

By varying parameters such as the angle and phase of the coin matrix and the initial coin state, the quantum random walk can produce asymmetric distributions. The paper also shows that, for some settings computed over a limited number of steps (100 in the figures), distributions with a higher probability of extreme values than a normal distribution (geometric Brownian motion) are obtained. The main claim is that it has the flexibility to express distributions that classical models struggle to produce.

“A higher probability of extreme values” here refers to a comparison with a classical random walk of the same number of steps (close to a normal distribution), not to a comparison of tail shape at equal variance. Because a quantum random walk spreads in proportion to t, comparing at the same number of steps raises the probability far from the centre partly through the difference in spreading alone. For example, in the most basic case — the symmetric Hadamard walk with no measurement along the way — rescaling to the same standard deviation and comparing with a normal distribution shows, by calculation from Konno’s limiting distribution, that for large t almost all the probability lies within about 1.3 standard deviations of the centre, so the tails are if anything shorter (4). The distributions the paper shows to have a higher probability of extreme values come from settings other than this basic form, but in every setting the distribution after t steps has no probability outside −t to t. The tails seen in the previous note are well approximated by a power law and decline slowly towards large values. The peaks at the edges are a concentration of probability produced by interference, not power-law tails. Whether quantum random walks can account for the power-law tails of actual returns is not addressed in the paper.

The authors are also clear about two limits. One is that the flexibility comes at the cost of additional parameters. The other is that the paper examines theoretical potential, and that testing performance against actual price data is left to future work. They also state explicitly that using methods from quantum mechanics does not imply that markets are quantum systems.

Summary

• A quantum random walk replaces the coin toss of a classical random walk with a quantum-mechanical operation and proceeds as amplitudes without measurement along the way.
• Interference between paths produces a distribution peaked at the edges that spreads in proportion to t rather than √t.
• The support of the distribution is finite (−t to t); the mechanism differs from that of power-law tails.
• When interference is lost, the walk returns to the classical random walk.
• A quantum random walk on a line can be computed on an ordinary computer. The two applied studies seen here use the mathematics of quantum mechanics as a model of probability distributions.
• Research applying it to finance is at an early stage, and testing against real data is still to come.

The next note divides finance research that carries the word “quantum” into three kinds, and sets out what “quantum-inspired” does and does not refer to.