What happens to the Kyle model when the asset's value follows a quantum random walk? Reading Liew and Rebentrost (2026)

This note reviews “Informed trading model with quantum walks” by H. W. Shawn Liew and Patrick Rebentrost (1), a preprint posted to arXiv on 5 October 2026 that has not yet been peer-reviewed. Both authors are at the Centre for Quantum Technologies, National University of Singapore; Rebentrost is also in the university’s School of Computing.

The paper works inside Kyle’s 1985 model (2), a standard reference point in the theory of market microstructure. It places the quantum random walk inside an equilibrium model, in which the outcome is fixed by participants anticipating one another’s behaviour.

The “insider” or “informed trader” of the model is a theoretical role: a participant who holds information the others do not. Nothing here concerns the legal treatment of insider dealing.

Why this paper

The studies of quantum random walks in finance covered in earlier notes have mainly asked what distribution prices or returns follow: De Backer and colleagues use the quantum random walk as a model of return distributions (3), and Orrell as a model of option prices (4). This paper asks a different question — how the orders of an informed trader are absorbed into prices, that is, price formation and market depth. Among the studies covered in these notes, in the view of the author of this note, it can be read as an example of the questions put to quantum random walks in finance widening from the description of distributions to the mechanics of price formation.

A second reason is that the design is easy to read. Within the Kyle model, the paper swaps only the stochastic process that supplies the future value, between a classical and a quantum random walk, and keeps the rest of the model in common (the numerical settings differ between the two). The author of this note therefore considers it easier to attribute differences in the results to the swap — although, as the section “A reading: separating width from shape” shows, those differences mix effects of the distribution’s width with effects of its shape (as well as the question of how lambda is measured in the quantum case), and the two need to be read separately.

At the same time, the paper is a preprint that has not been peer-reviewed, its computations cover only t ≤ 23, and it uses no market data. This note therefore presents it not as a claim that quantum random walks explain markets, but as a study of the properties of a model.

What the Kyle model is for

Kyle opens his paper with a set of questions. How quickly is new private information incorporated into prices? How valuable is private information to the person who holds it? How does noise trading affect price volatility? What determines the liquidity of a market (2)? His approach is to derive the answers from a model of price formation in which the trading strategy of an informed participant is optimised explicitly.

There are three kinds of participant:

• a single risk-neutral informed trader (Kyle’s “insider”), who alone knows the asset’s future (liquidation) value;
• noise traders, who buy and sell random quantities unrelated to that information;
• competitive, risk-neutral market makers, who see only the combined quantity ordered (the order flow) and set the price.

Each round of trading (an “auction”, in Kyle’s terms) has two steps. First, the informed trader and the noise traders choose their order sizes simultaneously. Then the market makers observe the total and set the price at which all orders are filled. Because market makers cannot tell the informed order from the noise, noise trading acts as camouflage. Competition between market makers drives the price to the expected future value conditional on the order flow, so that market makers break even on average (2). The informed trader, knowing that the order will itself move the price, chooses the size that maximises expected profit.

When the future value and the noise-trading quantity are both normally distributed, Kyle derives an equilibrium in which the informed trader’s strategy and the market makers’ pricing rule are both linear, and shows that it is the only linear equilibrium (2). The price takes the form “initial price + λ × order flow”. The coefficient λ measures how far a given order moves the price — its price impact — and is known as Kyle’s lambda (1). It equals half the ratio of the standard deviation of the future value to that of noise trading. Its reciprocal, 1/λ, is the order flow needed to move the price by one unit: the depth of the market. In this equilibrium half of the insider’s private information is incorporated into the price. The insider’s expected profit is half the product of the two standard deviations, and, as Kyle puts it, is proportional to each of them (2). This last property matters later.

Kyle goes on to a sequential version with many auctions and to its continuous-trading limit. In the continuous equilibrium prices follow Brownian motion, market depth is constant over time, and all private information is in prices by the end of trading (2). The paper reviewed here uses only the single-auction version.

The linearity of the equilibrium rests on normality. Kyle’s proof states explicitly that it is normality that makes the regression of the future value on the order flow linear (2). Replace the normal distribution of the future value with something else and there is no guarantee of a linear equilibrium. That is exactly the assumption this paper changes.

The continuous-time quantum random walk

An earlier note dealt mainly with the discrete-time quantum random walk, which uses a coin, and mentioned the coinless continuous-time version only briefly. The paper uses the continuous-time version.

Farhi and Gutmann formulated it in 1998 as a quantum system evolving in continuous time on a graph (5). There is no coin: the state evolves under an operator (a Hamiltonian) that moves probability amplitude between neighbouring vertices. Started at the origin of the integer line, the amplitude at each position at time t can be written in closed form with Bessel functions of the first kind, and the probability of each position is the square of the corresponding Bessel function. The paper derives this from first principles (1).

Its behaviour resembles the discrete-time walk.

• The spread grows in proportion to t. The standard deviation of a classical random walk grows as √t; that of the quantum random walk grows as t. This is called ballistic, as opposed to diffusive, spreading, and is treated in detail for the discrete-time walk in (6). For the continuous-time walk, Farhi and Gutmann showed that the amplitudes are Bessel functions that spread at a constant speed and fall off rapidly beyond the front (5). Under the normalisation of the Hamiltonian used in the paper (1), the recurrence relation and sum rules for Bessel functions (DLMF 10.6.1, 10.23.3 and 10.23.4 (7)) give a mean squared position of exactly t²/2 (the calculation is this note’s own), so the standard deviation of the position is t/√2.
• Probability gathers near both edges. In the paper’s figure the probability is concentrated near the two outer edges of the distribution and oscillates from site to site in between (1).
• The support is not finite. In the discrete-time walk, no probability lies outside −t to t after t steps. In the continuous-time walk, at the integer values of t that the paper computes, no position has zero probability (1) (at non-integer t there are moments when a position’s probability vanishes, at the zeros of the Bessel functions (DLMF §10.21 (7))). Beyond the front the probability falls off rapidly (5). The paper truncates the line to a finite range for the sake of numerical convergence, and reports that the range it arrived at contained more than 1 − 10⁻⁸ of the probability (1) (see below).

As the earlier note explained, these properties come from the interference of probability amplitudes. The position of a classical random walk, by contrast, is binomially distributed and approaches a normal distribution as t grows. The paper asks what changes in the Kyle model when one of these distributions is swapped for the other as the distribution of the asset’s future value.

What the paper replaces and computes

Set-up

The candidate future values sit on a lattice with fixed spacing s: initial price + s × j, for integer j. The probability that the future value takes a given lattice value is set equal to the probability that a random walk is at position j at time t — binomial in the classical case, a squared Bessel function in the quantum case. In the computations, s = 0.1 and the noise-trading quantity is normal with mean 0 and variance 1 (1).

Note that t is neither the number of trades nor a trading time. The model has a single auction throughout. t is a parameter that sets how widely the future value is dispersed before the auction takes place. Changing t mainly plays the role that changing the variance of the future value plays in Kyle’s normal model. The shape of the distribution changes at the same time, however: in the classical case the binomial distribution draws closer to a normal one, and in the quantum case the number of oscillations within the distribution increases.

One analytical step

The authors start with an analytical approximation. If the market maker uses a linear pricing rule, the informed trader’s best order is linear too. Recomputing the market maker’s pricing rule on the basis of that order gives, in the quantum case, a rule that is no longer linear: a ratio of weighted sums of hyperbolic functions. The paper then shows that the slope of this updated rule — the price impact — is proportional to the uncertainty about the future value that remains for the market maker after seeing the order flow (the conditional variance). Its reading is that price impact is greatest where the market maker is least sure of the asset’s value (1).

This is only the first step of an iteration, not the equilibrium itself. The authors note that the analysis quickly becomes intractable and turn to numerical methods.

The numerical computation

An equilibrium is a pair — the informed trader’s strategy and the market maker’s pricing rule — in which each is the best response to the other. The paper treats it as a fixed point and computes it iteratively: from the pricing rule, compute the informed trader’s best order; from that order, recompute the pricing rule; repeat. The iteration is accelerated with Anderson acceleration (8), and the convergence criterion is a residual of at most 10⁻⁴ (1).

For the quantum case, the authors report that the quality of convergence was highly sensitive to the settings of the iteration: the grid, the initial pricing rule, the acceleration parameters and the range over which future values are truncated. They therefore ran 100 trials of settings with the hyperparameter optimisation software Optuna (9) and, among those meeting the convergence criterion, chose the one with the widest truncation range. The truncation range was thus an outcome of this search; for every t, the range selected turned out to contain more than 1 − 10⁻⁸ of the probability (1).

The computations cover integer t from 1 to 23. In the quantum case no equilibrium was found for larger t, and the classical case was restricted to the same range for comparison. The informed trader’s expected profit is the average over 200,000 simulated rounds under the computed equilibrium (1).

All of this is ordinary numerical computation on a conventional computer. Because the distribution of the quantum random walk has a closed form, no quantum computer is needed, and the paper reports no computation on one. “Quantum” here refers to the mathematics that supplies the shape of the future-value distribution.

What the computation shows

The classical random walk

The informed trader’s strategy is close to linear: a straight-line fit gives a coefficient of determination above 0.996 for every t. The pricing rule is close to a straight line for order flow near zero and levels off for large orders, so that overall it resembles a logistic curve. It levels off because a binomial future value lies within ±st of the initial price, and the price cannot leave that range. Kyle’s lambda, measured as the slope near zero order flow, grows roughly as t to the power 0.491 for t from 2 to 23 (t = 1 is an outlier) (1).

The quantum random walk

The informed trader’s strategy breaks into several clusters separated by wide gaps, and the pricing rule shows a corresponding series of jumps. The authors remark that the numerically computed pricing rule bears some resemblance to the one from the analytical step. Kyle’s lambda grows almost in proportion to t (a straight-line fit gives a coefficient of determination of about 0.9999) (1).

For the informed trader’s expected profit — an expectation for a participant inside the model, not a performance figure from any actual market — the paper reports a dependence on t of roughly t to the power 0.69 in the classical case and close to proportional to t in the quantum case (1). The authors’ emphasis is that this dependence is close to linear even though the quantum equilibrium becomes strongly non-linear.

The authors attribute the difference in how lambda grows to the difference between the ballistic spreading of the quantum random walk and the diffusive spreading of the classical one (1).

A reading: separating width from shape

That explanation becomes clearer against Kyle’s normal model. The check that follows is not in the paper; it is this note’s own calculation from the paper’s formulae and reported numbers.

In Kyle’s equilibrium, lambda is half the standard deviation of the future value divided by that of noise trading, and the insider’s expected profit is half the product of the two (2). In the paper’s set-up (lattice spacing 0.1, noise-trading variance 1), the standard deviation of the future value is 0.1 × √t for the classical random walk and 0.1 × t/√2 for the quantum one (see the section on the continuous-time walk). Call the values obtained by inserting these into Kyle’s formulae the normal benchmark. The benchmark lambda is 0.05 × √t in the classical case and about 0.0354 × t in the quantum case.

First, compare the rates of growth — the power of t. The classical fit (t to the power 0.491) is close to √t, and the quantum lambda is proportional to t. The difference in how lambda grows is therefore largely accounted for by the difference in distribution width at the same t, which is consistent with the authors’ explanation.

Next, compare levels. In the classical case the paper’s fitted formula (1) matches the benchmark down to the coefficient (at t = 10, for instance, about 0.161 against a benchmark of about 0.158). In the quantum case, however, the slope of lambda (about 0.0517) is about 1.46 times the benchmark slope (about 0.0354). The slope of expected profit in the quantum case (about 0.0407) — an expectation within the model, not a market performance figure — is likewise about 15% above its benchmark slope (about 0.0354). Per unit of standard deviation, the quantum equilibrium departs from what Kyle’s normal model would lead one to expect, and width alone does not account for the gap. For lambda, two candidate explanations suggest themselves. One is an effect of the distribution’s shape. The other is measurement: the paper does not state over what range of the pricing rule the quantum lambda was taken as a slope, and for a pricing rule with jumps the slope depends on the range (see “Assumptions and limitations” below). The gap in expected profit (about 15%), by contrast, is an average over 200,000 simulated rounds and is not affected by how lambda is measured. Either way, the paper does not allow one to decide whether the gaps are an effect of the distribution’s shape.

The results are therefore best read by separating what is due to width from what is due to shape. The growth rate of lambda is mostly a matter of width; the gap in level per unit of standard deviation, and the non-linear form of the equilibrium, involve something beyond width. That the comparison is made at equal t rather than equal variance is the same kind of caveat raised in the earlier note on quantum random walks, where heavier-looking extremes were compared at an equal number of steps.

If there is an effect specific to the quantum random walk, it shows in the shape of the equilibrium rather than in how its scale grows with t. Non-linearity as such is not peculiar to the quantum case. Kyle-type equilibria under non-normal distributions have been studied for some time (10), including work on non-linear equilibria (11, 12). In this paper’s classical case too, the pricing rule levels off and departs from a straight line, because the future value lies within a finite range; once the normality assumption is dropped, a linear equilibrium is not guaranteed. What is distinctive in the quantum case is the appearance of multiple clusters and jumps, which the paper presents as a manifestation of oscillations arising from the quantum walk (1). How the shape of the distribution — peaks near both edges, site-to-site oscillation in between — maps onto particular jumps in the pricing rule is not analysed in detail.

One further point deserves checking. Under the normal benchmark, lambda and expected profit are proportional to the same standard deviation, so in the classical case both would be expected to grow roughly as √t. In the paper’s classical results the fit for lambda matches the benchmark not only in its exponent (0.491) but down to the coefficient, whereas the exponent for expected profit (0.69) is well away from 0.5. For the profit fit, the paper does not state the form of the fitted expression — whether it includes an intercept, or whether t = 1 is included. Where the gap comes from is not discussed in the text.

Assumptions and limitations

The paper’s claims should be read together with the following.

• It is a preprint (v1) that has not been peer-reviewed. Results and interpretation may change in review.
• The computations cover only t = 1 to 23. In the quantum case no equilibrium was found for larger t. The paper says nothing about behaviour at large t.
• The equilibria are numerical fixed points. The authors themselves report that convergence was sensitive to the iteration settings. Uniqueness of the equilibria is not shown.
• Parameters are fixed. Noise trading is normal with mean 0 and variance 1, and the lattice spacing is 0.1. Results for other values are not reported.
• How lambda is measured in the quantum case is not stated. In the classical case it is the slope near zero order flow. For the quantum case the introduction says only “in the linear regime”; for a non-linear pricing rule with jumps, the range over which the slope was taken is not given.
• It is the single-auction model. The quantum random walk is not introduced into Kyle’s sequential or continuous-trading versions.
• No market data are used. Nor does the paper ground the choice of a quantum-walk distribution for the future value in any observation of markets. What it offers is a comparison between the properties of two models.
• Code and data availability could not be confirmed. The text says nothing about releasing them, and the arXiv abstract page links to no code (accessed 26 October 2026).

What can and cannot be said

What the paper supports:

• In Kyle’s single-auction model, replacing the future-value distribution with that of a continuous-time quantum random walk yields, within numerical computations for t ≤ 23, equilibria in which the informed trader’s strategy breaks into clusters and the pricing rule has multiple jumps.
• The analytical step gives a relation in which price impact is proportional to the uncertainty that remains for the market maker after seeing the order flow. This relation holds whatever the distribution of the future value and is not specific to the quantum random walk; what is specific to the quantum case is the form that the conditional variance takes as a function of order flow.
• The difference in how lambda grows — proportional to t, or close to √t — is largely accounted for by the difference in how the two distributions spread. Per unit of standard deviation, however, the quantum lambda is about 1.5 times the normal benchmark, a gap that width alone does not explain (this note’s own check).

What it does not yet support:

• That future values in actual markets are distributed in this way. The paper does not test this.
• That the results hold for larger t, or for other noise levels or lattice spacings.
• Any conclusion that separates the effect of the quantum random walk from the effect of distribution width. The comparison is made at equal t, not at equal variance, and the paper does not allow one to decide whether the gap in level per unit of standard deviation comes from the shape of the distribution.
• Any claim that quantum computers make financial calculations faster. This is not that kind of research. In the classification of an earlier note on “quantum-inspired”, it belongs with research that uses the mathematics of quantum mechanics as a model of probability distributions.

The Kyle model has been extended in several directions — dropping normality, adding assets, introducing risk-averse participants (1). This paper can be read as an attempt to bring the quantum random walk into the distribution of the future value. Its results are best taken with the range of the computations and the conditions of the comparison in view.

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This series explains research in quantum finance and econophysics from the published literature. It offers no view on any particular market or asset.