A quantum walk replaces the Kyle model's Gaussian prior
The Kyle model describes a single-asset, single-auction market with three agents: an informed trader who knows the asset's future price z, noise traders whose orders u follow a Gaussian distribution, and a risk-neutral market maker who earns zero profit on average and clears orders at p(y) = E[z | y], where y is total order volume. Under Gaussian assumptions the equilibrium is linear, p(y) = p0 + lambda*y, and lambda is the Kyle lambda that measures price impact.
This paper keeps that structure but changes how the future price is generated. Instead of a Gaussian z, the future price is tied to the position of a random walker on a line of nodes indexed by j, with each node mapped to a possible future price z_j = p0 + s*j. The authors study two versions: a classical random walk, where the probability of node j after t steps is binomial, and a continuous-time quantum walk, where the probability is the squared Bessel function J_j^2(t).
The classical walk keeps a linear strategy and a logistic price rule
With the classical random walk, the authors solve for the equilibrium numerically using a fixed-point iterator with Anderson acceleration, over t from 1 to 23, with noise variance Sigma_u = 1 and step size s = 0.1. The informed trader's strategy x*(z) is linear in z, confirmed by a linear regression fit with R^2 above 0.996 for every plotted t.
The market maker's pricing rule is linear near y = 0 but tapers off for large |y|, reaching Delta p*(y) = s*t, the minimum and maximum achievable value of z in the model. The Kyle lambda for t from 2 to 23 grows sub-linearly and is described by lambda(t) approximately 0.0514*t^0.491 + 0.00167; t = 1 is an outlier at about 0.0100. The informed trader's average profit per round also grows sub-linearly, scaling as t^0.69.
The quantum walk produces clustered orders and jumpy prices
In the continuous-time quantum walk setting, the informed trader's equilibrium strategy forms several distinct clusters with large gaps between them, and the market maker's clearing price shows corresponding jumps. The authors note that this clearing price resembles the one derived from their analytical iteration, in which the price impact dp/dy equals (s^2 / 2*lambda*Sigma_u) times Var(j | y), meaning price impact is largest when the market maker has low confidence in the asset's future value.
The quantum equilibrium is therefore highly nonlinear, in contrast to the classical case where the informed trader's strategy was linear. The authors also report that the Kyle lambda grows linearly with t in this setting, described by lambda(t) approximately 0.0517*t - 0.00905 with R^2 about 0.9999, which they attribute to the difference between ballistic and diffusive transport.
Profit growth separates the two regimes more than the strategy shape
Despite the equilibrium becoming increasingly nonlinear, the informed trader's average profit per round in the quantum setting grows linearly with t, fitted as approximately 0.0407*t + 0.0159 with R^2 about 0.9999. In the classical setting the same quantity grew as t^0.69. The paper's discussion frames this as a key difference: the Kyle lambda grows quadratically faster with t in the quantum setting (lambda proportional to t) than in the classical setting (lambda proportional to t^0.491).
These two results answer different questions. The Kyle lambda measures price impact per unit of order flow near y = 0, while average profit per round measures the informed trader's realized gains across 200,000 simulated rounds. A reader comparing the regimes should track both, since the quantum setting changes them in the same direction but through different mechanisms.
Numerical convergence, not closed form, identifies the quantum equilibrium
The authors first attempt an analytical iteration assuming a linear pricing rule p(y) = p0 + lambda*y and an informed order x(z) = (z - p0)/2*lambda, then update the pricing rule. They show this approach becomes intractable, so they switch to numerical methods. The classical case uses a fixed-point iterator with a discretized order-flow grid from -30 to +30 with step 0.01, an initial pricing rule built from tanh terms, and Anderson acceleration with m = 10 and beta = 1.0.
The quantum case is more sensitive: convergence quality depended heavily on input hyperparameters, so the authors first run an Optuna-based tuning procedure over 100 trials to select discretization, initial pricing rule, Anderson parameters, and the terminal value bound M, then feed the best hyperparameters into the fixed-point iterator. They set Delta j = 1 in this setting and keep Sigma_u = 1 and s = 0.1.
The comparison holds only over t from 1 to 23
Both regimes are reported for t from 1 to 23. The authors state that in the quantum regime the algorithm failed to identify an equilibrium for large t, so they restricted the quantum results to that range and imposed the same range on the classical case to make the two comparable. The quantum computation also truncates the future-price set to a finite window {z_j} from -M to M, chosen so that the summed probability exceeds 1 - 10^-8 for every t in 1 to 23, because the iterator failed to converge for large M.
The paper describes its contribution as a first consideration of financial insider information in a quantum setting, and it presents the quantum equilibrium as a numerical result rather than a closed-form solution. The linear and sub-linear growth fits for the Kyle lambda and profit are regression descriptions of the computed equilibria over this finite time range, not extrapolations beyond it.
Track the Kyle lambda and profit growth together
The most decision-relevant split in this paper is not classical versus quantum in the abstract, but which quantity a reader is tracking. If the question is how much price impact a unit of order flow carries near y = 0, the quantum walk roughly doubles the growth exponent, from about t^0.491 to about t. If the question is how much the informed trader earns per round, the quantum walk changes the scaling from about t^0.69 to about t, while the equilibrium itself becomes clustered and jumpy.
A reader evaluating whether quantum dynamics matter for informed trading should therefore compare the two regimes on the Kyle lambda and on average profit per round separately, and should treat both comparisons as valid only within the t from 1 to 23 window where the numerical equilibrium was actually identified.
Sources used in this answer
Informed Trading model with quantum walks
Modeling the asset's future value with a continuous-time quantum walk instead of a classical random walk makes the Kyle model's equilibrium highly nonlinear, with clustered informed-trader orders and jumpy pricing, while the Kyle lambda grows linearly with t and the informed trader's average profit per round grows linearly rather than as t^0.69.