Quantum Speedup for PDEs Arising from Option Pricing

The methodology is based on reformulating option-pricing PDEs as discrete Markov-chain evolutions and then implementing those evolutions on a quantum computer using Quantum Fast Forwarding (QFF). We begin from the pricing PDE (for example under the Black–Scholes model), transform it into a heat equation, and discretise space and time using the Explicit Finite Difference Method. This produces a linear update rule in which each time step corresponds to multiplication by a transition matrix representing a random walk on a spatial lattice. 

For homogeneous Dirichlet boundary conditions, the resulting Markov chain is non-symmetric because of absorbing boundary states, which prevents direct application of standard QFF. To overcome this, we introduce a discrete reflection method: the spatial grid is duplicated, the initial payoff vector is encoded as a discrete odd extension, and the dynamics are embedded into a symmetric random walk with periodic boundary conditions. This enlarged transition operator is compatible with Szegedy quantum walks and therefore with QFF. The algorithm then simulates t effective Markov steps in approximately \sqrt{t} quantum walk steps, giving quadratic acceleration in the transient evolution regime.

The quantum register stores the lattice index in computational basis states, with additional ancilla qubits used for coin/workspace registers required by the Szegedy walk construction and polynomial approximation steps required for QFF. State preparation consists of loading the discretised payoff profile (or transformed initial condition) into amplitudes over lattice basis states. For European options with time-dependent boundaries, an explicit lifting transformation is first applied so that the PDE satisfies homogeneous boundary conditions and can be processed by the same pipeline. 

After the QFF circuit evolves the state to maturity, measurement of the position register yields amplitudes corresponding to the discretised solution on the pricing grid. Repeated sampling or amplitude-estimation subroutines can then be used to extract specific option prices or selected grid values with prescribed precision. The methodology is benchmarked against classical finite-difference solvers and the state of the art quantum algorithms for solving PDEs, with full complexity analysis from input to output state.

The correct functioning of the approach is demonstrated through a combination of quantum-circuit construction, mathematical proofs forc complexity analysis and classical numerical validation. First, we provide a fully worked-out quantum circuit for the core Quantum Fast Forwarding (QFF) routine based on Szegedy quantum walks. This circuit explicitly specifies the state registers, ancilla qubits, walk operators, reflection steps, and measurement protocol required to implement the accelerated Markov-chain evolution corresponding to the discretised pricing PDE. This addresses point (c) directly.

Second, we validate the mathematical correctness of the construction through classical simulations. In particular, we numerically compare the original discrete heat equation with homogeneous Dirichlet boundary conditions against the enlarged periodic evolution obtained through our discrete reflection method. These simulations verify that restricting the evolved doubled system back to the physical lattice reproduces the same solution as the original boundary-value problem at each time step. This confirms that the symmetric periodic embedding used to enable QFF preserves the desired pricing dynamics. This addresses point (b) at the algorithmic level.

Third, the circuit architecture is already designed for execution on real quantum hardware, and implementation on available gate-based devices is the next experimental stage of the project. Initial runs will focus on small lattice sizes compatible with current qubit and noise constraints, allowing empirical verification of output distributions against simulator and classical benchmarks.

The approach is designed with scalability in mind and is supported by a full end-to-end complexity analysis of the quantum circuit. The core computational task is the time evolution of a discretised Markov chain arising from the finite-difference formulation of pricing PDEs. Classically, evolving the system for t time steps requires O(t) sequential applications of the transition operator. By contrast, Quantum Fast Forwarding (QFF) implements the same effective evolution in approximately O(\sqrt{t}) quantum walk steps, yielding a quadratic improvement in the dominant time-propagation cost.

This speedup is particularly relevant in option pricing, where fine spatial grids and long maturities require many discretisation steps and one must generate an algorithm that scales will with system size. In large-scale derivatives books, the same PDE must also be solved repeatedly across strikes, maturities, and market scenarios. As the number of required time steps grows, the asymptotic advantage of QFF becomes more significant, especially for products such as barrier options and interest-rate derivatives where boundary effects increase computational burden.