quantum-pde-option-pricing

name

quantum-pde-option-pricing

description

End-to-end quantum PDE framework for derivative pricing. Use when: designing quantum algorithms for option pricing, solving high-dimensional financial PDEs on quantum hardware, comparing quantum vs classical pricing complexity, implementing Black-Scholes or Heston models on quantum circuits, or evaluating quantum advantage for financial derivative pricing. Covers: quantum PDE solvers, finite-difference discretization, gate complexity analysis, Clifford+T resource estimation, implied volatility extraction.

Quantum PDE Option Pricing

End-to-end quantum PDE framework for European option pricing under local- and stochastic-volatility models (arXiv: 2605.26610).

Core Framework

1. Classical Input: Contract + model data (Black-Scholes / Heston)
2. Discretization: Finite-difference on spatial grids (N=2^n points per direction)
3. Quantum PDE Solver: Solve pricing PDE on quantum hardware
4. Classical Output: Option value estimates at selected points

Gate Complexity

For d assets and N grid points per direction:

• Black-Scholes (local vol): Õ(d² N^(2+d/2))
• Heston (stochastic vol): Õ(d² N^(d+2))

Polynomial improvement over classical finite-difference baselines: N^(d/2) and N^d respectively.

Implementation Steps

1. Problem Formulation

• Write PDE for option price V(S,t) under chosen model
• Black-Scholes: ∂V/∂t + ½σ²S²∂²V/∂S² + rS∂V/∂S - rV = 0
• Heston: Add stochastic variance with mean-reversion

2. Discretization

• Finite-difference on uniform grid
• Convert to matrix equation: dV/dt = AV
• Map to quantum state: |V(t)⟩ encoding discretized prices

3. Quantum State Preparation

• Encode initial/boundary conditions as quantum states
• Use quantum linear system algorithms for time evolution

4. Solution Recovery

• Single-point recovery via amplitude estimation
• Multi-point: repeated measurements or tomography

5. Resource Accounting

• Compile to Clifford+T via standard techniques
• Account for CNOT gates + single-qubit Pauli rotations

Key Insight

The quantum advantage comes from the grid-size dependence: quantum algorithms scale polynomially better than classical for high-dimensional PDEs, making them particularly relevant for multi-asset options where d > 2.

Classical Benchmark

Compare against:

• Monte Carlo simulation
• Finite-difference methods (ADI, Crank-Nicolson)
• FFT-based methods (for simpler cases)

Activation Keywords

• quantum pde option pricing
• quantum derivative pricing
• quantum black scholes
• quantum heston model
• 量子期权定价
• quantum financial pde