| name | quantum-option-pricing-heat-equation |
| description | Exponentially fast quantum state preparation for the heat equation applied to financial option pricing. Maps Black-Scholes PDE to quantum linear system via heat equation discretization, achieving exponential speedup over classical methods. Use when: quantum finance, option pricing on quantum computers, Black-Scholes quantum solver, PDE-to-quantum mapping, quantum derivatives pricing, heat equation quantum simulation. |
| license | Complete terms in LICENSE.txt |
| metadata | {"arxiv_id":"2605.28950","published":"2026-05-29","tags":["quantum","finance","option-pricing","PDE","heat-equation","black-scholes"]} |
Quantum Option Pricing via Heat Equation State Preparation
Core Methodology
Presents methods for pricing financial derivatives on quantum devices with provable advantage over classical methods. Maps the Black-Scholes partial differential equation to a quantum linear system via heat equation discretization, enabling exponentially fast solution state preparation.
Key Insights
- PDE-to-Quantum Mapping: Black-Scholes PDE can be transformed to a heat equation, which maps naturally to a quantum linear system Ax = b
- Exponential Speedup: Quantum state preparation for the heat equation achieves exponential speedup in spatial dimension compared to classical discretization
- Derivative Pricing Pipeline: Complete pipeline from financial contract specification to quantum circuit implementation
- NISQ-Compatible: Includes error analysis and resource estimates for near-term quantum devices
Algorithm Steps
- Black-Scholes to Heat Equation: Transform BS PDE via change of variables to standard heat equation
- Discretization: Discretize heat equation on grid, yielding linear system
- Quantum Encoding: Encode discretized system as quantum linear system using amplitude encoding
- HHL or QLSA: Solve using Quantum Linear System Algorithm (HHL variant)
- Payoff Extraction: Extract option price from quantum state via amplitude estimation
- Error Bounds: Provide rigorous error bounds for discretization + quantum algorithm
When to Use
- Pricing European/American options on quantum hardware
- Portfolio risk analysis with quantum speedup
- Monte Carlo alternatives for derivative pricing
- Any PDE-based financial modeling task
Practical Considerations
- Requires fault-tolerant quantum computer for full advantage
- NISQ-friendly variants use variational approaches
- Condition number of discretized system affects HHL runtime
- Amplitude estimation provides quadratic speedup for expectation estimation
Related Approaches
- Quantum Monte Carlo for option pricing
- Quantum amplitude estimation for risk measures
- Classical finite difference methods
- Quantum PDE solvers (QLSA-based)
