quantum-option-pricing-heat-equation

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

1. PDE-to-Quantum Mapping: Black-Scholes PDE can be transformed to a heat equation, which maps naturally to a quantum linear system Ax = b
2. Exponential Speedup: Quantum state preparation for the heat equation achieves exponential speedup in spatial dimension compared to classical discretization
3. Derivative Pricing Pipeline: Complete pipeline from financial contract specification to quantum circuit implementation
4. NISQ-Compatible: Includes error analysis and resource estimates for near-term quantum devices

Algorithm Steps

1. Black-Scholes to Heat Equation: Transform BS PDE via change of variables to standard heat equation
2. Discretization: Discretize heat equation on grid, yielding linear system
3. Quantum Encoding: Encode discretized system as quantum linear system using amplitude encoding
4. HHL or QLSA: Solve using Quantum Linear System Algorithm (HHL variant)
5. Payoff Extraction: Extract option price from quantum state via amplitude estimation
6. 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)