Simulating the non-Hermitian dynamics of financial option pricing with quantum computers

Area of Science:

• Quantum Computing
• Computational Physics
• Financial Mathematics

Background:

• The Schrödinger equation governs quantum state evolution.
• Hermitian Hamiltonians ensure unitary dynamics.
• Anti-Hermitian Hamiltonians are used for imaginary time evolution to find ground states.

Purpose of the Study:

• To generalize the quantum imaginary time evolution (QITE) algorithm.
• To remove the restriction of QITE to anti-Hermitian Hamiltonians.
• To solve partial differential equations (PDEs) equivalent to the Schrödinger equation with arbitrary non-Hermitian Hamiltonians.

Main Methods:

• Developed a generalized QITE methodology.
• Applied the method to solve PDEs, including the Black-Scholes equation.
• Simulated normalized dynamics of non-unitary evolution on a quantum computer.

Main Results:

• Successfully broadened the scope of QITE for non-Hermitian Hamiltonians.
• Demonstrated the feasibility of the generalized QITE for pricing European option contracts.
• Showcased a quantum approach to solving the Black-Scholes equation.

Conclusions:

• The generalized QITE offers a powerful new tool for quantum computation.
• This method provides a feasible quantum approach for real-world applications in finance.
• Quantum computing can be leveraged to solve complex financial models like the Black-Scholes equation.