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Quantum Circuits for the Black-Scholes equations via Schrödingerisation

Shi Jin, Zihao Tang, Xu Yin, Lei Zhang

TL;DR

This work tackles high-dimensional derivative pricing by solving Black-Scholes PDEs on quantum computers through Schrödingerisation, a warped-phase transformation that converts non-unitary dynamics into unitary, Hermitian evolution. The authors derive 1D and $d$-dimensional quantum-circuit constructions using a first-order Lie–Trotter–Suzuki decomposition, achieve explicit complexity and error bounds, and demonstrate polynomial-speedup potential in high dimensions with numerical validation on Qiskit. The key contributions include rigorous error analyses, detailed circuit designs (including $U_1$, $U_2$, and multi-dimensional extensions), and evidence of convergence to classical solutions, highlighting substantial memory advantages over classical finite-difference methods for multi-asset pricing. The work advances quantum financial mathematics by providing scalable quantum algorithms for option pricing in settings where traditional methods suffer from the curse of dimensionality, potentially enabling practical quantum-assisted pricing for portfolios with many underlying assets.

Abstract

In this paper, we construct quantum circuits for the Black-Scholes equations, a cornerstone of financial modeling, based on a quantum algorithm that overcome the cure of high dimensionality. Our approach leverages the Schrödingerisation technique, which converts linear partial and ordinary differential equations with non-unitary dynamics into a system evolved by unitary dynamics. This is achieved through a warped phase transformation that lifts the problem into a higher-dimensional space, enabling the simulation of the Black-Scholes equation on a quantum computer. We will conduct a thorough complexity analysis to highlight the quantum advantages of our approach compared to existing algorithms. The effectiveness of our quantum circuit is substantiated through extensive numerical experiments.

Quantum Circuits for the Black-Scholes equations via Schrödingerisation

TL;DR

This work tackles high-dimensional derivative pricing by solving Black-Scholes PDEs on quantum computers through Schrödingerisation, a warped-phase transformation that converts non-unitary dynamics into unitary, Hermitian evolution. The authors derive 1D and -dimensional quantum-circuit constructions using a first-order Lie–Trotter–Suzuki decomposition, achieve explicit complexity and error bounds, and demonstrate polynomial-speedup potential in high dimensions with numerical validation on Qiskit. The key contributions include rigorous error analyses, detailed circuit designs (including , , and multi-dimensional extensions), and evidence of convergence to classical solutions, highlighting substantial memory advantages over classical finite-difference methods for multi-asset pricing. The work advances quantum financial mathematics by providing scalable quantum algorithms for option pricing in settings where traditional methods suffer from the curse of dimensionality, potentially enabling practical quantum-assisted pricing for portfolios with many underlying assets.

Abstract

In this paper, we construct quantum circuits for the Black-Scholes equations, a cornerstone of financial modeling, based on a quantum algorithm that overcome the cure of high dimensionality. Our approach leverages the Schrödingerisation technique, which converts linear partial and ordinary differential equations with non-unitary dynamics into a system evolved by unitary dynamics. This is achieved through a warped phase transformation that lifts the problem into a higher-dimensional space, enabling the simulation of the Black-Scholes equation on a quantum computer. We will conduct a thorough complexity analysis to highlight the quantum advantages of our approach compared to existing algorithms. The effectiveness of our quantum circuit is substantiated through extensive numerical experiments.
Paper Structure (12 sections, 8 theorems, 85 equations, 15 figures, 1 table)

This paper contains 12 sections, 8 theorems, 85 equations, 15 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notations for finite difference operators
  3. The one-dimensional Black-Scholes equation
  4. Schrödingerisation
  5. Discretization
  6. Quantum circuit
  7. The d-dimension Black-Scholes equation
  8. Schrödingerisation
  9. Quantum circuit
  10. Complexity analysis
  11. Numerical results
  12. Conclusion

Key Result

Lemma 5.1

For the Schrödinger equation $\frac{d|{\bf u}(t)\rangle}{dt} = iH_{BS}|{\bf u}(t)\rangle,$ with the Hamiltonian $H_{BS}$ as specified in Equation hbs, the time evolution operator $U_{BS}(\tau) = \exp(iH_{BS}\tau)$ over a time step $\tau$ can be effectively approximated by the unitary operator $V_{BS where $N_p = 2^{n_p}$ and $N_x - 1 = 2^{n_x}$ correspond to the number of grid points for the varia

Figures (15)

  • Figure 3.1: Quantum circuit for $W_{j}(\gamma \tau,\lambda)$
  • Figure 3.2: Quantum circuit for $\tilde{V}_1(\tau)$
  • Figure 3.3: Quantum circuit for $\tilde{V}_2(\tau)$
  • Figure 3.4: Quantum circuit for ${\tilde{V} }_{BS}$
  • Figure 3.5: Quantum circuit for the Schrödingerisation method, where the measurement requires only projection onto $P>0$ and $\mathcal{QFT (IQFT )}$ denotes the (inverse) quantum Fourier transform.
  • ...and 10 more figures

Theorems & Definitions (26)

  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 4.1
  • Remark 4.2
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • proof
  • ...and 16 more