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Quantum Circuits for partial differential equations via Schrödingerisation

Junpeng Hu, Shi Jin, Nana Liu, Lei Zhang

TL;DR

The paper develops an explicit quantum-circuit framework to solve general linear PDEs via Schrödingerisation, transforming non-Hamiltonian PDEs into Schrödinger-type dynamics in one higher dimension using a warped phase transformation. It provides detailed unitary circuit constructions for the heat equation and the upwind advection equation, including continuous formulations, discretizations, and resource estimates that avoid oracles. Through complexity analysis and numerical experiments with Qiskit, the work demonstrates quantum advantages in high dimensions and validates the approach against classical Schrödingerisation and matrix-exponential solutions. The results offer a practical pathway for quantum PDE solvers and motivate future work on higher-order schemes, boundary conditions, and real hardware implementations.

Abstract

Quantum computing has emerged as a promising avenue for achieving significant speedup, particularly in large-scale PDE simulations, compared to classical computing. One of the main quantum approaches involves utilizing Hamiltonian simulation, which is directly applicable only to Schrödinger-type equations. To address this limitation, Schrödingerisation techniques have been developed, employing the warped transformation to convert general linear PDEs into Schrödinger-type equations. However, despite the development of Schrödingerisation techniques, the explicit implementation of the corresponding quantum circuit for solving general PDEs remains to be designed. In this paper, we present detailed implementation of a quantum algorithm for general PDEs using Schrödingerisation techniques. We provide examples of the heat equation, and the advection equation approximated by the upwind scheme, to demonstrate the effectiveness of our approach. Complexity analysis is also carried out to demonstrate the quantum advantages of these algorithms in high dimensions over their classical counterparts.

Quantum Circuits for partial differential equations via Schrödingerisation

TL;DR

The paper develops an explicit quantum-circuit framework to solve general linear PDEs via Schrödingerisation, transforming non-Hamiltonian PDEs into Schrödinger-type dynamics in one higher dimension using a warped phase transformation. It provides detailed unitary circuit constructions for the heat equation and the upwind advection equation, including continuous formulations, discretizations, and resource estimates that avoid oracles. Through complexity analysis and numerical experiments with Qiskit, the work demonstrates quantum advantages in high dimensions and validates the approach against classical Schrödingerisation and matrix-exponential solutions. The results offer a practical pathway for quantum PDE solvers and motivate future work on higher-order schemes, boundary conditions, and real hardware implementations.

Abstract

Quantum computing has emerged as a promising avenue for achieving significant speedup, particularly in large-scale PDE simulations, compared to classical computing. One of the main quantum approaches involves utilizing Hamiltonian simulation, which is directly applicable only to Schrödinger-type equations. To address this limitation, Schrödingerisation techniques have been developed, employing the warped transformation to convert general linear PDEs into Schrödinger-type equations. However, despite the development of Schrödingerisation techniques, the explicit implementation of the corresponding quantum circuit for solving general PDEs remains to be designed. In this paper, we present detailed implementation of a quantum algorithm for general PDEs using Schrödingerisation techniques. We provide examples of the heat equation, and the advection equation approximated by the upwind scheme, to demonstrate the effectiveness of our approach. Complexity analysis is also carried out to demonstrate the quantum advantages of these algorithms in high dimensions over their classical counterparts.
Paper Structure (30 sections, 11 theorems, 107 equations, 16 figures)

This paper contains 30 sections, 11 theorems, 107 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Contribution
  3. Organization
  4. Notation
  5. Representation of finite difference operator
  6. The heat equation
  7. Schrödingerisation
  8. Continuous formulation
  9. Discretization
  10. Quantum circuit
  11. The advection equation
  12. Schrödingerisation
  13. Continuous formulation
  14. Discretization
  15. Quantum circuit
  16. ...and 15 more sections

Key Result

Lemma 1

sato2024hamiltonian Given an operator of the form $S = \ket{a}\bra{b} + \ket{b}\bra{a}$ with $\ket{a}$, $\ket{b} \in \mathbb{C}^{2^n}$ and $\langle a\ket{b}=0$, it can be decomposed into where $\langle a^\prime\ket{b^\prime}=0$. Then, there exists a unitary matrix $B$ such that the following relation holds true, and $S$ can be written as The time evolution operator $\exp(iSt)$ can be implemente

Figures (16)

  • Figure 1: Quantum circuit for $W_j(\gamma\tau, \lambda)$.
  • Figure 2: Quantum circuit for $V_0(\tau)$.
  • Figure 3: Quantum circuit for $\tilde{V}_0(\tau)$.
  • Figure 4: Quantum circuit for $V_{\text{heat}}(\tau)$.
  • Figure 5: Quantum circuit for the Schrödingerisation method, where the measurement requires only projection onto $\ket{k}, p_k > 0$, and $\mathcal{QFT}$ ($\mathcal{IQFT}$) denotes the (inverse) quantum Fourier transform.
  • ...and 11 more figures

Theorems & Definitions (26)

  • Lemma 1
  • Remark 1
  • Remark 2
  • Lemma 2
  • Lemma 3
  • Remark 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 16 more