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arXiv:2403.10032 · Quantum

Quantum Circuits for partial differential equations via Schrödingerisation

Junpeng Hu, Shi Jin, Nana Liu, Lei Zhang

Published Dec 12, 2024• 27 citations
arXivHTMLPDF

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.

DOI
10.22331/q-2024-12-12-1563

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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.
We study a recently introduced simple method [S. Jin, N. Liu, and Y. Yu, Quantum simulation of partial differential equations via Schr\"odingerisation, arXiv:2212.13969] for solving general linear partial differential equations with quantum simulation. This method converts linear partial differential equations into a Hamiltonian system, using a simple transformation called the warped phase transformation. Here we provide a more-in-depth technical discussion and expand on this approach in a more detailed and pedagogical way. We apply this to examples of partial differential equations, including heat, convection, Fokker-Planck, linear Boltzmann, and Black-Scholes equations. This approach can also be extended to general linear partial differential equations, including the Vlasov-Fokker-Planck equation and the Liouville representation equation for nonlinear ordinary differential equations. Extension to higher-order time derivatives is also possible.
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.
Solving partial differential equations for extremely large-scale systems within a feasible computation time serves in accelerating engineering developments. Quantum computing algorithms, particularly the Hamiltonian simulations, present a potential and promising approach to achieve this purpose. Actually, there are several oracle-based Hamiltonian simulations with potential quantum speedup, but their detailed implementations and accordingly the detailed computational complexities are all unclear. This paper presents a method that enables us to explicitly implement the quantum circuit for Hamiltonian simulation; the key technique is the explicit gate construction of differential operators contained in the target partial differential equation discretized by the finite difference method. Moreover, we show that the space and time complexities of the constructed circuit are exponentially smaller than those of conventional classical algorithms. We also provide numerical experiments and an experiment on a real device for the wave equation to demonstrate the validity of our proposed method. Published by the American Physical Society 2024
We present a novel approach to solving the advection equation under arbitrary velocity fields using a quantuminspired ‘Schrödingerisation’ technique for Hamiltonian simulation. Although numerous methods exist for solving partial differential equations (PDEs), Hamiltonian simulation remains a relatively underexplored yet promising direction-particularly in the context of long-term, fault-tolerant quantum computing. Building on this potential, our quantum algorithm is designed to accommodate non-trivial, spatially varying velocity fields and is applicable to both 2D and 3D advection problems. To ensure numerical stability, the algorithm incorporates an upwinding discretization scheme, which we adapt for quantum implementation through a combination of approximation and optimization techniques. We demonstrate the algorithm's effectiveness on benchmark cases involving rotational and shear flows in two and three dimensions. Moreover, we successfully implement the 2D advection equation using 16 qubits on IBM Quantum hardware, validating the approach and showcasing its practical applicability and robustness.
Quantum algorithms offer new avenues for solving partial differential equations (PDEs). While the potential for end-to-end quantum advantage is at present not well understood, recent literature presents explicit circuit constructions for solving certain classes of linear PDEs in the frequency domain and thus offers concrete examples to study. In this work, we develop end-to-end implementations of these quantum circuits compiled to machine-level instructions and benchmark them in both numerical simulations and IBMQ hardware experiments. We focus on the advection, wave, and Poisson equations and study quantum circuits that propagate the dynamics in frequency space via the quantum Fourier transform using approximate methods based on a first-order approximation which offer compact representations with uncontrollable approximation error, and polynomial approximation methods based on quantum signal processing (QSP) leading to deeper circuits with tunable algorithmic error. In addition, we experimentally demonstrate that the QSP-augmented algorithm can provide accurate solutions under realistic hardware constraints. Finally, we extend our method to address non-homogeneous Dirichlet boundary conditions and verify it numerically for a Poisson equation with source term obtained from high-fidelity physics simulations of a capacitively coupled plasma.
We introduce a simple and stable computational method for ill-posed partial differential equation (PDE) problems. The method is based on Schrödingerization, introduced in [S. Jin, N. Liu and Y. Yu, arXiv:2212.13969][S. Jin, N. Liu and Y. Yu, Phys. Rev. A, 108 (2023), 032603], which maps all linear PDEs into Schrödinger-type equations in one higher dimension, for quantum simulations of these PDEs. Although the original problem is ill-posed, the Schrödingerized equations are Hamiltonian systems and time-reversible, allowing stable computation both forward and backward in time. The original variable can be recovered by data from suitably chosen domain in the extended dimension. We will use the backward heat equation and the linear convection equation with imaginary wave speed as examples. Error analysis of these algorithms are conducted and verified numerically. The methods are applicable to both classical and quantum computers, and we also lay out quantum algorithms for these methods. Moreover, we introduce a smooth initialization for the Schrödingerized equation which will lead to essentially spectral accuracy for the approximation in the extended space, if a spectral method is used. Consequently, the extra qubits needed due to the extra dimension, if a qubit based quantum algorithm is used, for both well-posed and ill-posed problems, becomes almost log⁡log⁡1/ε\log\log {1/\varepsilon}loglog1/ε where ε\varepsilonε is the desired precision. This optimizes the complexity of the Schrödingerization based quantum algorithms for any non-unitary dynamical system introduced in [S. Jin, N. Liu and Y. Yu, arXiv:2212.13969][S. Jin, N. Liu and Y. Yu, Phys. Rev. A, 108 (2023), 032603].