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arXiv:2312.14747 · Geophysical Journal International

A quantum computing concept for 1-D elastic wave simulation with exponential speedup

Malte Schade, Cyrill Bösch, Václav Hapla, Andreas Fichtner

Published May 2, 2024• 8 citations
arXivHTMLPDF

Abstract

SUMMARY Quantum computing has attracted considerable attention in recent years because it promises speedups that conventional supercomputers cannot offer, at least for some applications. Though existing quantum computers (QCs) are, in most cases, still too small to solve significant problems, their future impact on domain sciences is already being explored now. Within this context, we present a quantum computing concept for 1-D elastic wave propagation in heterogeneous media with two components: a theoretical formulation and an implementation on a real QC. The method rests on a finite-difference approximation, followed by a sparsity-preserving transformation of the discrete elastic wave equation to a Schrödinger equation, which can be simulated directly on a gate-based QC. An implementation on an error-free quantum simulator verifies our approach and forms the basis of numerical experiments with small problems on the real QC IBM Brisbane. The latter produce simulation results that qualitatively agree with the error-free version but are contaminated by quantum decoherence and noise effects. Complementing the discrete transformation to the Schrödinger equation by a continuous version allows the replacement of finite differences by other spatial discretization schemes, such as the spectral-element method. Anticipating the emergence of error-corrected quantum chips, we analyse the computational complexity of the best quantum simulation algorithms for future QCs. This analysis suggests that our quantum computing approach may lead to wavefield simulations that run exponentially faster than simulations on classical computers.

DOI
10.1093/gji/ggae160

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In this paper we study quantum simulation algorithms on the elastic wave equations using the Schrödingerisation method. The Schrödingerisation method transforms any linear PDEs into a system of Schrödinger-type PDEs–with unitary evolution–using the warped phase transformation that maps the equations in one higher dimension. This makes them suitable for quantum simulations. We explore the application in two forms of the elastic wave equations. For the velocity-stress equation in isotropic media, we explore the symmetric matrix form under the external forcing via Schrödingerisation combined with spectral method. For problems with variable medium parameters, we apply Schrödingerisation method based on the staggered grid method to simulate velocity and stress fields, and give the complexity estimates. For the wave displacement equation, we transform it into a hyperbolic system and apply the Schrödingerisation method, which is then discretized by the spectral method and central difference scheme. Details of the quantum algorithms will be provided, along with the complexity analysis which demonstrates an exponential quantum advantage in space dimension over the classical algorithms if only considering the quantum subroutine which evolves the initial quantum state to the final quantum state.
Accurate modeling of seismic wave propagation in the subsurface of the earth is essential for understanding earthquake dynamics, characterizing seismic hazards on global scales and hydrocarbon reservoir exploration and monitoring on local scales. These are among the most challenging computational problems in geoscience. Despite algorithmic advances and the increasingly powerful computational resources currently available, including fast CPUs, GPUs and large volumes of computer memory, there are still daunting computational challenges in simulating 3D seismic wave propagation in complex earth environments. Recent advances in quantum computing are suggestive that geoscience may soon begin to benefit from this promising field. For example, Finite Difference (FD) modeling is the most widely used method to simulate seismic wave propagation. In the frequency domain, FD methods reduce solutions of the wave equation into systems of linear equations; such systems are just the type that quantum algorithms may be capable of solving with exponential speedup, in comparison with classical algorithms. For the computational geophysicist, to prepare to take advantage of these speed-ups, which could arrive in as few as 5-10 years, the tasks at hand are (1) to become familiar with the logic and concepts associated with quantum computing, and (2) to map our key computational algorithms (e.g., frequency domain FD) to this domain. Presentation Date: Monday, October 15, 2018 Start Time: 1:50:00 PM Location: 204B (Anaheim Convention Center) Presentation Type: Oral
We present an explicit quantum circuit construction for Hamiltonian simulation of a first-order velocity--stress formulation of the three-dimensional elastic wave equation in homogeneous isotropic media. Previous studies have shown how elastic wave equations can be cast into forms amenable to Hamiltonian simulation, but they typically rely on black box Hamiltonian access assumptions, making gate complexity estimation difficult. Starting from the first-order velocity--stress formulation, we discretize the system by finite differences, transform it into Schrödinger form, and exploit the separation between the component register and the spatial register to decompose the Hamiltonian into structured tensor product terms. This yields explicit implementations of first-order and second-order Trotter formulas for the resulting time evolution operator. We derive corresponding error bounds and constant sensitive qubit and CNOT complexity estimates in terms of the discretization parameter, simulation time, target accuracy, and material parameters. Numerical experiments validate the proposed framework through comparisons with the exact time evolution and reconstructed physical fields.
Numerical models are essential for investigating and understanding the physics and sound propagation and scattering, given the ocean’s complexity and the limitations of real-world field experiments. High-performance computing has become a primary tool for researchers to simulate long-range sound propagation. Quantum computing is an emerging technique that offers potential speedups over conventional binary computers. However, classical algorithms cannot be directly implemented in quantum systems due to fundamental differences in computing principles. This necessitates the development of quantum ocean acoustic models that remain interpretable within the same context as classical algorithms. This paper presents a quantum formulation for a finite-difference time-domain (FDTD) solver and a parabolic equation (PE) method. The results demonstrate that these quantum algorithms implemented on a simulator reproduce results identical to their classical counterparts, providing a foundation for further developments in quantum-centric ocean acoustics.
We construct explicit quantum circuits for the velocity-stress formulation of elastic wave equations in homogeneous media using Schrödingerisation. After rewriting the equations in symmetric matrix form(SMF), we use Fourier spectral discretization for source-driven periodic problems and staggered-grid finite differences for Dirichlet, Neumann, and impedance boundary conditions. The Schrödingerisation method transforms any linear PDEs into a system of Schrödinger-type PDEs -with unitary evolution- using the warped phase transformation that maps the equations in one higher dimension. For periodic problems, we develop circuits by combining controlled Fourier-mode rotations. For physical boundaries, interleaved encoding swaps the spatial and field-component registers, enabling a more efficient circuit construction using shift operators. We analyze pppth-order Trotter-Suzuki formulas and derive splitting-error and CNOT-gate estimates, under the assumptions of smooth auxiliary initialization and solution-state recovery. Comparison with classical partitioned splitting methods identifies a polynomial improvement in spatial-resolution dependence for the quantum state-evolution subroutine. Noiseless circuit simulations of a forced periodic problem and an impedance problem illustrate the constructions and quantify their errors against exact solutions and classical reference computations.
We present a quantum algorithmic framework for simulating linear, anti-Hermitian (lossless) wave equations in heterogeneous, anisotropic, and time-independent media. This framework encompasses a broad class of wave equations, including the acoustic wave equation, Maxwell’s equations, and the elastic wave equation. Our formulation is compatible with standard numerical discretization schemes and allows for the efficient implementation of multiple practically relevant time- and space-dependent sources. Furthermore, we demonstrate that subspace energies can be extracted and wave fields compared through an l 2 loss function, achieving optimal precision scaling with the number of samples taken. Additionally, we introduce techniques for incorporating boundary conditions and linear constraints that preserve the anti-Hermitian nature of the equations. Leveraging the Hamiltonian simulation algorithm, our framework achieves a quartic speedup over classical solvers in three-dimensional simulations, under conditions of sufficiently global measurements and compactly supported sources and initial conditions. This quartic speedup is optimal for time-domain solutions, as the Hamiltonian of the discretized wave equations has local couplings. In summary, our framework provides a versatile approach for simulating wave equations on quantum computers, offering substantial speedups over state-of-the-art classical methods.