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Quantum Circuit and Tensor Network Implementation of the 2D Acoustic Wave Equation
Tamas Nemeth, Gabor Vattay
TL;DR
Accurate seismic wave propagation is computationally demanding, motivating alternatives to classical propagation on large grids. The paper derives quantum-circuit and MPS formulations of the 2D acoustic wave equation and applies the MPS solver to Marmousi data. It produces valid 2D seismic wavefields with close similarity to explicit FDTD results, supporting MPS as a direction for computing and scaling wavefield propagation.
Problem
Accurate wavefield propagation through complex subsurface media is costly on classical high-performance computing systems, motivating lower-cost computational frameworks.
Method
The paper derives a quantum-circuit formulation of the 2D acoustic FDTD scheme, maps it to tensor-train/MPS form, and implements the MPS solver on Marmousi data.
Results
The MPS solver generates valid 2D seismic wavefields on Marmousi data, with close similarity to the FDTD wavefield.
Takeaways & Limitations
MPS representation is presented as a viable direction for deterministic computation and scaling of wavefield propagation on classical high-performance systems.
Takeaways & Limitations
The authors report limited experience with MPS wavefield simulation, and visible differences from FDTD are attributed to MPS parameterization under further analysis.
Abstract
from arXiv · showhide
We present a cohesive framework for simulating seismic wave propagation utilizing quantum computing paradigms and their classical tensor network equivalents. We detail a quantum circuit-based formulation for the explicit finite-difference time-domain (FDTD) solution of the two-dimensional acoustic wave equation and map this quantum architecture onto a tensor train representation, namely for Matrix Product State (MPS). The MPS solver enables deterministic simulation of large-scale wavefield dynamics on classical high-performance computing systems. We demonstrate the MPS representation by computing 2D seismic wavefields on the Marmousi model. Our results indicate that the MPS representation is a viable direction for computing and scaling wavefield propagation.
1 Introduction
The paper targets the high computational cost of accurate seismic wave propagation and develops quantum-circuit and MPS formulations of the 2D acoustic wave equation. It implements the MPS solver on 2D Marmousi data and compares it with explicit FDTD propagation.
- Accurate seismic wave propagation underpins advanced imaging methods including Reverse Time Migration and Full Waveform Inversion.
- High-resolution seismic surveys push classical high-performance computing toward architectural limits, motivating lower-cost propagation methods.
- Quantum computing offers a computational paradigm with potential exponential memory scaling for massive multidimensional seismic grids.
- Tensor networks, particularly tensor trains implemented as Matrix Product States, provide a classical emulation route for quantum wavefield algorithms.
- The study derives quantum and MPS formulations of the 2D acoustic wave equation, implements MPS on Marmousi data, and compares it with explicit FDTD.
2 Acoustic wave equation in 2D
The two-dimensional acoustic wave equation models scalar wavefield propagation through an inhomogeneous velocity medium. The paper discretizes it on a uniform Cartesian grid and obtains an explicit FDTD update subject to CFL stability.
- The scalar acoustic wavefield u(x, z, t) propagates through a two-dimensional inhomogeneous medium.
- The horizontal coordinate is x, depth is z, and c(x, z) denotes the spatially varying subsurface velocity.
- The computational domain uses a square Cartesian grid with Nx = Nz = 2^n points and uniform spacing Δh = Δx = Δz.
- Time is discretized with step Δt, which is constrained by the Courant-Friedrichs-Lewy stability criterion.
- A second-order central-difference discretization of temporal and spatial derivatives yields the explicit FDTD update equation, with λ = Δt/Δh as the mesh ratio.
3 Quantum Register Encoding
The quantum encoding maps the 2D grid and wavefield amplitudes into computational-basis states, represents the velocity model as a diagonal operator, and implements spatial shifts arithmetically. The resulting non-unitary wavefield update is block-encoded with LCU while retaining consecutive time-step registers.
- A 2^n × 2^n grid is mapped onto the computational basis of a 2n-qubit register.
- The acoustic wavefield at time step k is represented by an unnormalized quantum state, with x and z encoded as binary spatial indices.
- The spatially varying squared velocity field maps to a diagonal operator C^2 that scales computational-basis amplitudes by coordinate.
- For complex geophysical models, C^2 can use QROM architectures or a Walsh-series approximation, while c(x, z) > 0 makes it strictly positive definite.
- Computational-basis forward and backward shifts use incrementer and decrementer circuits to avoid quantum-Fourier-transform translation overhead.
- The forward shift cyclically maps |x⟩ to |(x + 1) mod 2^n⟩ as a ripple-carry addition implemented with cascaded multi-controlled Pauli-X gates.
- The backward shift is the adjoint of the forward shift, and their sum forms the discrete Laplacian in the quantum representation.
- The explicit FDTD update uses the two most recent wavefields and represents the generalized non-unitary operator A = 2I + λ^2C^2L through Linear Combination of Unitaries.
4 Motivation for Tensor Train Emulation of Subsurface Models
The tensor-train/MPS formulation provides a deterministic classical emulation of quantum wavefield operations, representing multidimensional fields compactly through tensor-network structure.
- Classical tensor-network emulation: Tensor Train methods provide a classical bridge to quantum algorithms while executing deterministically on high-performance computing architectures.The framework is mathematically isomorphic to MPS methods and exploits the same spatial-entanglement bounding principles.
- Field representation: A 2D discrete field with Nx = Nz = 2^n contains 2^2n grid points and maps to a 2n-qubit Hilbert-space state vector.This mapping replaces direct storage of the multidimensional grid with a many-body quantum spin-chain representation.
- Field representation: Sequential SVDs reshape discrete field amplitudes into an MPS whose bond dimension χ limits represented entanglement.The bond dimension also constrains spatial correlations and high-frequency structures that the network can encode.
- Matrix Product Operators: MPO shift operators implement periodic translations along the spatial axes, with forward and backward shifts representing quantum ripple-carry operations.The 2D operators extend one-dimensional MPOs using identity operators on the orthogonal register.
- Wave-equation evolution: The MPS update applies the discretized Laplacian and medium operator through MPO algebra equivalent to the wave-equation update.The velocity model is encoded as a diagonal MPO, while the Laplacian is represented as a sum of shift-operator terms.
- Compression: After each operator application, QR orthogonalization and SVD truncation discard small singular values and cap the bond dimension for numerical tractability.The cutoff preserves dominant spatial structures of the propagating wavefronts while controlling growth in representation size.
5 Results
The study evaluates FDTD and MPS wavefield propagation on a 256x256 Marmousi subset, using a centered impulsive source and 300 time steps. The MPS result closely resembles the FDTD wavefield, with visible parameterization-related differences.
- Experimental setup: A 256x256 Marmousi grid uses a unit impulsive source at (1280 m, 1280 m), 10 m spatial sampling, and 1 ms temporal stepping.The MPS bond dimension is set to χmax = 128.
- Experimental setup: 300 propagation steps are applied to the wavefield from the model center.The simulation uses an SVD cutoff of λmax = 10−7.
- Comparison with FDTD: The MPS wavefield exhibits close similarity to the FDTD wavefield, although differences remain visible upon closer inspection.The authors attribute these discrepancies to MPS-solver parameterization and report ongoing analysis of its capabilities.
6 Discussion
The results demonstrate that MPS can generate 2D seismic wavefields on the Marmousi model. The formulations share the same update structure while differing in grid and wavefield representation, and the authors identify scaling as a promising direction.
- Validated capability: The MPS representation produces valid 2D seismic wavefields on the Marmousi model.The authors further express confidence that MPS solvers can generate 3D seismic wavefields.
- Shared update structure: FDTD, quantum, and MPS formulations construct the next wavefield from current and previous wavefields through local and neighboring values.Their principal difference is how the grid and wavefield values are represented and accessed.
- Visual comparison: Figure 2 compares center-originating wavefields computed by FDTD and MPS on the Marmousi model.The upper field is FDTD and the lower field is MPS, providing a direct visual comparison.
- Discussion: The authors describe MPS wavefield simulation as an intriguing possibility for computing and scaling propagation, based on limited implementation experience.This conclusion is presented as provisional rather than as a completed scalability demonstration.