⚛️ quantum physics

Quantum simulation of dissipative acoustic and elastic waves by Schrödingerisation: absorbing boundaries and explicit polynomial-size circuits

This paper presents explicit, polynomial-size quantum circuits that simulate dissipative acoustic and elastic waves in one, two, and three dimensions by combining a graded sponge layer for absorbing boundaries with Schrödingerisation to efficiently convert the resulting non-unitary dynamics into unitary evolution.

Original author: Hoang Anh Nguyen, Ali Tura

Published 2026-11-03

📖 7 min read🧠 Deep dive

CC BY 4.0

Original author: Hoang Anh Nguyen, Ali Tura

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the author. For technical accuracy, refer to the original paper. Read full disclaimer

In the study of how sound travels through the air or how seismic waves ripple through the Earth, scientists rely on complex mathematical models to predict the future behavior of these waves. These models are essential for everything from designing concert halls to locating oil reserves and understanding earthquakes. A fundamental challenge in running these models on a computer is that the digital world is a closed box. When a simulated wave reaches the edge of the computer's grid, it cannot simply vanish into the distance as it would in the real world. Instead, it bounces back, creating a false echo that corrupts the entire simulation. To stop this, classical computers use a technique called a sponge layer, a special zone at the edge of the grid where the wave energy is gradually drained away, mimicking the effect of the wave traveling off into infinity. While this works well on traditional machines, simulating these same waves on a quantum computer presents a unique puzzle. Quantum computers are built to follow strict rules of conservation, where information and energy are never lost, only transformed. This makes it impossible for them to naturally model a process where waves simply disappear, creating a barrier between the physics of the real world and the mechanics of the quantum machine.

A team of researchers at the Colorado School of Mines has found a way to bridge this gap, creating a method that allows quantum computers to simulate waves that are absorbed and lost, just like in reality. They achieved this by combining three distinct ideas: a specific way to write the wave equations so they fit a quantum computer's structure, the use of a sponge layer to absorb the waves, and a mathematical trick that turns the problem of energy loss into a problem of movement. The researchers showed that by adding a single extra variable to the system, they could convert the non-conserving, dissipative behavior of a dying wave into a conserving, unitary motion that a quantum computer can handle. This transformation allows the computer to track the wave as it travels through the sponge layer and fades away, without breaking the fundamental rules of quantum mechanics.

The core of their work involves a technique called Schrödingerisation, which acts as a translator between two different types of physics. In the standard quantum world, a system evolves in a way that preserves the total amount of information, much like a perfectly elastic collision where no energy is lost. However, the real-world waves the researchers wanted to study are dissipative; they lose energy as they are absorbed by the sponge layer. To simulate this, the team introduced an auxiliary variable, a kind of extra dimension that the wave can move through. In this new framework, the act of the wave losing energy in the real world is reinterpreted as the wave moving along this extra dimension. The faster the wave is damped in the simulation, the faster it travels along this extra path. This clever rephrasing turns a problem that a quantum computer cannot solve directly into a family of problems it can solve perfectly, because the new version of the problem conserves energy in the expanded space.

To test this idea, the researchers built explicit instructions, known as quantum circuits, that a quantum computer could follow to simulate acoustic and elastic waves in one, two, and three dimensions. They started with the basic equations that describe how sound and seismic waves move and added a sponge layer to the edges of their simulated space. This sponge layer was designed to dampen every part of the wave field, not just the speed of the particles, but also the pressure and stress components. By doing so, they ensured that the waves were absorbed smoothly without reflecting back into the simulation, a common failure point in simpler methods. They then applied their Schrödingerisation technique to this damped system, deriving a set of quantum operations that could run on a machine with a limited number of qubits, the basic units of quantum information.

The results of their simulations were striking. When they ran these circuits on a classical computer to mimic a quantum device, the outcomes matched the expected behavior of damped waves with high precision. The simulations successfully reproduced the way waves travel and fade away in a sponge layer, even when the grid was as large as twenty-five qubits. A key finding was that the method is highly efficient. The researchers discovered that the specific mathematical structure of the sponge layer allowed them to break down the complex quantum operations into a manageable number of simple steps. Instead of requiring an exponential explosion of computational resources, which is a common hurdle in quantum computing, their method required only a polynomial number of steps. This means that as the size of the simulation grows, the number of operations needed grows at a manageable rate, making the approach scalable for larger, more realistic problems.

Another significant aspect of their work was the analysis of how well the method works in practice. They found that the probability of successfully retrieving the correct wave state from the quantum computer is directly tied to how much energy remains in the system. If the sponge has absorbed most of the wave, the chance of a successful measurement is lower, but this is a fair trade-off that reflects the physical reality of the simulation. The researchers proved that they could recover the exact state of the wave at any point in time, provided they measured the auxiliary variable correctly. They also showed that by measuring just one or two extra qubits, they could isolate the correct result with a high degree of certainty. This level of control is crucial for practical applications, as it means the simulation does not require an impractical amount of repeated trials to get a useful answer.

The team also compared their method of damping the entire wave field against older techniques that only dampened the velocity of the particles. Their classical simulations confirmed that damping all parts of the wave field is far superior, absorbing the waves much more effectively and leaving behind almost no residual energy. This validation was important because it ensured that the quantum algorithm was built on a physically sound foundation. The researchers demonstrated that their approach works for both simple sound waves and the more complex elastic waves found in solid materials, which involve multiple types of motion and stress. By handling these different types of waves with the same underlying framework, they showed the versatility of their method.

The paper concludes by providing a complete roadmap from the original physical equations to the final quantum circuit. This includes detailed instructions on how to set up the grid, how to apply the sponge layer, and how to execute the quantum operations step by step. The researchers made all their code and data available, allowing others to verify and build upon their work. Their findings suggest that it is possible to simulate open, dissipative systems on quantum computers without sacrificing the integrity of the quantum mechanical rules. This opens the door for future simulations of complex physical phenomena where energy loss is a key factor, such as the propagation of seismic waves through the Earth's crust or the behavior of sound in complex environments. The work represents a significant step forward in making quantum computers useful for real-world problems that involve the natural dissipation of energy.

Technical Summary: Quantum Simulation of Dissipative Acoustic and Elastic Waves by Schrödingerisation

Problem Statement
Quantum algorithms for partial differential equations (PDEs) often rely on Hamiltonian simulation, which naturally preserves energy and describes closed domains. For wave equations (acoustic and elastic), this unitary evolution prevents energy from leaving the computational region. Outgoing waves either reflect off the boundaries (in Dirichlet/Neumann setups) or wrap around (in periodic setups), contaminating the solution with artificial reflections. Classical geophysical simulations typically resolve this using absorbing boundary conditions or sponge layers—regions where wave fields are damped at a rate increasing toward the domain edge. However, damping introduces dissipation, rendering the evolution non-unitary ( $\dot{\psi} = A\psi$ with non-Hermitian $A$ ), which cannot be directly simulated on a quantum computer. While methods like Schrödingerisation exist to map non-unitary linear dynamics to unitary ones, explicit, gate-level quantum circuits for dissipative wave equations with absorbing boundaries have not been previously established.

Methodology
The authors combine three components to simulate dissipative wave equations: explicit quantum circuits for wave propagation, sponge layer damping, and the Schrödingerisation technique.

Hamiltonian Formulation: The acoustic and isotropic elastic wave equations are rewritten in first-order form. By scaling variables (e.g., $q = ic\nabla u$ for acoustic, and stress/velocity scaling for elastic), the semi-discrete generators become anti-Hermitian, allowing standard Hamiltonian simulation ( $\partial_t \psi = -iH\psi$ ). The discretization uses a staggered grid with forward/backward finite differences to ensure the discrete Hamiltonian is Hermitian without periodic boundary artifacts.
Sponge Layer Implementation: To absorb outgoing waves, a sponge layer is added where all wave fields (velocity and stress/gradient) are damped at a local rate $\sigma(x)$ . This modifies the generator to $A = -iH - \Sigma$ , where $\Sigma$ is a diagonal matrix of damping rates. Crucially, the Hermitian part of this generator, $H_1 = (A+A^\dagger)/2 = -\Sigma$ , is negative semidefinite.
Schrödingerisation: The non-unitary system $\dot{\psi} = A\psi$ is mapped to a unitary system by introducing an auxiliary real variable $p$ . A warped phase transformation $w(t, p) = e^{-p}\psi(t)$ converts the zeroth-order damping term into a first-order transport term in $p$ . The resulting system is a family of Schrödinger equations parameterized by the Fourier modes of $p$ , which can be simulated in parallel on a quantum register.
Circuit Construction: The time evolution is Trotterized. The wave part ( $e^{-i\tau H}$ ) uses existing ladder-operator decompositions. The damping part ( $e^{i\tau D_\eta \otimes \Sigma}$ ) is the critical innovation. Instead of generic synthesis (which would require exponential CNOT gates), the authors show that the damping step factorizes exactly into a polynomial number of multi-controlled phase gates.

Key Contributions
The paper makes four specific technical contributions:

Derivation of Dissipative Hamiltonian Systems: The authors derive the path from the PDE to a dissipative Hamiltonian system for acoustic and elastic waves in 1D, 2D, and 3D. They demonstrate that the sponge layer is reflectionless in the 1D continuum and that damping all fields (velocity and stress/gradient) is significantly more effective than damping velocity alone.
Sharp Analysis of Schrödingerisation for Sponges: Because the sponge's Hermitian part $H_1 = -\Sigma$ is negative semidefinite, the recovery threshold for Schrödingerisation vanishes ( $\lambda^+_{\max}(H_1) = 0$ ). This allows for exact recovery of the solution on the entire non-negative half of the auxiliary axis ( $p \ge 0$ ) at any time. The authors prove that the auxiliary axis can be truncated periodically based on the maximum damping rate and simulation time, and that the success probability is a fixed constant ( $\approx 0.4$ ) times the fraction of energy remaining.
Polynomial-Size Damping Circuits: The paper constructs explicit circuits where the damping step factorizes into polynomially many multi-controlled phase gates. This replaces the generic exponential cost of implementing diagonal unitaries, ensuring the entire Trotter step (wave + damping) scales polynomially with the number of qubits.
End-to-End Gate-Level Simulations: The authors performed simulations on up to 25 qubits (including 17–25 qubit runs with time loops containing only elementary gates and multi-controlled phases). These simulations reproduced classical solutions of damped wave equations with errors in the range of $5 \times 10^{-4}$ to $1.7 \times 10^{-2}$ .

Results

Absorption Efficiency: Independent classical finite-difference solvers confirmed that damping both velocity and stress/gradient fields absorbs outgoing waves far better than the common "velocity-only" damping mask. The velocity-only mask acts as a telegraph equation, causing wave spreading and reflection, whereas the full damping preserves the pulse shape while attenuating it.
Reflection Levels: In discrete simulations, the sponge layer achieved reflection levels of approximately $2.8 \times 10^{-2}$ to $6.3 \times 10^{-2}$ depending on layer width and damping amplitude. While these are percent-level reflections (the "price" of the simplest absorber), they are sufficient for many applications and are significantly lower than rigid boundaries.
Quantum Accuracy: Gate-level simulations on 17–25 qubits matched classical solutions. The error was dominated by the discretization of the auxiliary $p$ variable and the Trotterization, not by the circuit synthesis.
Success Probability: The probability of successfully post-selecting the correct state (by measuring the auxiliary qubits) is directly proportional to the energy remaining in the domain. The method effectively "pays" for the energy absorbed by the sponge.

Significance
The paper claims that this work provides the first explicit gate-level circuits for acoustic and elastic waves with absorbing boundaries. The significance lies in the "clean" quantum structure of the sponge layer: its negative semidefinite Hermitian part simplifies the Schrödingerisation pipeline, removing the need for large auxiliary registers or complex recovery conditions. The factorization of the damping step into polynomial-size circuits resolves a major bottleneck in simulating dissipative PDEs, making the simulation of open wave domains feasible on near-term quantum hardware. The authors emphasize that the method is rigorous, with exact recovery properties and explicit error bounds, rather than relying on heuristic approximations.

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