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

Abstract

Explicit quantum circuits for the acoustic and elastic wave equations, obtained by Hamiltonian simulation of their first-order forms, conserve energy and therefore describe closed domains: an outgoing wave reflects off the edge of the computational region or, with periodic boundaries, wraps around and re-enters from the opposite side. We open the domain with the simplest physical absorber, a graded sponge layer in which every wave field is damped at a rate that grows towards the outer edge, and implement the resulting dissipative dynamics on a quantum simulator by Schrödingerisation, which turns a non-unitary linear evolution into a family of unitary ones by means of a single auxiliary variable. We derive the whole chain from the partial differential equation to the circuit, in one, two and three dimensions. An independent finite-difference solver confirms that damping all wave fields absorbs far better than the common velocity-only damping mask. For the sponge the quantum analysis is sharp: recovery is exact on the whole non- negative half of the auxiliary axis at every time; that axis can be truncated periodically at a length set by the largest damping rate and the simulated time; and the success probability is a fixed constant, about 0.4, times the fraction of the energy not yet absorbed, with only one or two auxiliary qubits measured. On the circuit side, the damping step, which generic synthesis compiles into a number of controlled-NOT gates exponential in the number of qubits, factorises exactly into polynomially many multi-controlled phase gates. Gate-level simulations on up to 25 qubits agree with the classical solutions of the damped wave equations.

Hamiltonian simulation of discretised wave equations conserves energy, so finite domains produce reflections or periodic wrap-around. This paper develops explicit quantum circuits for acoustic and isotropic elastic waves with absorbing boundaries, combining a graded sponge layer with Schrödingerisation. The construction covers one, two and three dimensions. Energy-scaled first-order fields and paired forward/backward differences give a Hermitian wave Hamiltonian $H$; damping produces $A=-iH-\Sigma$, with $\Sigma\ge0$ diagonal. Schrödingerisation adds an auxiliary coordinate $p$. Half-line energy estimates establish exact continuum recovery $w(t,p)=e^{-p}\psi(t)$ for all $p\ge0$, and a domain-of-dependence argument gives the sufficient periodic-truncation condition $p_{\max}\ge p^*+\|\Sigma\|T$. Post-selection succeeds with probability $\kappa_g\|\psi(T)\|^2/\|\psi(0)\|^2$, where $\kappa_g\approx0.4$; one or two auxiliary qubits select the recovery window. For fixed-degree polynomial damping profiles on dyadic pieces, the damping unitary factorises exactly into polynomially many controlled phases, avoiding generic exponential-size diagonal synthesis. Combined with existing wave circuits and Strang splitting, this gives polynomial gate counts per step. Gate-level statevector simulations on 17–25 qubits reproduce classical damped-wave references with reported relative errors $5\times10^{-4}$–$1.7\times10^{-2}$. Independent classical solvers also compare full-field and velocity-only damping. The sponge is not perfectly matched: multidimensional reflections remain percent-level on tested grids. Recovery becomes expensive as energy is absorbed, auxiliary wrap-around can contaminate post-selection, and the resolution-dependent Trotter constant is measured rather than bounded. No hardware experiment is reported; enlarging a closed domain remains an alternative for short transients.