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.