Quantum simulation of dissipation for Maxwell equations in dispersive media

TL;DR

This paper develops quantum algorithms to simulate dissipative Maxwell equations in dispersive media, using probabilistic dilation methods to handle non-unitary operators efficiently on quantum computers.

Contribution

It introduces two probabilistic dilation algorithms for implementing dissipation in quantum simulations of Maxwell equations, improving efficiency and scalability.

Findings

The first dilation algorithm scales as O(2^{n-1}n^2) gates.

The second dilation algorithm, using LCU, reduces to O(2^{n}) gates.

For weak dissipation, the algorithms efficiently simulate transient dynamics.

Abstract

In dispersive media, dissipation appears in the Schr\"odinger representation of classical Maxwell equations as a sparse diagonal operator occupying an $r$-dimensional subspace. A first order Suzuki-Trotter approximation for the evolution operator enables us to isolate the non-unitary operators (associated with dissipation) from the unitary operators (associated with lossless media). The unitary operators can be implemented through qubit lattice algorithm (QLA) on $n$ qubits. However, the non-unitary-dissipative part poses a challenge on how it should be implemented on a quantum computer. In this paper, two probabilistic dilation algorithms are considered for handling the dissipative operators. The first algorithm is based on treating the classical dissipation as a linear amplitude damping-type completely positive trace preserving (CPTP) quantum channel where the combined system-environment must undergo unitary evolution in the dilated space. The unspecified environment can be modeled by just one ancillary qubit, resulting in an implementation scaling of $\textit{O}(2^{n-1}n^2)$ elementary gates for the dilated unitary evolution operator. The second algorithm approximates the non-unitary operators by the Linear Combination of Unitaries (LCU). We obtain an optimized representation of the non-unitary part, which requires $\textit{O}(2^{n})$ elementary gates. Applying the LCU method for a simple dielectric medium with homogeneous dissipation rate, the implementation scaling can be further reduced into $\textit{O}[poly(n)]$ basic gates. For the particular case of weak dissipation we show that our proposed post-selective dilation algorithms can efficiently delve into the transient evolution dynamics of dissipative systems by calculating the respective implementation circuit depth. A connection of our results with the non-linear-in-normalization-only (NINO) quantum channels is also presented.