Quantum simulation of dissipation for Maxwell equations in dispersive media

The dissipative character of an electromagnetic medium breaks the unitary evolution structure that is present in lossless, dispersive optical media. In dispersive media, dissipation appears in the Schrodinger representation of 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, based on the discretization and the dimensionality of the pertinent fields. However, the non-unitary-dissipative part poses a challenge both physically and computationally on how it should be implemented on a quantum computer. In this paper, two 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 an unspecified environment interacts with the system of interest and produces the non-unitary evolution. Therefore, the combined system-environment is now closed, and 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 O(2n−1n2) elementary gates for the total system-environment unitary evolution operator. The second algorithm approximates the non-unitary operators by the Linear Combination of Unitaries (LCU). On exploiting the diagonal structure of the dissipation, we obtain an optimized representation of the non-unitary part, which requires O(2n) elementary gates. A connection of our results with the non-linear-in-normalization-only (NINO) quantum channels is also presented.电磁介质的耗散特性会破坏无损色散光学介质中存在的幺正演化结构。在色散介质中，麦克斯韦方程的薛定谔表象里，耗散表现为占据r维子空间的稀疏对角算符。通过演化算符的一阶铃木-特罗特近似，我们可以将与耗散相关的非幺正算符，与对应无损介质的幺正算符分离开来。基于相关场的离散化与维度，幺正算符可通过n量子比特上的量子比特格点算法 (qubit lattice algorithm, QLA) 实现。然而，非幺正耗散部分在物理与计算层面都面临着如何在量子计算机上实现的挑战。本文针对耗散算符的处理问题，探讨了两种扩张算法。第一种算法将经典耗散建模为线性振幅阻尼型完全保迹 (completely positive trace preserving, CPTP) 量子信道，其中未指定的环境与目标系统相互作用，从而产生非幺正演化。此时系统与环境组成的复合系统成为封闭系统，需在扩张空间中进行幺正演化。未指定的环境仅需单个辅助量子比特即可建模，此时系统-环境复合系统的总幺正演化算符的实现复杂度标度为O(2^{n−1}n²)个基本量子门。第二种算法通过幺正线性组合 (Linear Combination of Unitaries, LCU) 对非幺正算符进行近似。通过利用耗散的对角结构，我们可得到非幺正部分的优化表示，其实现仅需O(2ⁿ)个基本量子门。此外，本文还探讨了所得结果与仅归一化非线性 (non-linear-in-normalization-only, NINO) 量子信道之间的关联。