AI 中文总结
本研究提出一种无需预言机的量子比特-量子模混合算法，通过扭曲相位变换和二分泡利分解求解非线性常微分方程，证明了电路开销与误差界，并经经典模拟验证了其精度优势。
AI 中文摘要
我们针对形如$\u200d\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$、漂移项多项式次数为$L$的非线性常微分方程，开发了一种量子比特-量子模混合算法。遵循Tennie和Magri提出的福克-普朗克（Fokker–Planck）思路，该算法对态密度进行传播，并在小噪声极限下将确定性轨迹作为该密度的峰值返回。通过Jin、Liu和Yu提出的扭曲相位变换，离散化的生成元被转化为一族参数化的薛定谔方程，该方程族的傅里叶模参数被加载到单个连续变量量子模上。我们的核心结构结论是，离散化生成元的厄米部分$H_1$和$H_2$可进行二分泡利分解，将非零泡利串划分为$\u200d\mathcal{O}(\log N)$个相互对易的族，且每个族可分解为次数不超过$L$的对角项与固定秩二键算符的张量积。这种分解使得每个族的指数算符可精确表示为$\u200d\mathcal{O}(n^{L})$个单项式控制的动量位移的乘积，不存在族内特罗特（Trotter）误差。在每个轴有$N=2^n$个点的$d$维网格上，每个特罗特步的电路开销为$\u200d\mathcal{O}(d^{L+1}n^{L+2})$个门。该算法无需调用稀疏访问预言机，也无需块编码：每个门都由漂移项的多项式系数以闭式形式确定。我们还证明了数值横坐标$\lambda_{\max}(H_1)$的界，该界确定了扭曲相位变换的恢复域和后选择开销。在两个非线性基准上的经典模拟验证了结构定理、偏移恢复特性，以及连续变量耦合相较于离散模寄存器的单位资源精度优势。
英文摘要
We develop a hybrid qubit--qumode algorithm for nonlinear ordinary differential equations of the form $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ with drift of polynomial degree~$L$. Following the Fokker--Planck route of Tennie and Magri, the algorithm propagates the state density and returns the deterministic trajectory as the peak of that density in the small-noise limit. The discretised generator is carried into a parametrised family of Schrödinger equations by the warped-phase transformation of Jin, Liu, and Yu, and the Fourier-mode parameter of that family is placed on a single continuous-variable qumode. Our central structural result is that the Hermitian parts $H_{1}$ and $H_{2}$ of the discretised generator admit a bipartite Pauli decomposition that sorts the non-zero Pauli strings into $\mathcal{O}(\log N)$ mutually commuting families and factorises each family into a diagonal of degree at most $L$ tensored with a fixed rank-two bond operator. The factorisation renders each family exponential an exact product of $\mathcal{O}(n^{L})$ monomial-controlled momentum displacements, with no intra-family Trotter error. On a $d$-dimensional grid of $N=2^{n}$ points per axis the circuit costs $\mathcal{O}(d^{L+1}n^{L+2})$ gates per Trotter step. No sparse-access oracle and no block encoding is invoked: every gate is fixed in closed form by the polynomial coefficients of the drift. We also prove a bound on the numerical abscissa $λ_{\max}(H_{1})$ that fixes the recovery domain of the warped-phase transform and the post-selection cost. A classical simulation on two nonlinear benchmarks confirms the structural theorems, the shifted recovery, and the accuracy-per-resource advantage of the continuous-variable coupling over a discretised mode register.
