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Quantum simulation in the semi-classical regime
Quantum ( IF 5.1 ) Pub Date : 2022-06-17 , DOI: 10.22331/q-2022-06-17-739
Shi Jin 1 , Xiantao Li 2 , Nana Liu 3
Affiliation  

Solving the time-dependent Schrödinger equation is an important application area for quantum algorithms. We consider Schrödinger's equation in the semi-classical regime. Here the solutions exhibit strong multiple-scale behavior due to a small parameter $\hbar$, in the sense that the dynamics of the quantum states and the induced observables can occur on different spatial and temporal scales. Such a Schrödinger equation finds many applications, including in Born-Oppenheimer molecular dynamics and Ehrenfest dynamics. This paper considers quantum analogues of pseudo-spectral (PS) methods on classical computers. Estimates on the gate counts in terms of $\hbar$ and the precision $\varepsilon$ are obtained. It is found that the number of required qubits, $m$, scales only logarithmically with respect to $\hbar$. When the solution has bounded derivatives up to order $\ell$, the symmetric Trotting method has gate complexity $\mathcal{O}\Big({ (\varepsilon \hbar)^{-\frac12} \mathrm{polylog}(\varepsilon^{-\frac{3}{2\ell}} \hbar^{-1-\frac{1}{2\ell}})}\Big),$ provided that the diagonal unitary operators in the pseudo-spectral methods can be implemented with $\mathrm{poly}(m)$ operations. When physical observables are the desired outcomes, however, the step size in the time integration can be chosen independently of $\hbar$. The gate complexity in this case is reduced to $\mathcal{O}\Big({\varepsilon^{-\frac12} \mathrm{polylog}( \varepsilon^{-\frac3{2\ell}} \hbar^{-1} )}\Big),$ with $\ell$ again indicating the smoothness of the solution.

中文翻译:

半经典状态下的量子模拟

求解瞬态薛定谔方程是量子算法的一个重要应用领域。我们考虑半经典状态下的薛定谔方程。在这里,由于一个小参数$\hbar$,解决方案表现出很强的多尺度行为,在某种意义上,量子态的动力学和诱导的可观测量可以发生在不同的空间和时间尺度上。这样的薛定谔方程有许多应用,包括在 Born-Oppenheimer 分子动力学和 Ehrenfest 动力学中。本文考虑了经典计算机上伪光谱 (PS) 方法的量子类似物。根据 $\hbar$ 和精度 $\varepsilon$ 估计门数。发现所需的量子比特数 $m$ 仅相对于 $\hbar$ 成对数缩放。当解具有高达 $\ell$ 阶的有界导数时,对称 Trotting 方法的门复杂度为 $\mathcal{O}\Big({ (\varepsilon \hbar)^{-\frac12} \mathrm{polylog}(\ varepsilon^{-\frac{3}{2\ell}} \hbar^{-1-\frac{1}{2\ell}})}\Big),$ 假设伪中的对角酉运算符谱方法可以用 $\mathrm{poly}(m)$ 操作来实现。然而,当物理可观测量是期望的结果时,时间积分中的步长可以独立于 $\hbar$ 来选择。这种情况下的门复杂度降低到 $\mathcal{O}\Big({\varepsilon^{-\frac12} \mathrm{polylog}( \varepsilon^{-\frac3{2\ell}} \hbar^{ -1} )}\Big),$ 和 $\ell$ 再次表示解的平滑度。对称小跑法的门复杂度为 $\mathcal{O}\Big({ (\varepsilon \hbar)^{-\frac12} \mathrm{polylog}(\varepsilon^{-\frac{3}{2\ell} } \hbar^{-1-\frac{1}{2\ell}})}\Big),$ 假设伪谱方法中的对角酉算子可以用 $\mathrm{poly}(m )$ 操作。然而,当物理可观测量是期望的结果时,时间积分中的步长可以独立于 $\hbar$ 来选择。这种情况下的门复杂度降低到 $\mathcal{O}\Big({\varepsilon^{-\frac12} \mathrm{polylog}( \varepsilon^{-\frac3{2\ell}} \hbar^{ -1} )}\Big),$ 和 $\ell$ 再次表示解的平滑度。对称小跑法的门复杂度为 $\mathcal{O}\Big({ (\varepsilon \hbar)^{-\frac12} \mathrm{polylog}(\varepsilon^{-\frac{3}{2\ell} } \hbar^{-1-\frac{1}{2\ell}})}\Big),$ 假设伪谱方法中的对角酉算子可以用 $\mathrm{poly}(m )$ 操作。然而,当物理可观测量是期望的结果时,时间积分中的步长可以独立于 $\hbar$ 来选择。这种情况下的门复杂度降低到 $\mathcal{O}\Big({\varepsilon^{-\frac12} \mathrm{polylog}( \varepsilon^{-\frac3{2\ell}} \hbar^{ -1} )}\Big),$ 和 $\ell$ 再次表示解的平滑度。$ 假设伪谱方法中的对角酉算子可以用 $\mathrm{poly}(m)$ 操作来实现。然而,当物理可观测量是期望的结果时,时间积分中的步长可以独立于 $\hbar$ 来选择。这种情况下的门复杂度降低到 $\mathcal{O}\Big({\varepsilon^{-\frac12} \mathrm{polylog}( \varepsilon^{-\frac3{2\ell}} \hbar^{ -1} )}\Big),$ 和 $\ell$ 再次表示解的平滑度。$ 假设伪谱方法中的对角酉算子可以用 $\mathrm{poly}(m)$ 操作来实现。然而,当物理可观测量是期望的结果时,时间积分中的步长可以独立于 $\hbar$ 来选择。这种情况下的门复杂度降低到 $\mathcal{O}\Big({\varepsilon^{-\frac12} \mathrm{polylog}( \varepsilon^{-\frac3{2\ell}} \hbar^{ -1} )}\Big),$ 和 $\ell$ 再次表示解的平滑度。
更新日期:2022-06-17
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