Quantum Power Method by a Superposition of Time-Evolved States
Quantum Power Method by a Superposition of Time-Evolved States
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DOI：
10.1103/prxquantum.2.010333
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发表时间：
2020-08
期刊：
影响因子：
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通讯作者：
K. Seki;S. Yunoki
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文献类型：
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作者：
K. Seki;S. Yunoki
提出了一种量子-经典的幂方法混合算法，这里称为量子幂方法，用于计算量子计算机中的{n$是非负整数，是一个与时间无关的哈密顿量，是一个量子态。量子幂方法是基于这样一个事实，即哈密顿幂通常可以由时间演化算符的线性组合$\hat{U}(T)=\mathm{e}^{-\mathm{i}\hat{\mathcal{H}}t}$在$n+1$不同的时间($t$)变量上近似。该方法基于高阶导数的时间离散化形式，结合中心有限差分格式和对称Suzuki-Trotter分解，使得近似的哈密顿幂在可控精度下保持Herm性.在量子功率方法中，在量子计算机上利用Suzuki-Trotter分解在接近于零的不同时间分别计算每个时间演化算符HAT{U}(T)，而在经典计算机上处理这些时间演化算符的线性组合。近似$\HAT{\Mathcal{H}}^n$所需的门数目为$O(N N)$，其中$N$是假设局部哈密顿量$\HAT{\Mathcal{H}}$的量子比特数。作为量子功率方法的应用，我们将该方法与多参考Krylov子空间对角化方案相结合，并通过对哈密顿幂为$\mathcal{H}}^{n}$的自旋$1/2$Heisenberg模型的无声数值模拟表明，随着幂的增加，估计的基态能量和基态保真度随着变分量子本征值方案的增加而系统地提高。
We propose a quantum-classical hybrid algorithm of the power method, here dubbed as quantum power method, to evaluate $\hat{\mathcal{H}}^{n}|\psi\rangle$ with quantum computers, where $n$ is a nonnegative integer, $\hat{\mathcal{H}}$ is a time-independent Hamiltonian of interest, and $|\psi\rangle$ is a quantum state. The quantum power method is formulated on the fact that the Hamiltonian power $\hat{\mathcal{H}}^{n}$ can generally be approximated by a linear combination of time-evolution operators $\hat{U}(t)=\mathrm{e}^{-\mathrm{i} \hat{\mathcal{H}}t}$ at $n+1$ different time ($t$) variables. The formalism is based on a time-discretized form of the higher-order derivative $\hat{\mathcal{H}}^{n}={\rm i}^n \mathrm{d}^n\hat{U}(t)/\mathrm{d} t^n|_{t=0}$ incorporated with the central-finite-difference scheme and the symmetric Suzuki-Trotter decomposition, by which the approximated Hamiltonian power retains its Hermiticity under a controlled accuracy. In the quantum power method, the Suzuki-Trotter decomposition is employed to evaluate each time-evolution operator $\hat{U}(t)$ separately at different time $t$ close to zero on quantum computers, while a linear combination of these time-evolution operators is treated on classical computers. The number of gates required for approximating $\hat{\mathcal{H}}^n$ is $O(N n)$, where $N$ is the number of qubits assuming a local Hamiltonian $\hat{\mathcal{H}}$. For an application of the quantum power method, we combine this method with a multireference Krylov-subspace diagonalization scheme and show, by noiseless numerical simulations for a spin-$1/2$ Heisenberg model with the Hamiltonian power $\hat{\mathcal{H}}^{n}$ up to $n=11$ (but not limited), that the estimated ground-state energy and the ground-state fidelity over a variational-quantum-eigensolver scheme is systematically improved with increasing the power $n$.