Approximating the $k$-th spectral gap $Δ_k=|λ_k-λ_{k+1}|$ and the corresponding midpoint $μ_k=\frac{λ_k+λ_{k+1}}{2}$ of an $N\times N$ Hermitian matrix with eigenvalues $λ_1\geqλ_2\geq\ldots\geqλ_N$, is an important special case of the eigenproblem with numerous applications in science and engineering. In this work, we present a quantum algorithm which approximates these values up to additive error $εΔ_k$ using a logarithmic number of qubits. Notably, in the QRAM model, its total complexity (queries and gates) is bounded by $O\left( \frac{N^2}{ε^{2}Δ_k^2}\mathrm{polylog}\left( N,\frac{1}{Δ_k},\frac{1}ε,\frac{1}δ\right)\right)$, where $ε,δ\in(0,1)$ are the accuracy and the failure probability, respectively. For large gaps $Δ_k$, this provides a speed-up against the best-known complexities of classical algorithms, namely, $O \left( N^ω\mathrm{polylog} \left( N,\frac{1}{Δ_k},\frac{1}ε\right)\right)$, where $ω\lesssim 2.371$ is the matrix multiplication exponent. A key technical step in the analysis is the preparation of a suitable random initial state, which ultimately allows us to efficiently count the number of eigenvalues that are smaller than a threshold, while maintaining a quadratic complexity in $N$. In the black-box access model, we also report an $Ω(N^2)$ query lower bound for deciding the existence of a spectral gap in a binary (albeit non-symmetric) matrix.

翻译：近似计算$N\times N$埃尔米特矩阵（特征值满足$\lambda_1\geq\lambda_2\geq\ldots\geq\lambda_N$）的第$k$个谱间隙$\Delta_k=|\lambda_k-\lambda_{k+1}|$及对应中点$\mu_k=\frac{\lambda_k+\lambda_{k+1}}{2}$，是特征值问题的重要特例，在科学与工程领域具有广泛应用。本文提出一种量子算法，利用对数数量级的量子比特即可在加性误差$\epsilon\Delta_k$范围内逼近这些数值。值得注意的是，在QRAM模型下，该算法的总复杂度（查询与门操作）被限制为$O\left( \frac{N^2}{\epsilon^{2}\Delta_k^2}\mathrm{polylog}\left( N,\frac{1}{\Delta_k},\frac{1}{\epsilon},\frac{1}{\delta}\right)\right)$，其中$\epsilon,\delta\in(0,1)$分别表示精度与失败概率。对于大谱间隙$\Delta_k$，该算法相较于经典算法目前最优复杂度$O \left( N^\omega\mathrm{polylog} \left( N,\frac{1}{\Delta_k},\frac{1}{\epsilon}\right)\right)$（其中$\omega\lesssim 2.371$为矩阵乘法指数）实现了加速。分析中的关键技术步骤是制备合适的随机初始态，这使我们最终能够在对$N$保持二次复杂度的同时，高效计数小于阈值的特征值数量。在黑盒访问模型下，我们还证明了判定二元（虽非对称）矩阵是否存在谱间隙的$\Omega(N^2)$查询下界。