Spectral Gaps with Quantum Counting Queries and Oblivious State Preparation

Approximating the $k$-th spectral gap $\Delta_k=|\lambda_k-\lambda_{k+1}|$ and the corresponding midpoint $\mu_k=\frac{\lambda_k+\lambda_{k+1}}{2}$ of an $N\times N$ Hermitian matrix with eigenvalues $\lambda_1\geq\lambda_2\geq\ldots\geq\lambda_N$, 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 $\epsilon\Delta_k$ using a logarithmic number of qubits. Notably, in the QRAM model, its total complexity (queries and gates) is bounded by $O\left( \frac{N^2}{\epsilon^{2}\Delta_k^2}\mathrm{polylog}\left( N,\frac{1}{\Delta_k},\frac{1}{\epsilon},\frac{1}{\delta}\right)\right)$, where $\epsilon,\delta\in(0,1)$ are the accuracy and the failure probability, respectively. For large gaps $\Delta_k$, this provides a speed-up against the best-known complexities of classical algorithms, namely, $O \left( N^{\omega}\mathrm{polylog} \left( N,\frac{1}{\Delta_k},\frac{1}{\epsilon}\right)\right)$, where $\omega\lesssim 2.371$ 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 $\Omega(N^2)$ query lower bound for deciding the existence of a spectral gap in a binary (albeit non-symmetric) matrix.