Butterfly Factorization

The paper introduces the butterfly factorization as a data-sparse approximation for the matrices that satisfy a complementary low-rank property. The factorization can be constructed efficiently if either fast algorithms for applying the matrix and its adjoint are available or the entries of the matrix can be sampled individually. For an $N \times N$ matrix, the resulting factorization is a product of $O(\log N)$ sparse matrices, each with $O(N)$ non-zero entries. Hence, it can be applied rapidly in $O(N\log N)$ operations. Numerical results are provided to demonstrate the effectiveness of the butterfly factorization and its construction algorithms

On low-depth quantum algorithms for robust multiple-phase estimation

This paper is an algorithmic study of quantum phase estimation with multiple eigenvalues. We present robust multiple-phase estimation (RMPE) algorithms with Heisenberg-limited scaling that are particularly suitable for early fault-tolerant quantum computers in the following senses: (1) a minimal number of ancilla qubits are used, (2) an imperfect initial state with a significant residue is allowed, (3) the prefactor in the maximum runtime can be arbitrarily small given that the residue is sufficiently small and a gap among the dominant eigenvalues is known in advance. Even if the eigenvalue gap does not exist, the proposed RMPE algorithms are able to achieve the Heisenberg limit while maintaining the aforementioned benefits (1) and (2). In addition, our method handles both the {\em integer-power} model, where the unitary $U$ is given as a black box with only integer powers accessible, and the {\em real-power} model, where the unitary $U$ is defined through a Hamiltonian $H$ with $U = \exp(-2\pi\mathrm{i} H)$.Comment: 20 pages, 3 figure

On low-depth algorithms for quantum phase estimation

Quantum phase estimation is one of the key building blocks of quantum computing. For early fault-tolerant quantum devices, it is desirable for a quantum phase estimation algorithm to (1) use a minimal number of ancilla qubits, (2) allow for inexact initial states with a significant mismatch, (3) achieve the Heisenberg limit for the total resource used, and (4) have a diminishing prefactor for the maximum circuit length when the overlap between the initial state and the target state approaches one. In this paper, we prove that an existing algorithm from quantum metrology can achieve the first three requirements. As a second contribution, we propose a modified version of the algorithm that also meets the fourth requirement, which makes it particularly attractive for early fault-tolerant quantum devices

A note on spike localization for line spectrum estimation

This note considers the problem of approximating the locations of dominant spikes for a probability measure from noisy spectrum measurements under the condition of residue signal, significant noise level, and no minimum spectrum separation. We show that the simple procedure of thresholding the smoothed inverse Fourier transform allows for approximating the spike locations rather accurately