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.2010 Mathematics Subject Classification. 81P60. Key words and phrases. Quantum phase estimation, early fault-tolerant quantum devices. We thank Lin Lin for communicating the recent development of quantum phase 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.
Block encoding lies at the core of many existing quantum algorithms. Meanwhile, efficient and explicit block encodings of dense operators are commonly acknowledged as a challenging problem. This paper presents a comprehensive study of the block encoding of a rich family of dense operators: the pseudo-differential operators (PDOs). First, a block encoding scheme for generic PDOs is developed. Then we propose a more efficient scheme for PDOs with a separable structure. Finally, we demonstrate an explicit and efficient block encoding algorithm for PDOs with a dimension-wise fully separable structure. Complexity analysis is provided for all block encoding algorithms presented. The application of theoretical results is illustrated with worked examples, including the representation of variable coefficient elliptic operators and the computation of the inverse of elliptic operators without invoking quantum linear system algorithms (QLSAs).
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