Abstract
We develop a phase-estimation method with a distinct feature: its maximal run time (which determines the circuit depth) is , where is the target precision, and the preconstant can be arbitrarily close to as the initial state approaches the target eigenstate. The total cost of the algorithm satisfies the Heisenberg-limited scaling . As a result, our algorithm may significantly reduce the circuit depth for performing phase-estimation tasks on early fault-tolerant quantum computers. The key technique is a simple subroutine called quantum complex exponential least squares (QCELS). Our algorithm can be readily applied to reduce the circuit depth for estimating the ground-state energy of a quantum Hamiltonian, when the overlap between the initial state and the ground state is large. If this initial overlap is small, we can combine our method with the Fourier-filtering method developed in [Lin and Tong, PRX Quantum 3, 010318, 2022], and the resulting algorithm provably reduces the circuit depth in the presence of a large relative overlap compared to . The relative-overlap condition is similar to a spectral-gap assumption but it is aware of the information in the initial state and is therefore applicable to certain Hamiltonians with small spectral gaps. We observe that the circuit depth can be reduced by around 2 orders of magnitude in numerical experiments under various settings.
1 More- Received 6 December 2022
- Revised 21 February 2023
- Accepted 24 April 2023
DOI:https://doi.org/10.1103/PRXQuantum.4.020331
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
Phase estimation is one of the most important quantum primitives. This paper focuses on designing phase-estimation algorithms that are suitable for early fault-tolerant quantum computers. Compared with full fault-tolerant computers, early fault-tolerant quantum computers have a limited number of logical qubits and limited circuit depths. Thus, algorithms on early fault-tolerant quantum computers should have a small number of qubits and small circuit depths.
In our paper, we develop a phase-estimation method that requires only one ancilla qubit and a small maximal run time. In the meantime, the total cost of our algorithm still satisfies the Heisenberg-limited scaling and is similar to that of other phase-estimation methods. As a result, our algorithm can significantly reduce the circuit depth for performing phase-estimation tasks on early fault-tolerant quantum computers. The key technique is a simple subroutine called quantum complex exponential least squares. We first transfer the phase-estimation problem to a fitting problem and then approximate the phase by solving an optimization problem. As an application, we use the idea to estimate the ground-state energy of a quantum Hamiltonian and justify the efficiency of our methods theoretically and numerically.