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  • Open Access

Heisenberg-Limited Ground-State Energy Estimation for Early Fault-Tolerant Quantum Computers

Lin Lin1,2,3,* and Yu Tong1,†

  • 1Department of Mathematics, University of California, Berkeley, California 94720, USA
  • 2Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
  • 3Challenge Institute of Quantum Computation, University of California, Berkeley, California 94720, USA

  • *linlin@math.berkeley.edu
  • †yu_tong@berkeley.edu
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PRX Quantum 3, 010318 – Published 2 February, 2022

DOI: https://doi.org/10.1103/PRXQuantum.3.010318

Abstract

Under suitable assumptions, the quantum-phase-estimation (QPE) algorithm is able to achieve Heisenberg-limited precision scaling in estimating the ground-state energy. However, QPE requires a large number of ancilla qubits and a large circuit depth, as well as the ability to perform inverse quantum Fourier transform, making it expensive to implement on an early fault-tolerant quantum computer. We propose an alternative method to estimate the ground-state energy of a Hamiltonian with Heisenberg-limited precision scaling, which employs a simple quantum circuit with one ancilla qubit, and a classical postprocessing procedure. Besides the ground-state energy, our algorithm also produces an approximate cumulative distribution function of the spectral measure, which can be used to compute other spectral properties of the Hamiltonian.

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Fault-tolerant quantum computers have the potential to significantly accelerate quantum many-body simulations. This paper concerns the estimation of the ground-state energy of a Hamiltonian on early fault-tolerant quantum devices. These devices are expected to be much more robust and powerful than the current noisy intermediate-scale quantum devices but have a limited number of logical qubits and limited circuit depths. Most existing fault-tolerant quantum algorithms for this task require a significant number of qubits and/or large circuit depths, which can be impractical in the early fault-tolerant setting.

Provided that a good initial guess of the ground state is available, our work can estimate the ground-state energy using only one ancilla qubit. Our algorithm achieves the optimal Heisenberg-limited precision scaling both in terms of the circuit depth and the total evolution time. The key technique is to perform the time evolution with a random duration followed by measurements. This allows us to stochastically compute a cumulative distribution function of the spectral measure, which encodes the information of the ground-state energy. The techniques developed in this work can be useful in performing many other important tasks on early fault-tolerant quantum computers, such as computing the spectral measure and the many-body Green’s function.

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Outline

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PRX Quantum 3, 010318– Published 2 February, 2022
  • Received 27 February 2021
  • Revised 6 October 2021
  • Accepted 6 January 2022
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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.

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