Abstract

Preparing the ground state of a given Hamiltonian and estimating its ground energy are important but computationally hard tasks. However, given some additional information, these problems can be solved efficiently on a quantum computer. We assume that an initial state with non-trivial overlap with the ground state can be efficiently prepared, and the spectral gap between the ground energy and the first excited energy is bounded from below. With these assumptions we design an algorithm that prepares the ground state when an upper bound of the ground energy is known, whose runtime has a logarithmic dependence on the inverse error. When such an upper bound is not known, we propose a hybrid quantum-classical algorithm to estimate the ground energy, where the dependence of the number of queries to the initial state on the desired precision is exponentially improved compared to the current state-of-the-art algorithm proposed in [Ge et al. 2019]. These two algorithms can then be combined to prepare a ground state without knowing an upper bound of the ground energy. We also prove that our algorithms reach the complexity lower bounds by applying it to the unstructured search problem and the quantum approximate counting problem.

Highlights

  • Estimating ground energy and obtaining information on the ground state of a given quantum Hamiltonian are of immense importance in condensed matter physics, quantum chemistry, and quantum information

  • The key assumption is that we have an initial state |φ0 which can be efficiently prepared by an oracle UI, and has some overlap with the ground state |ψ0 lower bounded by γ

  • In this work we first develop a filtering method that filters out all eigenstates corresponding to eigenvalues above a certain threshold. This filtering method enables us to prepare the ground state of a Hamiltonian with spectral gap bounded away from zero when only an upper bound of the ground energy is known, unlike in the filtering methods discussed above which all require either exact value or high-precision estimate of the ground energy

Read more

Summary

Introduction

Estimating ground energy and obtaining information on the ground state of a given quantum Hamiltonian are of immense importance in condensed matter physics, quantum chemistry, and quantum information. In this work we first develop a filtering method that filters out all eigenstates corresponding to eigenvalues above a certain threshold This filtering method enables us to prepare the ground state of a Hamiltonian with spectral gap bounded away from zero when only an upper bound of the ground energy is known, unlike in the filtering methods discussed above which all require either exact value or high-precision estimate of the ground energy. Our filtering method has an exponentially improved dependence on precision compared to Kitaev’s phase estimation [28] and uses fewer qubits compared to other variants of the phase estimation algorithm [21, 42] This filtering method, applied to the initial state given in our assumption, enables us to tell whether the ground energy is smaller than a or greater than b for some b > a, with high probability.

Block-encoding of reflector and projector
Algorithm with a priori ground energy bound
Algorithm without a priori ground energy bound
Optimality of the query complexities
Low-energy state preparation
Discussions
A An example of block-encoding and constructing the reflector
B Gap and overlap in the unstructured search problem

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.