Quantum Information Processing

, Volume 14, Issue 1, pp 103–118 | Cite as

A quantum algorithm for obtaining the lowest eigenstate of a Hamiltonian assisted with an ancillary qubit system

  • Jeongho Bang
  • Seung-Woo Lee
  • Chang-Woo Lee
  • Hyunseok Jeong
Article
  • 263 Downloads

Abstract

We propose a quantum algorithm to obtain the lowest eigenstate of any Hamiltonian simulated by a quantum computer. The proposed algorithm begins with an arbitrary initial state of the simulated system. A finite series of transforms is iteratively applied to the initial state assisted with an ancillary qubit. The fraction of the lowest eigenstate in the initial state is then amplified up to \(\simeq \)1. We prove that our algorithm can faithfully work for any arbitrary Hamiltonian in the theoretical analysis. Numerical analyses are also carried out. We firstly provide a numerical proof-of-principle demonstration with a simple Hamiltonian in order to compare our scheme with the so-called “Demon-like algorithmic cooling (DLAC)”, recently proposed in Xu (Nat Photonics 8:113, 2014). The result shows a good agreement with our theoretical analysis, exhibiting the comparable behavior to the best ‘cooling’ with the DLAC method. We then consider a random Hamiltonian model for further analysis of our algorithm. By numerical simulations, we show that the total number \(n_c\) of iterations is proportional to \(\simeq \mathcal{O}(D^{-1}\epsilon ^{-0.19})\), where \(D\) is the difference between the two lowest eigenvalues and \(\epsilon \) is an error defined as the probability that the finally obtained system state is in an unexpected (i.e., not the lowest) eigenstate.

Keywords

Quantum algorithm Algorithmic cooling Ground state 

Notes

Acknowledgments

The authors thank Sunwhan Jo, Chanhyoup Lee, and Junghee Ryu for helpful discussions. We acknowledge the support of the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (No. 2010-0018295 and No. 2010-0015059).

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Jeongho Bang
    • 1
    • 4
  • Seung-Woo Lee
    • 1
  • Chang-Woo Lee
    • 2
  • Hyunseok Jeong
    • 1
    • 3
  1. 1.Department of Physics and Astronomy, Center for Macroscopic Quantum ControlSeoul National UniversitySeoulKorea
  2. 2.Department of PhysicsTexas A&M University at QatarDohaQatar
  3. 3.Centre for Quantum Computation and Communication Technology, School of Mathematics and PhysicsUniversity of QueenslandBrisbaneAustralia
  4. 4.Department of PhysicsHanyang UniversitySeoulKorea

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