Exact and efficient Lanczos method on a quantum computer

William Kirby; Mario Motta; Antonio Mezzacapo

doi:10.22331/q-2023-05-23-1018

Exact and efficient Lanczos method on a quantum computer

William Kirby^1,2, Mario Motta³, and Antonio Mezzacapo⁴

¹IBM Quantum, IBM Research Cambridge, Cambridge, MA 02142, USA
²Department of Physics and Astronomy, Tufts University, Medford, MA 02155, USA
³IBM Quantum, IBM Research Almaden, San Jose, CA 95120, USA
⁴IBM Quantum, T. J. Watson Research Center, Yorktown Heights, NY 10598, USA

Published:	2023-05-23, volume 7, page 1018
Eprint:	arXiv:2208.00567v4
Doi:	https://doi.org/10.22331/q-2023-05-23-1018
Citation:	Quantum 7, 1018 (2023).

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Abstract

We present an algorithm that uses block encoding on a quantum computer to exactly construct a Krylov space, which can be used as the basis for the Lanczos method to estimate extremal eigenvalues of Hamiltonians. While the classical Lanczos method has exponential cost in the system size to represent the Krylov states for quantum systems, our efficient quantum algorithm achieves this in polynomial time and memory. The construction presented is exact in the sense that the resulting Krylov space is identical to that of the Lanczos method, so the only approximation with respect to the exact method is due to finite sample noise. This is possible because, unlike previous quantum Krylov methods, our algorithm does not require simulating real or imaginary time evolution. We provide an explicit error bound for the resulting ground state energy estimate in the presence of noise. For our method to be successful efficiently, the only requirement on the input problem is that the overlap of the initial state with the true ground state must be $\Omega(1/\text{poly}(n))$ for $n$ qubits.

Featured image: The circuits required to generate a Krylov space of dimension $D$, expressed in terms of the input block encoding operators.

► BibTeX data

@article{Kirby2023exactefficient,
  doi = {10.22331/q-2023-05-23-1018},
  url = {https://doi.org/10.22331/q-2023-05-23-1018},
  title = {Exact and efficient {L}anczos method on a quantum computer},
  author = {Kirby, William and Motta, Mario and Mezzacapo, Antonio},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {7},
  pages = {1018},
  month = may,
  year = {2023}
}

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