Quantum eigenvalue estimation for irreducible non-negative matrices
Abstract
Quantum phase estimation algorithm (PEA) has been successfully adapted as a sub frame of many other algorithms applied to a wide variety of applications in different fields. However, the requirement of a good approximate eigenvector given as an input to the algorithm hinders the application of the algorithm to the problems where we do not have any prior knowledge about the eigenvector. In this paper, we show that the principal eigenvalue of an irreducible non-negative operator can be determined by using an equal superposition initial state in the PEA. This removes the necessity of the existence of an initial good approximate eigenvector. Moreover, we show that the success probability of the algorithm is related to the closeness of the operator to a stochastic matrix. Therefore, we draw an estimate for the success probability by using the variance of the column sums of the operator. This provides a priori information which can be used to know the success probability of the algorithm beforehand for the non-negative matrices and apply the algorithm only in cases when the estimated probability is reasonably high. Finally, we discuss the possible applications and show the results for random symmetric matrices and 3-local Hamiltonians with non-negative off-diagonal elements.
References
- 1. , Proc. Roy. Soc. London. A. Math. Phys. Sci. 400 (1985) 97. Crossref, Web of Science, Google Scholar
- 2. , SIAM J. Comput 26 (1997) 1510. Crossref, Web of Science, Google Scholar
- 3. , Phys. Rev. Lett. 83 (1999) 5162. Crossref, Web of Science, Google Scholar
- 4. , ECCC 3 (1996). Google Scholar
- 5. , J. Phys. A Math. Gen. 37 (2004) 4607. Crossref, Google Scholar
- 6. , Phys. Rev. A 82 (2010) 062303. Crossref, Web of Science, Google Scholar
- 7. , J. Chem. Phys. 134 (2011) 144112. Crossref, Web of Science, Google Scholar
- 8. , Quantum Inf. Process. 13 (2014) 333. Crossref, Web of Science, Google Scholar
- 9. , Sci. 309 (2005) 1704. Crossref, Web of Science, Google Scholar
- 10. , PNAS 105 (2008) 18681. Crossref, Web of Science, Google Scholar
- 11. , Phys. Chem. Chem. Phys. 10 (2008) 5388. Crossref, Web of Science, Google Scholar
- 12. , J. Chemical Phys. 133 (2010) 194106. Crossref, Web of Science, Google Scholar
- 13. , Nat. Chem. 2 (2010) 106. Crossref, Web of Science, Google Scholar
- 14. , Phys. Rev. A 85 (2012) 062304. Crossref, Web of Science, Google Scholar
- 15. , Phys. Chem. Chem. Phys. 14 (2012) 9411. Crossref, Web of Science, Google Scholar
- 16. , Ann. Rev. Phys. Chem. 62 (2011) 185. Crossref, Web of Science, Google Scholar
- 17. , Rev. Mod. Phys. 86 (2014) 153. Crossref, Web of Science, Google Scholar
- 18. , Matrix Analysis and Applied Linear Algebra Book and Solutions Manual, Vol. 2, (Society for Industrial and Applied Mathematics, USA, 2000). Crossref, Google Scholar
- 19. , Nonnegative Matrices and Applications, Vol. 64, (Cambridge University Press, Cambridge, 1997). Crossref, Google Scholar
- 20. , Math. Sci. Classics Appl. Math. 9 (1979). Google Scholar
- 21. , SIAM J. Numer. Anal. 7 (1970) 424. Crossref, Web of Science, Google Scholar
- 22. , J. Chem. Inf. Comput. Sci. 41 (2001) 739. Crossref, Google Scholar
- 23. , Quantum Inf. Processing 13 (2014) 2653. Crossref, Web of Science, Google Scholar
- 24. , Electron. J. Linear Al. 16 (2007) 366. Web of Science, Google Scholar
- 25. ,
Order structure and topological methods in nonlinear partial differential equations , Krein-Rutman Theorem and the Principal Eigenvalue, Vol. 2, (world Scientific, Singapore, 2006). Link, Google Scholar - 26. , Quantum Inf. Comput. 8 (2008) 361. Web of Science, Google Scholar
- 27. , SIAM J. Comput. 39 (2010) 1462. Crossref, Web of Science, Google Scholar
- 28. , SIAM J. Comput. 35 (2006) 1070. Crossref, Web of Science, Google Scholar
- 29. , Phys. Rev. A 81 (2010) 032331. Crossref, Web of Science, Google Scholar
- 30. , Phys. Rev. Lett. 79 (1997) 491. Crossref, Web of Science, Google Scholar
- 31. , Math. Magazine 48 (1975) 192. Crossref, Google Scholar
- 32. ,
Matrix nearness problems and applications , Applications of Matrix Theory, (Oxford University Press, Oxford, 1989), pp. 1. Google Scholar - 33. , Closest normal matrix finally found! BIT Numer. Math. 27 (1987) 585. Crossref, Web of Science, Google Scholar
- 34. , Phys. Rev. Lett. 92 (2004) 177902. Crossref, Web of Science, Google Scholar
- 35. , Comput. Phys. Commun. 183 (2012) 1760. Crossref, Web of Science, Google Scholar
- 36. , Comput. Phys. Commun. 184 (2013) 1234. Crossref, Web of Science, Google Scholar
| Remember to check out the Most Cited Articles! |
|---|
|
Check out Annual Physics Catalogue 2019 and recommend us to your library! |
