A generalized circuit for the Hamiltonian dynamics through the truncated series

  • Ammar Daskin
  • Sabre Kais


In this paper, we present a method for Hamiltonian simulation in the context of eigenvalue estimation problems, which improves earlier results dealing with Hamiltonian simulation through the truncated Taylor series. In particular, we present a fixed-quantum circuit design for the simulation of the Hamiltonian dynamics, \({\mathcal {H}}(t)\), through the truncated Taylor series method described by Berry et al. (Phys Rev Lett 114:090502, 2015). The circuit is general and can be used to simulate any given matrix in the phase estimation algorithm by only changing the angle values of the quantum gates implementing the time variable t in the series. The circuit complexity depends on the number of summation terms composing the Hamiltonian and requires O(Ln) number of quantum gates for the simulation of a molecular Hamiltonian. Here, n is the number of states of a spin orbital, and L is the number of terms in the molecular Hamiltonian and generally is bounded by \(O(n^4)\). We also discuss how to use the circuit in adaptive processes and eigenvalue-related problems along with a slightly modified version of the iterative phase estimation algorithm. In addition, a simple divide-and-conquer method is presented for mapping a matrix which are not given as sums of unitary matrices into the circuit. The complexity of the circuit is directly related to the structure of the matrix and can be bounded by \(O(\mathrm{poly}(n))\) for a matrix with \(\mathrm{poly}(n)\)-sparsity.


Quantum circuit design Taylor expansion Quantum phase estimation 



We would like to thank two anonymous reviewers for their help in improving the clarity of the paper and the complexity analysis of the circuit.


  1. 1.
    Kitaev, A.: Quantum measurements and the Abelian stabilizer problem. Electron. Colloq. Comput. Complex. (ECCC) 3, 5 (1996)Google Scholar
  2. 2.
    Somma, R., Ortiz, G., Gubernatis, J.E., Knill, E., Laflamme, R.: Simulating physical phenomena by quantum networks. Phys. Rev. A 65, 042323 (2002)ADSCrossRefGoogle Scholar
  3. 3.
    Brown, K.L., Munro, W.J., Kendon, V.M.: Using quantum computers for quantum simulation. Entropy 12, 2268–2307 (2010)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Kassal, I., Whitfield, J.D., Perdomo-Ortiz, A., Yung, M.-H., Aspuru-Guzik, A.: Simulating chemistry using quantum computers. Annu. Rev. Phys. Chem. 62, 185–207 (2011)ADSCrossRefGoogle Scholar
  5. 5.
    Wecker, D., Bauer, B., Clark, B.K., Hastings, M.B., Troyer, M.: Gate-count estimates for performing quantum chemistry on small quantum computers. Phys. Rev. A 90, 022305 (2014)ADSCrossRefGoogle Scholar
  6. 6.
    Montanaro, A.: Quantum algorithms: an overview. NPJ Quant. Inf. 2, 15023 (2016)ADSCrossRefGoogle Scholar
  7. 7.
    Nielsen, M.A., Chuang, I.: Quantum computation and quantum information (2002)Google Scholar
  8. 8.
    Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Rev. 41, 303–332 (1999)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Harrow, A.W., Hassidim, A., Lloyd, S.: Quantum algorithm for linear systems of equations. Phys. Rev. Lett. 103, 150502 (2009)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Trotter, H.F.: On the product of semi-groups of operators. Proc. Am. Math. Soc. 10, 545–551 (1959)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Suzuki, M.: Generalized trotter’s formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems. Commun. Math. Phys. 51, 183–190 (1976)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Poulin, D., Hastings, M.B., Wecker, D., Wiebe, N., Doberty, A.C., Troyer, M.: The trotter step size required for accurate quantum simulation of quantum chemistry. Quant. Inf. Comput. 15, 361–384 (2015)MathSciNetGoogle Scholar
  13. 13.
    Berry, D.W., Childs, A.M., Cleve, R., Kothari, R., Somma, R.D.: Simulating hamiltonian dynamics with a truncated taylor series. Phys. Rev. Lett. 114, 090502 (2015)ADSCrossRefGoogle Scholar
  14. 14.
    Daskin, A., Kais, S.: An Ancilla-based quantum simulation framework for non-unitary matrices. Quant. Inf. Process. 16, 33 (2017)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Low, G.H., Chuang, I.L.: Optimal hamiltonian simulation by quantum signal processing. Phys. Rev. Lett. 118, 010501 (2017)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Poulin, D., Kitaev, A., Steiger, D.S., Hastings, M.B., Troyer, M.: Fast quantum algorithm for spectral properties. arXiv preprint arXiv:1711.11025 (2017)
  17. 17.
    Daskin, A., Kais, S.: Direct application of the phase estimation algorithm to find the eigenvalues of the hamiltonians. arXiv preprint arXiv:1703.03597 (2017)
  18. 18.
    Babbush, R., Berry, D.W., Sanders, Y.R., Kivlichan, I.D., Scherer, A., Wei, A.Y., Love, P.J., Aspuru-Guzik, A.: Exponentially more precise quantum simulation of fermions in the configuration interaction representation. Quant. Sci. Technol. 3, 015006 (2018)ADSCrossRefGoogle Scholar
  19. 19.
    Daskin, A., Grama, A., Kollias, G., Kais, S.: Universal programmable quantum circuit schemes to emulate an operator. J. Chem. Phys. 137, 234112 (2012)ADSCrossRefGoogle Scholar
  20. 20.
    Babbush, R., Berry, D., Kieferová, M., Low, H.G., Sanders, Y., Scherer, A., Wiebe, N.: Improved techniques for preparing eigenstates of fermionic hamiltonians. arXiv preprint arXiv:1711.10460 (2017)
  21. 21.
    Low, H.G., Chuang, I.L.: Hamiltonian simulation by uniform spectral amplification. arXiv preprint arXiv:1707.05391 (2017)
  22. 22.
    Low, H.G., Chuang, I.L.: Hamiltonian simulation by qubitization. arXiv preprint arXiv:1610.06546 (2016)
  23. 23.
    Pei, W.: Additive Combinations of Special Operators, pp. 337–361. Banach Center Publications, Warszawa (1994)zbMATHGoogle Scholar
  24. 24.
    Möttönen, M., Vartiainen, J.J., Bergholm, V., Salomaa, M.M.: Quantum circuits for general multiqubit gates. Phys. Rev. Lett. 93, 130502 (2004)ADSCrossRefGoogle Scholar
  25. 25.
    Bullock, S.S., OLeary, D.P., Brennen, G.K.: Asymptotically optimal quantum circuits for d-level systems. Phys. Rev. Lett. 94, 230502 (2005)ADSCrossRefGoogle Scholar
  26. 26.
    Urías, J.: Householder factorizations of unitary matrices. J. Math. Phys. 51, 072204 (2010)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Novo, L., Berry, D.W.: Improved Hamiltonian simulation via a truncated Taylor series and corrections. arXiv preprint arXiv:1611.10033 (2016)
  28. 28.
    Babbush, R., Berry, D.W., Kivlichan, I.D., Wei, A.Y., Love, P.J., Aspuru-Guzik, A.: Exponentially more precise quantum simulation of fermions in second quantization. N. J. Phys. 18, 033032 (2016)CrossRefGoogle Scholar
  29. 29.
    Berry, D.W., Ahokas, G., Cleve, R., Sanders, B.C.: Efficient quantum algorithms for simulating sparse Hamiltonians. Commun. Math. Phys. 270, 359–371 (2007)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Childs, A.M., Kothari, R.: Simulating sparse Hamiltonians with star decompositions. In: Conference on Quantum Computation, Communication, and Cryptography pp. 94–103, Springer, Berlin (2010)CrossRefGoogle Scholar
  31. 31.
    Lanyon, B.P., Whitfield, J.D., Gillett, G.G., Goggin, M.E., Almeida, M.P., Kassal, I., Biamonte, J.D., Mohseni, M., Powell, B.J., Barbieri, M., et al.: Towards quantum chemistry on a quantum computer. Nat. Chem. 2, 106–111 (2010)CrossRefGoogle Scholar
  32. 32.
    Whitfield, J.D., Biamonte, J., Aspuru-Guzik, A.: Simulation of electronic structure hamiltonians using quantum computers. Mol. Phys. 109, 735–750 (2011)ADSCrossRefGoogle Scholar
  33. 33.
    Seeley, J.T., Richard, M.J., Love, P.J.: The Bravyi–Kitaev transformation for quantum computation of electronic structure. J. Chem. Phys. 137, 224109 (2012)ADSCrossRefGoogle Scholar
  34. 34.
    Jordan, P., Wigner, E.: Über das paulische äquivalenzverbot. Zeitschrift für Physik 47, 631–651 (1928)ADSCrossRefGoogle Scholar
  35. 35.
    Bravyi, S.B., Kitaev, A.Y.: Fermionic quantum computation. Ann. Phys. 298, 210–226 (2002)ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    Fassbender, H., Kressner, D.: Structured eigenvalue problems. GAMM Mitt. 29, 297–318 (2006)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Guhr, T., Müller-Groeling, A., Weidenmüller, H.A.: Random-matrix theories in quantum physics: common concepts. Phys. Rep. 299, 189–425 (1998)ADSMathSciNetCrossRefGoogle Scholar
  38. 38.
    Escher, B.M., de Matos Filho, R.L., Davidovich, L.: General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology. Nat. Phys. 7, 406–411 (2011)CrossRefGoogle Scholar
  39. 39.
    Romero, J., Olson, J.P., Aspuru-Guzik, A.: Quantum autoencoders for efficient compression of quantum data. Quant. Sci. Technol. 2, 045001 (2017)ADSCrossRefGoogle Scholar
  40. 40.
    Widrow, B., McCool, J.M., Larimore, M.G., Richard Johnson, C.: Stationary and nonstationary learning characteristics of the LMS adaptive filter. Proc. IEEE 64, 1151–1162 (1976)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Computer EngineeringIstanbul Medeniyet UniversityÜsküdarTurkey
  2. 2.Department of Chemistry, Department of Physics and Birck Nanotechnology CenterPurdue UniversityWest LafayetteUSA

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