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Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets


Quantum computers can be used to address electronic-structure problems and problems in materials science and condensed matter physics that can be formulated as interacting fermionic problems, problems which stretch the limits of existing high-performance computers1. Finding exact solutions to such problems numerically has a computational cost that scales exponentially with the size of the system, and Monte Carlo methods are unsuitable owing to the fermionic sign problem. These limitations of classical computational methods have made solving even few-atom electronic-structure problems interesting for implementation using medium-sized quantum computers. Yet experimental implementations have so far been restricted to molecules involving only hydrogen and helium2,3,4,5,6,7,8. Here we demonstrate the experimental optimization of Hamiltonian problems with up to six qubits and more than one hundred Pauli terms, determining the ground-state energy for molecules of increasing size, up to BeH2. We achieve this result by using a variational quantum eigenvalue solver (eigensolver) with efficiently prepared trial states that are tailored specifically to the interactions that are available in our quantum processor, combined with a compact encoding of fermionic Hamiltonians9 and a robust stochastic optimization routine10. We demonstrate the flexibility of our approach by applying it to a problem of quantum magnetism, an antiferromagnetic Heisenberg model in an external magnetic field. In all cases, we find agreement between our experiments and numerical simulations using a model of the device with noise. Our results help to elucidate the requirements for scaling the method to larger systems and for bridging the gap between key problems in high-performance computing and their implementation on quantum hardware.

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Figure 1: Quantum chemistry on a superconducting quantum processor.
Figure 2: Experimental implementation of six-qubit optimization.
Figure 3: Application to quantum chemistry.
Figure 4: Application to quantum magnetism.

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  1. National Energy Research Scientific Computing Center 2015 Annual Report (2015)

  2. Lanyon, B. P. et al. Towards quantum chemistry on a quantum computer. Nat. Chem. 2, 106–111 (2010)

    Article  CAS  PubMed  Google Scholar 

  3. Du, J. et al. NMR implementation of a molecular hydrogen quantum simulation with adiabatic state preparation. Phys. Rev. Lett. 104, 030502 (2010)

    Article  ADS  PubMed  Google Scholar 

  4. Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nat. Commun. 5, 4213 (2014)

    Article  ADS  CAS  PubMed  Google Scholar 

  5. Wang, Y. et al. Quantum simulation of helium hydride cation in a solid-state spin register. ACS Nano 9, 7769–7774 (2015)

    Article  CAS  PubMed  Google Scholar 

  6. O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016)

    Article  Google Scholar 

  7. Shen, Y. et al. Quantum implementation of the unitary coupled cluster for simulating molecular electronic structure. Phys. Rev. A 95, 020501 (2017)

    Article  ADS  Google Scholar 

  8. Paesani, S. et al. Experimental Bayesian quantum phase estimation on a silicon photonic chip. Phys. Rev. Lett. 118, 100503 (2017)

    Article  ADS  CAS  PubMed  Google Scholar 

  9. Bravyi, S., Gambetta, J. M., Mezzacapo, A. & Temme, K. Tapering off qubits to simulate fermionic hamiltonians. Preprint at (2017)

  10. Spall, J. C. Multivariate stochastic approximation using a simultaneous perturbation gradient approximation. IEEE Trans. Automat. Contr. 37, 332–341 (1992)

    Article  MathSciNet  Google Scholar 

  11. Bravyi, S. & Kitaev, A. Fermionic quantum computation. Ann. Phys. 298, 210–226 (2002)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  12. Kempe, J., Kitaev, A. & Regev, O. The complexity of the local Hamiltonian problem. SIAM J. Comput. 35, 1070–1097 (2006)

    Article  MathSciNet  Google Scholar 

  13. Abrams, D. S. & Lloyd, S. Simulation of many-body Fermi systems on a universal quantum computer. Phys. Rev. Lett. 79, 2586–2589 (1997)

    Article  ADS  CAS  Google Scholar 

  14. Aspuru-Guzik, A., Dutoi, A. D., Love, P. J. & Head-Gordon, M. Simulated quantum computation of molecular energies. Science 309, 1704–1707 (2005)

    Article  ADS  CAS  PubMed  Google Scholar 

  15. Kitaev, A. Y. Quantum measurements and the Abelian stabilizer problem. Preprint at (1995)

  16. Farhi, E., Goldstone, J. & Gutmann, S. A quantum approximate optimization algorithm. Preprint at (2014)

  17. Farhi, E., Goldstone, J., Gutmann, S. & Neven, H. Quantum algorithms for fixed qubit architectures. Preprint at (2017)

  18. Yung, M.-H. et al. From transistor to trapped-ion computers for quantum chemistry. Sci. Rep. 4, 3589 (2014)

    Article  PubMed  PubMed Central  Google Scholar 

  19. McClean, J., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New J. Phys. 18, 023023 (2016)

    Article  ADS  Google Scholar 

  20. Wecker, D., Hastings, M. B. & Troyer, M. Progress towards practical quantum variational algorithms. Phys. Rev. A 92, 042303 (2015)

    Article  ADS  Google Scholar 

  21. Romero, J. et al. Strategies for quantum computing molecular energies using the unitary coupled cluster ansatz. Preprint at (2017)

  22. Hutchings, M. et al. Tunable superconducting qubits with flux-independent coherence. Preprint at (2017)

  23. Sheldon, S., Magesan, E., Chow, J. M. & Gambetta, J. M. Procedure for systematically tuning up cross-talk in the cross-resonance gate. Phys. Rev. A 93, 060302 (2016)

    Article  ADS  Google Scholar 

  24. McKay, D. C., Wood, C. J., Sheldon, S., Chow, J. M. & Gambetta, J. M. Efficient Z-gates for quantum computing. Preprint at (2017)

  25. McClean, J. R., Schwartz, M. E., Carter, J. & de Jong, W. A. Hybrid quantum-classical hierarchy for mitigation of decoherence and determination of excited states. Phys. Rev. A 95, 042308 (2017)

    Article  ADS  Google Scholar 

  26. Li, Y. & Benjamin, S. C. Efficient variational quantum simulator incorporating active error minimisation. Phys. Rev. X 7, 021050 (2017)

    Google Scholar 

  27. Temme, K., Bravyi, S. & Gambetta, J. M. Error mitigation for short depth quantum circuits. Preprint at (2017)

  28. Lanyon, B. P. et al. Universal digital quantum simulation with trapped ions. Science 334, 57–61 (2011)

    Article  ADS  CAS  PubMed  Google Scholar 

  29. Bultink, C. C. et al. Active resonator reset in the nonlinear dispersive regime of circuit QED. Phys. Rev. Appl. 6, 034008 (2016)

    Article  ADS  Google Scholar 

  30. Spall, J. C. Adaptive stochastic approximation by the simultaneous perturbation method. IEEE Trans. Automat. Contr. 45, 1839–1853 (2000)

    Article  MathSciNet  Google Scholar 

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We thank J. Chavez-Garcia, A. D. Córcoles and J. Rozen for experimental contributions, J. Hertzberg and S. Rosenblatt for room temperature characterization, B. Abdo for design and characterization of the Josephson Parametric Converters, S. Brayvi, J. Smolin, E. Magesan, L. Bishop, S. Sheldon, N. Moll, P. Barkoutsos and I. Tavernelli for discussions, and W. Shanks for assistance with the experimental control software. We thank A. D. Córcoles for edits to the manuscript. We acknowledge support from the IBM Research Frontiers Institute. The research is based on work supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via the Army Research Office contract W911NF-10-1-0324.

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Authors and Affiliations



A.K. and A.M. contributed equally to this work. J.M.G. and K.T. designed the experiments. A.K. and M.T. characterized the device and A.K. performed the experiments. M.B. fabricated the devices. A.M. developed the theory and the numerical simulations. A.K., A.M. and J.M.G. interpreted and analysed the experimental data. A.K., A.M., K.T., J.M.C. and J.M.G. contributed to writing the manuscript.

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Correspondence to Abhinav Kandala or Antonio Mezzacapo.

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The authors declare no competing financial interests.

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Reviewer Information Nature thanks N. Linke and the other anonymous reviewer(s) for their contribution to the peer review of this work.

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Kandala, A., Mezzacapo, A., Temme, K. et al. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature 549, 242–246 (2017).

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