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Scalable Quantum Simulation of Molecular Energies

P. J. J. O’Malley1,*, R. Babbush2,†, I. D. Kivlichan3, J. Romero3, J. R. McClean4, R. Barends5, J. Kelly5, P. Roushan5, A. Tranter6,7 et al.

N. Ding2, B. Campbell1, Y. Chen5, Z. Chen1, B. Chiaro1, A. Dunsworth1, A. G. Fowler5, E. Jeffrey5, E. Lucero5, A. Megrant5, J. Y. Mutus5, M. Neeley5, C. Neill1, C. Quintana1, D. Sank5, A. Vainsencher1, J. Wenner1, T. C. White5, P. V. Coveney7, P. J. Love6, H. Neven2, A. Aspuru-Guzik3, and J. M. Martinis5,1,‡

  • 1Department of Physics, University of California, Santa Barbara, California 93106, USA
  • 2Google Inc., Venice, California 90291, USA
  • 3Department of Chemistry, Harvard University, Cambridge, Massachusetts 02138, USA
  • 4Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
  • 5Google Inc., Santa Barbara, California 93117, USA
  • 6Department of Physics, Tufts University, Medford, Massachusetts 02155, USA
  • 7Center for Computational Science and Department of Chemistry, University College London, London WC1H 0AJ, United Kingdom

  • *Corresponding author. pomalley@physics.ucsb.edu
  • †Corresponding author. babbush@google.com
  • ‡Corresponding author. martinis@google.com
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Phys. Rev. X 6, 031007 – Published 18 July, 2016

DOI: https://doi.org/10.1103/PhysRevX.6.031007

Abstract

We report the first electronic structure calculation performed on a quantum computer without exponentially costly precompilation. We use a programmable array of superconducting qubits to compute the energy surface of molecular hydrogen using two distinct quantum algorithms. First, we experimentally execute the unitary coupled cluster method using the variational quantum eigensolver. Our efficient implementation predicts the correct dissociation energy to within chemical accuracy of the numerically exact result. Second, we experimentally demonstrate the canonical quantum algorithm for chemistry, which consists of Trotterization and quantum phase estimation. We compare the experimental performance of these approaches to show clear evidence that the variational quantum eigensolver is robust to certain errors. This error tolerance inspires hope that variational quantum simulations of classically intractable molecules may be viable in the near future.

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Popular Summary

Universal simulation of physical systems is among the most compelling applications of quantum computing. In particular, quantum simulation of molecular energies promises significant advances in our understanding of chemistry, including precise predictions of chemical reaction rates. These advances, which may require relatively few qubits to achieve, may have broad industrial impact given the wide importance of chemistry. Here, we report the first electronic structure calculation performed on a quantum computer without exponentially costly precompilation.

We use a programmable array of superconducting qubits held at a base temperature of 20 mK in a dilution refrigerator; each qubit is composed of a superconducting quantum interference device (SQUID) and a capacitor, and is coupled to its nearest neighbors. We compute the energy surface of molecular hydrogen (i.e., the potential energy curve) using two distinct quantum algorithms: the variational quantum eigensolver and the phase estimation algorithm with Trotterization. We compare the experimental performance of these approaches and show that the variational quantum eigensolver yields chemically accurate results. Our experiments also reveal that the variational quantum eigensolver is robust to certain systematic errors.

Our findings suggest that the quantum simulation of classically intractable molecules may be possible without the overhead of quantum error correction.

Article Text

References (50)

  1. S. Lloyd, Universal Quantum Simulators, Science 273, 1073 (1996).
  2. A. Aspuru-Guzik, A. D. Dutoi, P. J. Love, and M. Head-Gordon, Simulated Quantum Computation of Molecular Energies, Science 309, 1704 (2005).
  3. R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, Y. C. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O’Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and J. Martinis, Superconducting Quantum Circuits at the Surface Code Threshold for Fault Tolerance, Nature (London) 508, 500 (2014).
  4. J. Kelly, R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, A. N. Cleland, and J. M. Martinis, State Preservation by Repetitive Error Detection in a Superconducting Quantum Circuit, Nature (London) 519, 66 (2015).
  5. A. D. Córcoles, E. Magesan, S. J. Srinivasan, A. W. Cross, M. Steffen, J. M. Gambetta, and J. M. Chow, Demonstration of a Quantum Error Detection Code Using a Square Lattice of Four Superconducting Qubits, Nat. Commun. 6, 6979 (2015).
  6. D. Ristè, S. Poletto, M.-Z. Huang, A. Bruno, V. Vesterinen, O.-P. Saira, and L. DiCarlo, Detecting Bit-Flip Errors in a Logical Qubit Using Stabilizer Measurements, Nat. Commun. 6, 6983 (2015).
  7. R. Barends, L. Lamata, J. Kelly, L. García-Álvarez, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, E. Solano, and J. M. Martinis, Digital Quantum Simulation of Fermionic Models with a Superconducting Circuit, Nat. Commun. 6, 7654 (2015).
  8. J. D. Whitfield, J. Biamonte, and A. Aspuru-Guzik, Simulation of Electronic Structure Hamiltonians Using Quantum Computers, Mol. Phys. 109, 735 (2011).
  9. Ivan Kassal, James Whitfield, A. Perdomo-Ortiz, M.-H. Yung, and A. Aspuru-Guzik, Simulating Chemistry Using Quantum Computers, Annu. Rev. Phys. Chem. 62, 185 (2011).
  10. N. C. Jones, James D. Whitfield, P. L. McMahon, M.-H. Yung, R. V. Meter, A. Aspuru-Guzik, and Y. Yamamoto, Faster Quantum Chemistry Simulation on Fault-Tolerant Quantum Computers, New J. Phys. 14, 115023 (2012).
  11. D. Wecker, B. Bauer, B. K. Clark, M. B. Hastings, and M. Troyer, Gate-Count Estimates for Performing Quantum Chemistry on Small Quantum Computers, Phys. Rev. A 90, 022305 (2014).
  12. D. Poulin, M. B. Hastings, D. Wecker, N. Wiebe, A. C. Doherty, and M. Troyer, The Trotter Step Size Required for Accurate Quantum Simulation of Quantum Chemistry, Quantum Inf. Comput. 15, 361 (2015).
  13. R. Babbush, J. McClean, D. Wecker, A. Aspuru-Guzik, and N. Wiebe, Chemical Basis of Trotter-Suzuki Errors in Chemistry Simulation, Phys. Rev. A 91, 022311 (2015).
  14. J. D. Whitfield, Spin-Free Quantum Computational Simulations and Symmetry Adapted States, J. Chem. Phys. 139, 021105 (2013).
  15. L. Veis and J. Pittner, Adiabatic State Preparation Study of Methylene, J. Chem. Phys. 140, 214111 (2014).
  16. L. Veis, J. Višák, H. Nakai, and J. Pittner, Quantum Chemistry beyond Born-Oppenheimer Approximation on a Quantum Computer: A Simulated Phase Estimation Study, arXiv:1507.03271.
  17. A. Tranter, S. Sofia, J. Seeley, M. Kaicher, J. McClean, R. Babbush, P. V. Coveney, F. Mintert, F. Wilhelm, and P. J. Love, The Bravyi-Kitaev Transformation: Properties and Applications, Int. J. Quantum Chem. 115, 1431 (2015).
  18. R. Babbush, P. J. Love, and A. Aspuru-Guzik, Adiabatic Quantum Simulation of Quantum Chemistry, Sci. Rep. 4, 6603 (2014).
  19. A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’Brien, A Variational Eigenvalue Solver on a Photonic Quantum Processor, Nat. Commun. 5, 1 (2014).
  20. M.-H. Yung, J. Casanova, A. Mezzacapo, J. McClean, L. Lamata, A. Aspuru-Guzik, and E. Solano, From Transistor to Trapped-Ion Computers for Quantum Chemistry, Sci. Rep. 4, 9 (2014).
  21. J. R. McClean, R. Babbush, P. J. Love, and A. Aspuru-Guzik, Exploiting Locality in Quantum Computation for Quantum Chemistry, J. Phys. Chem. Lett. 5, 4368 (2014).
  22. D. Wecker, M. B. Hastings, and M. Troyer, Progress towards Practical Quantum Variational Algorithms, Phys. Rev. A 92, 042303 (2015).
  23. J. R. McClean, J. Romero, R. Babbush, and A. Aspuru-Guzik, The Theory of Variational Hybrid Quantum-Classical Algorithms, New J. Phys. 18, 023023 (2016).
  24. B. Toloui and P. J. Love, Quantum Algorithms for Quantum Chemistry Based on the Sparsity of the CI-Matrix, arXiv:1312.2579.
  25. R. Babbush, D. W. Berry, I. D. Kivlichan, A. Y. Wei, P. J. Love, and A. Aspuru-Guzik, Exponentially More Precise Quantum Simulation of Fermions in Second Quantization, New J. Phys. 18, 033032 (2016).
  26. R. Babbush, D. W. Berry, I. D. Kivlichan, A. Y. Wei, P. J. Love, and A. Aspuru-Guzik, Exponentially More Precise Quantum Simulation of Fermions II: Quantum Chemistry in the CI Matrix Representation, arXiv:1506.01029.
  27. C. J. Trout and K. R. Brown, Magic State Distillation and Gate Compilation in Quantum Algorithms for Quantum Chemistry, Int. J. Quantum Chem. 115, 1296 (2015).
  28. N. Moll, A. Fuhrer, P. Staar, and I. Tavernelli, Optimizing Qubit Resources for Quantum Chemistry Simulations in Second Quantization on a Quantum Computer, J. Phys. A 49, 295301 (2016).
  29. L. Mueck, Quantum Reform, Nat. Chem. 7, 361 (2015).
  30. J. M Martinis, Qubit Metrology for Building a Fault-Tolerant Quantum Computer, arXiv:1502.01406.
  31. M. W. Johnson et al., Quantum Annealing with Manufactured Spins, Nature (London) 473, 194 (2011).
  32. R. Barends et al., Digitized Adiabatic Quantum Computing with a Superconducting Circuit, Nature (London) 534, 222 (2016).
  33. J. A. Smolin, Graeme Smith, and A. Vargo, Oversimplifying Quantum Factoring, Nature (London) 499, 163 (2013).
  34. H. F. Trotter, On the Product of Semi-Groups of Operators, Proc. Am. Math. Soc. 10, 545 (1959).
  35. A. Y. Kitaev, Quantum Measurements and the Abelian Stabilizer Problem, arXiv:quant-ph/9511026.
  36. B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, Towards Quantum Chemistry on a Quantum Computer, Nat. Chem. 2, 106 (2010).
  37. J. Du, N. Xu, X. Peng, P. Wang, S. Wu, and D. Lu, NMR Implementation of a Molecular Hydrogen Quantum Simulation with Adiabatic State Preparation, Phys. Rev. Lett. 104, 030502 (2010).
  38. Y. Wang, F. Dolde, J. Biamonte, R. Babbush, V. Bergholm, S. Yang, I. Jakobi, P. Neumann, A. Aspuru-Guzik, J. D Whitfield, and J. Wrachtrup, Quantum Simulation of Helium Hydride Cation in a Solid-State Spin Register, ACS Nano 9, 7769 (2015).
  39. Y. Shen, X. Zhang, S. Zhang, J.-N. Zhang, M.-H. Yung, and K. Kim, Quantum Implementation of Unitary Coupled Cluster for Simulating Molecular Electronic Structure, arXiv:1506.00443.
  40. R. D. Somma, G. Ortiz, J. E. Gubernatis, E. Knill, and R. Laflamme, Simulating Physical Phenomena by Quantum Networks, Phys. Rev. A 65, 042323 (2002).
  41. S. Bravyi and A. Kitaev, Fermionic Quantum Computation, Ann. Phys. (Amsterdam) 298, 210 (2002).
  42. J. T. Seeley, M. J. Richard, and P. J. Love, The Bravyi-Kitaev Transformation for Quantum Computation of Electronic Structure, J. Chem. Phys. 137, 224109 (2012).
  43. T Helgaker, P Jorgensen, and J. Olsen, Molecular Electronic Structure Theory (Wiley, New York, 2002).
  44. M. R. Hoffmann and J. Simons, A Unitary Multiconfigurational Coupled-Cluster Method: Theory and Applications, J. Chem. Phys. 88, 993 (1988).
  45. R. J. Bartlett, S. A. Kucharski, and J. Noga, Alternative Coupled-Cluster Ansätze II. The Unitary Coupled-Cluster Method, Chem. Phys. Lett. 155, 133 (1989).
  46. A. G. Taube and R. J. Bartlett, New Perspectives on Unitary Coupled-Cluster Theory, Int. J. Quantum Chem. 106, 3393 (2006).
  47. C. M. Bishop, Pattern Recognition and Machine Learning (Springer, New York, 2006).
  48. R. Barends, J. Kelly, A. Megrant, D. Sank, E. Jeffrey, Y. Chen, Y. Yin, B. Chiaro, J. Mutus, C. Neill, P. O’Malley, P. Roushan, J. Wenner, T. C. White, A. N. Cleland, and J. M. Martinis, Coherent Josephson Qubit Suitable for Scalable Quantum Integrated Circuits, Phys. Rev. Lett. 111, 080502 (2013).
  49. J. Koch, T. Yu, J. Gambetta, A. Houck, D. Schuster, J. Majer, A. Blais, M. Devoret, S. Girvin, and R. Schoelkopf, Charge-Insensitive Qubit Design Derived from the Cooper Pair Box, Phys. Rev. A 76, 042319 (2007).
  50. J. Kelly et al., Optimal Quantum Control Using Randomized Benchmarking, Phys. Rev. Lett. 112, 240504 (2014).

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Phys. Rev. X 6, 031007– Published 18 July, 2016
  • Received 7 April 2016
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This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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