Witnessing eigenstates for quantum simulation of Hamiltonian spectra

We introduce the concept of an eigenstate witness and use it to find energies of quantum systems with quantum computers. The efficient calculation of Hamiltonian spectra, a problem often intractable on classical machines, can find application in many fields, from physics to chemistry. We introduce the concept of an “eigenstate witness” and, through it, provide a new quantum approach that combines variational methods and phase estimation to approximate eigenvalues for both ground and excited states. This protocol is experimentally verified on a programmable silicon quantum photonic chip, a mass-manufacturable platform, which embeds entangled state generation, arbitrary controlled unitary operations, and projective measurements. Both ground and excited states are experimentally found with fidelities >99%, and their eigenvalues are estimated with 32 bits of precision. We also investigate and discuss the scalability of the approach and study its performance through numerical simulations of more complex Hamiltonians. This result shows promising progress toward quantum chemistry on quantum computers.

[1]  H. Terai,et al.  Silicon photonic processor of two-qubit entangling quantum logic , 2017, 1709.00214.

[2]  J. Gambetta,et al.  Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets , 2017, Nature.

[3]  J. Rarity,et al.  Experimental quantum Hamiltonian learning , 2017, Nature Physics.

[4]  S. Paesani,et al.  Experimental Bayesian Quantum Phase Estimation on a Silicon Photonic Chip. , 2017, Physical review letters.

[5]  J. McClean,et al.  Strategies for quantum computing molecular energies using the unitary coupled cluster ansatz , 2017, Quantum Science and Technology.

[6]  Mikhail Smelyanskiy,et al.  Practical optimization for hybrid quantum-classical algorithms , 2017, 1701.01450.

[7]  Ying Li,et al.  Efficient Variational Quantum Simulator Incorporating Active Error Minimization , 2016, 1611.09301.

[8]  Matthias Troyer,et al.  Solving the quantum many-body problem with artificial neural networks , 2016, Science.

[9]  M. Troyer,et al.  Elucidating reaction mechanisms on quantum computers , 2016, Proceedings of the National Academy of Sciences.

[10]  Jeremy L O'Brien,et al.  Chip-to-chip quantum photonic interconnect by path-polarization interconversion , 2016 .

[11]  J. Carter,et al.  Hybrid Quantum-Classical Hierarchy for Mitigation of Decoherence and Determination of Excited States , 2016, 1603.05681.

[12]  Nicolò Spagnolo,et al.  Photonic simulation of entanglement growth and engineering after a spin chain quench , 2016, Nature Communications.

[13]  Geoff J Pryde,et al.  A quantum Fredkin gate , 2016, Science Advances.

[14]  P. Coveney,et al.  Scalable Quantum Simulation of Molecular Energies , 2015, 1512.06860.

[15]  M Sanz,et al.  Genetic Algorithms for Digital Quantum Simulations. , 2015, Physical review letters.

[16]  Alán Aspuru-Guzik,et al.  The theory of variational hybrid quantum-classical algorithms , 2015, 1509.04279.

[17]  Michael G. Tanner,et al.  Quantum Photonic Interconnect , 2015, 1508.03214.

[18]  M. Hastings,et al.  Progress towards practical quantum variational algorithms , 2015, 1507.08969.

[19]  Annie Y. Wei,et al.  Exponentially more precise quantum simulation of fermions in second quantization , 2015, 1506.01020.

[20]  Michael J. Strain,et al.  Qubit entanglement between ring-resonator photon-pair sources on a silicon chip , 2014, Nature Communications.

[21]  Alán Aspuru-Guzik,et al.  On the Chemical Basis of Trotter-Suzuki Errors in Quantum Chemistry Simulation , 2014, 1410.8159.

[22]  David Poulin,et al.  The Trotter step size required for accurate quantum simulation of quantum chemistry , 2014, Quantum Inf. Comput..

[23]  Alán Aspuru-Guzik,et al.  A variational eigenvalue solver on a photonic quantum processor , 2013, Nature Communications.

[24]  Dmitri Maslov,et al.  Fast and efficient exact synthesis of single-qubit unitaries generated by clifford and T gates , 2012, Quantum Inf. Comput..

[25]  J. Dalibard,et al.  Quantum simulations with ultracold quantum gases , 2012, Nature Physics.

[26]  B. Lanyon,et al.  Universal Digital Quantum Simulation with Trapped Ions , 2011, Science.

[27]  R. Blatt,et al.  Quantum simulations with trapped ions , 2011, Nature Physics.

[28]  Alán Aspuru-Guzik,et al.  Solving Quantum Ground-State Problems with Nuclear Magnetic Resonance , 2011, Scientific reports.

[29]  Giuseppe Vallone,et al.  Polarization entangled states measurement on a chip , 2011, Optics + Optoelectronics.

[30]  A. Beskos,et al.  On the stability of sequential Monte Carlo methods in high dimensions , 2011, 1103.3965.

[31]  Giuseppe Vallone,et al.  Polarization entangled state measurement on a chip , 2010, CLEO: 2011 - Laser Science to Photonic Applications.

[32]  T. Ralph,et al.  Adding control to arbitrary unknown quantum operations , 2010, Nature communications.

[33]  Jiangfeng Du,et al.  NMR implementation of a molecular hydrogen quantum simulation with adiabatic state preparation. , 2010, Physical review letters.

[34]  S. Aaronson.  Computational complexity: Why quantum chemistry is hard , 2009 .

[35]  B. Lanyon,et al.  Towards quantum chemistry on a quantum computer. , 2009, Nature chemistry.

[36]  P. Høyer,et al.  Higher order decompositions of ordered operator exponentials , 2008, 0812.0562.

[37]  M. Lipson,et al.  Generation of correlated photons in nanoscale silicon waveguides. , 2006, Optics express.

[38]  G. Agrawal,et al.  Silicon waveguides for creating quantum-correlated photon pairs. , 2006, Optics letters.

[39]  Göran Wendin,et al.  Arbitrary accuracy iterative quantum phase estimation algorithm using a single ancillary qubit: A two-qubit benchmark , 2006, quant-ph/0610214.

[40]  A. Warshel,et al.  Transition state theory can be used in studies of enzyme catalysis: lessons from simulations of tunnelling and dynamical effects in lipoxygenase and other systems , 2006, Philosophical Transactions of the Royal Society B: Biological Sciences.

[41]  Jürgen Köhler,et al.  The architecture and function of the light-harvesting apparatus of purple bacteria: from single molecules to in vivo membranes , 2006, Quarterly Reviews of Biophysics.

[42]  I. Chuang,et al.  Limitations of quantum simulation examined by simulating a pairing Hamiltonian using nuclear magnetic resonance. , 2006, Physical review letters.

[43]  Manuela Merchán,et al.  Quantum chemistry of the excited state: 2005 overview , 2005 .

[44]  M. Head‐Gordon,et al.  Simulated Quantum Computation of Molecular Energies , 2005, Science.

[45]  O. Regev,et al.  The Complexity of the Local Hamiltonian Problem , 2004, FSTTCS.

[46]  Massimiliano Di Ventra,et al.  Targeting specific eigenvectors and eigenvalues of a given Hamiltonian using arbitrary selection criteria , 2002 .

[47]  T. Crawford,et al.  Some surprising failures of Brueckner coupled cluster theory , 2000 .

[48]  R. Feynman.  Simulating physics with computers , 1999 .

[49]  Daniel A. Lidar,et al.  Calculating the thermal rate constant with exponential speedup on a quantum computer , 1998, quant-ph/9807009.

[50]  Seth Lloyd,et al.  Universal Quantum Simulators , 1996, Science.

[51]  Lin-wang Wang,et al.  Solving Schrödinger’s equation around a desired energy: Application to silicon quantum dots , 1994 .

[52]  J. Pople,et al.  Self‐Consistent Molecular‐Orbital Methods. I. Use of Gaussian Expansions of Slater‐Type Atomic Orbitals , 1969 .

[53]  Harrison Shull,et al.  NATURAL ORBITALS IN THE QUANTUM THEORY OF TWO-ELECTRON SYSTEMS , 1956 .

[54]  M. Plesset,et al.  Note on an Approximation Treatment for Many-Electron Systems , 1934 .

[55]  S. Thurber.  English , 1894 .

[56]  Volker Bach,et al.  Many-Electron Approaches in Physics, Chemistry and Mathematics , 2014 .

[57]  D. Sebastiani,et al.  Linear Response Methods in Quantum Chemistry , 2014 .

[58]  Zhihao Lan,et al.  Quantum simulations with ultracold quantum gases , 2012 .