Quantum generative model for sampling many-body spectral functions

Quantum phase estimation is at the heart of most quantum algorithms with exponential speedup. In this paper we demonstrate how to utilize it to compute the dynamical response functions of many-body quantum systems. Specifically, we design a circuit that acts as an efficient quantum generative model, providing samples out of the spectral function of high rank observables in polynomial time. This includes many experimentally relevant spectra such as the dynamic structure factor, the optical conductivity, or the NMR spectrum. Experimental realization of the algorithm, apart from logarithmic overhead, requires doubling the number of qubits as compared to a simple analog simulator.

[1]  Jstor,et al.  Proceedings of the American Mathematical Society , 1950 .

[2]  E. Knill,et al.  Power of One Bit of Quantum Information , 1998, quant-ph/9802037.

[3]  Man-Duen Choi.  Completely positive linear maps on complex matrices , 1975 .

[4]  Pankaj Mehta,et al.  Reinforcement Learning in Different Phases of Quantum Control , 2017, Physical Review X.

[5]  M. Rigol,et al.  From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics , 2015, 1509.06411.

[6]  P. Cejnar,et al.  Excited state quantum phase transitions in many-body systems , 2007, 0707.0325.

[7]  Yaliang Li,et al.  SCI , 2021, Proceedings of the 30th ACM International Conference on Information & Knowledge Management.

[8]  U. Schollwoeck.  The density-matrix renormalization group , 2004, cond-mat/0409292.

[9]  A. Roggero,et al.  Dynamic linear response quantum algorithm , 2018, Physical Review C.

[10]  Scott Aaronson.  The Equivalence of Sampling and Searching , 2013, Theory of Computing Systems.

[11]  H. Neven,et al.  Characterizing quantum supremacy in near-term devices , 2016, Nature Physics.

[12]  Immanuel Bloch,et al.  Colloquium : Many-body localization, thermalization, and entanglement , 2018, Reviews of Modern Physics.

[13]  A. Cleland,et al.  Electron spin resonance of nitrogen-vacancy centers in optically trapped nanodiamonds , 2012, Proceedings of the National Academy of Sciences.

[14]  Kenneth R. Brown,et al.  Two-qubit entangling gates within arbitrarily long chains of trapped ions , 2019, Physical Review A.

[15]  M. Lukin,et al.  Probing real-space and time-resolved correlation functions with many-body Ramsey interferometry. , 2013, Physical review letters.

[16]  Hideo Aoki,et al.  Nonequilibrium dynamical mean-field theory and its applications , 2013, 1310.5329.

[17]  Alessandro Silva,et al.  Colloquium: Nonequilibrium dynamics of closed interacting quantum systems , 2010, 1007.5331.

[18]  Animesh Datta,et al.  Role of entanglement and correlations in mixed-state quantum computation , 2007 .

[19]  Universal adiabatic dynamics in the vicinity of a quantum critical point , 2003, cond-mat/0312144.

[20]  John Preskill,et al.  Quantum computing and the entanglement frontier , 2012, 1203.5813.

[21]  S. Gopalakrishnan,et al.  Anomalous relaxation and the high-temperature structure factor of XXZ spin chains , 2019, Proceedings of the National Academy of Sciences.

[22]  Raymond Laflamme,et al.  Quantum computing and quadratically signed weight enumerators , 2001, Inf. Process. Lett..

[23]  E. Demler,et al.  Far-from-equilibrium spin transport in Heisenberg quantum magnets. , 2014, Physical review letters.

[24]  T. Gasenzer,et al.  Tuning universality far from equilibrium , 2013, Scientific Reports.

[25]  Travis S. Humble,et al.  Quantum supremacy using a programmable superconducting processor , 2019, Nature.

[26]  Lance Fortnow,et al.  Proceedings of the forty-third annual ACM symposium on Theory of computing , 2011, STOC 2011.

[28]  E. Demler,et al.  Quantum correlations at infinite temperature: The dynamical Nagaoka effect , 2017, 1703.09231.

[29]  A. Polkovnikov.  Phase space representation of quantum dynamics , 2009, 0905.3384.

[30]  F. Verstraete,et al.  Computational complexity of interacting electrons and fundamental limitations of density functional theory , 2007, 0712.0483.

[31]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[32]  L. Cugliandolo,et al.  Scaling and super-universality in the coarsening dynamics of the 3D random field Ising model , 2008, 0803.0664.

[33]  M. Birkner,et al.  Blow-up of semilinear PDE's at the critical dimension. A probabilistic approach , 2002 .

[34]  Tommaso Calarco,et al.  Dressing the chopped-random-basis optimization: A bandwidth-limited access to the trap-free landscape , 2015, 1506.04601.

[35]  M. Lukin,et al.  Probing many-body dynamics on a 51-atom quantum simulator , 2017, Nature.

[36]  J. Herskowitz,et al.  Proceedings of the National Academy of Sciences, USA , 1996, Current Biology.

[37]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[38]  B. Terhal,et al.  Quantum phase estimation of multiple eigenvalues for small-scale (noisy) experiments , 2018, New Journal of Physics.

[39]  Antonio-José Almeida,et al.  NAT , 2019, Springer Reference Medizin.

[40]  L. Vandersypen,et al.  NMR techniques for quantum control and computation , 2004, quant-ph/0404064.

[41]  Dmitri Maslov,et al.  Experimental comparison of two quantum computing architectures , 2017, Proceedings of the National Academy of Sciences.

[42]  E. Altman,et al.  A Universal Operator Growth Hypothesis , 2018, Physical Review X.

[43]  M. Mézard,et al.  Journal of Statistical Mechanics: Theory and Experiment , 2011 .

[44]  I. Chuang,et al.  Quantum Computation and Quantum Information: Bibliography , 2010 .