Quantum Simulators: Where do we stand?

1. Quantum Simulators: Where do we stand?

4. Quantum simulators: Where do we stand? - Outline • Ideology/introduction • Quantum simulators • Quantum annealers • Analog versus digital • What do we want to/can we simulate? • Platforms/models • Some physics – simulating interesting phenomena • Quantum droplets in dipolar Bose gases and Bose-Bose mixtures • High Tc- superconductivity and Fermi Hubbard model • Many body localization * • Errors, validation and certification • Energy/cost function • Validation • Certification of entanglement and non-locality • Quantum simulators and machine learning *

5. Quantum Simulators: Ideology I • There exist many interesting quantumphenomena (such as superconductivity). • These phenomena may have important applications! • These phenomena are often difficult to be described and understood with the help of standard computers. • Maybe we can use another, simpler and better controllable quantum system to simulate, understand and control these phenomena (R.P. Feynman)? Such a system would thus work as quantum computer of special purpose, i.e. QUANTUM SIMULATOR

6. Interesting quantum phenomenon: Superconductivity • Superconductors are “ideal” conductors, they conduct electric currents without resistance. • They conduct strong currents without losses. • They can generate strong magnetic fields outside of them, but due to the Meissner- Ochsenfeld effect, they do not let magnetic fields enter them! • Unfortunately, superconductors exist only at low temperatures: normally close to absolute zero, -270º Celcius, “high temperature SC” at T>-166º (boiling of liquid nitrogen).

7. Superconductivity: Theory • We (some of us) believe there exist a simple model that captures the phenomenon of high Tc superconductivity, the Hubbard model. • Unfortunately, even this simple model is far too hard to simulate, and hard to understand with existing computers. • So, why not trying to mimick it with atoms…

8. (Fermi) Hubbard model On site interactions Tunneling Simple Hubbard model First band

9. 3D Lattice Potential (by courtesy of M. Greiner, O. Mandel, T. Esslinger, I. Bloch, and T. Hänsch) V0 up to 22 Erecoil wrup to 2p ´ 30 kHz n 1-5 atoms on average per site • Resulting potential consists of a simple cubic lattice • BEC coherently populates more than 100,000 lattice sites

10. Quantum Simulators: Ideology II • There exist many interesting classical optimization problems (such as spin glasses, travelling salesman…) • These phenomena may have important applications! • These phenomena are often difficult to be described and understood with the help of standard computers • Maybe we can use another, simpler and better controllable quantum system to solve, understand and control these classical problems (D-Wave computers)? Such a system would thus work as quantum computer of special purpose, i.e. QUANTUM SIMULATOR/ANNEALER

11. Quantum simulators/annealers May be: • analog (no error correction) • digital (error correction possible)

12. Quantum simulators/annealers What shall we simulate? • Statics at zero temperature – ground state and its properties. • Statics/equilibrium dynamics at non-zero temperature, or low energies. • Dynamics (Hamiltonian, but out of equilibrium) • Dissipative dynamics

13. Platforms/models • Ultracold atoms in traps/on chips (QFT with Ψ4 interactions, or more…) • Ultracold atoms in optical lattices (Hubbard models, spin models) • Ultracold trapped ions (spin models with long range interactions) • Rydberg atoms • Circuit QED • Josephson/superconducting qubits arrays (Ising spin glasses) • Photonic platforms (boson sampling)

14. Navon, N., Nascimbène, S., Chevy, F. & Salomon, C. The equation of state of a low-temperature Fermi gas with tunable interactions. Science 328, 729732 (2010). Ku, M. J. H., Sommer, A. T., Cheuk, L. W. & Zwierlein, M. W. Revealing thesuperfluid Lambda transition in the universal thermodynamics of a unitary Fermi gas. Science 335, 563567 (2012).

15. Platforms/models • Ultracold atoms in traps/on chips (QFT with Ψ4 interactions, or more…) • Ultracold atoms in optical lattices (Hubbard models, spin models) • Ultracold trapped ions (spin models with long range interactions) • Rydberg atoms • Circuit QED • Josephson/superconducting qubits arrays (Ising spin glasses) • Polariton condensates • Photonic platforms (boson sampling)

17. Quantum simulators Ultracold atoms in optical lattices: Simulating quantum many-body physics M. Lewenstein, A. Sanpera, V. Ahufinger, Oxford University Press (2012, corrected soft-cover 2017)

18. Platforms/models • Ultracold atoms in traps/on chips (QFT with Ψ4 interactions, or more…) • Ultracold atoms in optical lattices (Hubbard models, spin models) • Ultracold trapped ions (spin models with long range interactions) • Rydberg atoms • Circuit QED • Josephson/superconducting qubits arrays (Ising spin glasses) • Polariton condensates • Photonic platforms (boson sampling)

22. Platforms/models • Ultracold atoms in traps/on chips (QFT with Ψ4 interactions, or more…) • Ultracold atoms in optical lattices (Hubbard models, spin models) • Ultracold trapped ions (spin models with long range interactions) • Rydberg atoms • Circuit QED • Josephson/superconducting qubits arrays (Ising spin glasses) • Polariton condensates • Photonic platforms (boson sampling)

25. Platforms/models • Ultracold atoms in traps/on chips (QFT • with Ψ4 interactions, or more…) • Ultracold atoms in optical lattices • (Hubbard models, spin models) • Ultracold trapped ions (spin models with • long range interactions) • Rydberg atoms • Circuit QED • Josephson/superconducting qubits arrays • (Ising spin glasses) • Polariton condensates • Photonic platforms (boson sampling)

27. Platforms/models • Ultracold atoms in traps/on chips (QFT with Ψ4 interactions, or more…) • Ultracold atoms in optical lattices (Hubbard models, spin models) • Ultracold trapped ions (spin models with long range interactions) • Rydberg atoms • Circuit QED • Josephson/superconducting qubits arrays (Ising spin glasses) • Polariton condensates • Photonic platforms (boson sampling)

29. Platforms/models • Ultracold atoms in traps/on chips (QFT with Ψ4 interactions, or more…) • Ultracold atoms in optical lattices (Hubbard models, spin models) • Ultracold trapped ions (spin models with long range interactions) • Rydberg atoms • Circuit QED • Josephson/superconducting qubits arrays (Ising spin glasses) • Polariton condensates • Photonic platforms (boson sampling)

31. Simulating interesting systems – quantum droplets

32. Simulating interesting systems – quantum droplets

33. Simulating interesting systems – High Tc- superconductivity

34. Simulating interesting systems – High Tc- superconductivity

35. Errors, validation and certification Energy/Cost functions • Variational upper bounds • “Semidefinite programming” lower bounds arXiv:0903.4368 Convergent relaxations of polynomial optimization problems with non-commuting variables, Stefano Pironio, Miguel Navascues, Antonio Acin, SIAM J. Optim. Volume 20, Issue 5, pp. 2157-2180 (2010); Miguel Navascués et al 2008 New J. Phys. 10 073013; Tillmann Baumgratz and Martin B Plenio 2012 New J. Phys. 14 023027.

36. Errors, validation and certification Validation • Go to parameters regime where classical simulations are efficient • Go to small systems and use finite size scaling

37. Errors, validation and certification

38. Errors, validation and certification

39. Errors, validation and certification

40. Errors, validation and certification Robustness • Add controlled disorder and check • Add controlled noise and check

41. Errors, validation and certification

42. Errors, validation and certification

43. Errors, validation and certification

44. Errors, validation and certification Certification • Certifying entanglement • Certifying nonlocality

45. Problem: certification of complex many-body systems Is this device a quantum computer or simulator? Are its results correct? Is this system entangled? Which quantum correlations does it display? • New methods for entanglement and non-locality detection. • In progress: new numerical methods for certifying classical and quantum simulators.

46. Errors, validation and certification Certification • Certifying entanglement • Certifying nonlocality

47. Errors, validation and certification

48. Errors, validation and certification

49. Errors, validation and certification

50. Errors, validation and certification