1. CLARENDON LABORATORY PHYSICS DEPARTMENT UNIVERSITY OF OXFORD and CENTRE FOR QUANTUM TECHNOLOGIES NATIONAL UNIVERSITY OF SINGAPORE Quantum Simulation Dieter Jaksch

2. Outline  Lecture 1: Introduction  What defines a quantum simulator? Quantum simulator criteria. Strongly correlated quantum systems.  Lecture 2: Optical lattices  Bose-Einstein condensation, adiabatic loading of an optical lattice. Hamiltonian  Lecture 3: Quantum simulation with ultracold atoms  Analogue simulation: Bose-Hubbard model and artificial gauge fields. Digital simulation: using cold collisions or Rydberg atoms.  Lecture 4: Tensor Network Theory (TNT)  Tensors and contractions, matrix product states, entanglement properties  Lecture 5: TNT applications  TNT algorithms, variational optimization and time evolution

3. INTRODUCTION Quantum simulation

4. Early simulators Orrery simulating the solar system Differential analyzer in 1938 T.H. Johnson, S.R. Clark and DJ, EPJ Quantum Technology 1, 10 (2014)

5. Difficult simulations © Prof Michael Engel

6. Quantum systems as simulators Cold atomsTrapped ionsSuperconductors © Nature publishing group

7. “... we never experiment with single particles inasmuch as we cannot raise an Ichtyosaurus in the zoo.” E. Schrödinger, 1952 “More is different.” P.W. Anderson, 1972

8. WHAT IS QUANTUM SIMULATION Quantum simulation

9. What are simulators?

10. Simulation

11. The process of simulation Physical Simulation Quantitative Model

12. What is a Quantum Simulator?  Every system is quantum if considered at sufficiently short length and/or time scales

13. Diagonal density matrix

14. A beam splitter photon BS photon BS atom 1 atom 2 photon 1 BS photon 2 All of these are quantum systems But only (b) and (c) are quantum states that could be used to violate a Bell inequality (a) (b) (c)

15. When are simulators trustworthy?

16. Simulation part of scientific method Quantitaive Model Only require model for system of interest and simulator to be falsifiable TRUST

17. Quantum simulator criteria  Quantum system  Large number of degrees of freedom, lattice system or confined in space  Initialization  Prepare a known quantum state, pure or mixed, e.g. thermal  Hamiltonian engineering  Set of interactions with external fields or between different particles  Interactions either local or of longer range  Detection  Perform measurement on the system, particles individually or collectively. Single shot which can be repeated several times.  Verification  Increase confidence about result, benchmark by running known limiting cases, run backward and forward, adjust time in adiabatic simulations. J. Ignacio Cirac and Peter Zoller, Nature Physics 8, 264 (2010)

18. Digital quantum simulation  Assume a discrete lattice in space time  First envisaged by R. Feynman in 1982 site empty site filled time space a † |  = |  a |  = |  n |  = 1 |  a † |  = 0 n |  = 0 |  a |  = 0 Creation operator Destruction operator Counting operator Any quantum system can be simulated by a Hamiltonian which couples these sites locally! n= a † a

19. using a classical computer: ‘… if you want to make a simulation of nature you’d better make it quantum mechanical, and by golly it’s a wonderful problem, because it doesn’t look so easy.’ using a quantum computer: ‘… I believe it is rather simple to answer that question and to find the class, but I just haven’t done it.’ Done: S. Lloyd, Science 273, 1073 (1996) Feynman on simulation in 1982

20. Analogue quantum simulator TRUST

21. Why do we need quantum simulators?  The curse of dimensionality in quantum systems  Hard classical problems # config = 4,722,366,482,869,645,213,696 Travelling salesman Ising model

22. STRONGLY CORRELATED SYSTEMS Quantum simulation

23. Strong correlations? Can we view “correlations” as a synonym for “interactions”? weak interactions can extrapolate en masse behaviour from one particle strong interactions particles do not move independently

24. Dramatic consequences Interplay of microscopic interactions with external influences can lead to abrupt macroscopic changes … + - 0 1 1 + + + - Simple “classical” Ising magnet:

25. Strong correlations Beyond physics strong correlations appear pervasive …

26. Add quantum mechanics … Major interest in quantum many-body problems arises in lattice systems – trying to understand the remarkable properties of electrons in some materials … High-temperature superconductivity What is the pairing mechanism? Quantum Hall effect What are the topological properties of fractional QH states? These are seminal strongly-correlated phenomena.

27. THE MANY BODY PROBLEM Quantum simulation

28. Strong or weak correlations? It would seem from the offset that condensed matter physics should be a very strongly correlated quantum problem? + + + + + + + + + – – – –– – – – – – + + + + + + + + + – – – – – – – – A problem described by the “theory of everything”?

29. Weak correlations Paradoxically, many solid state systems, like metals, display weak correlations despite being composed of strongly interacting particles. Why? First, the Born-Oppenheimer approx. decouples ions and e’s Locality means that only these single-particle elements are relevant: on-site potential: n.n. hopping: ++ + Neighbouring localised orbitals overlap. Electrons see fixed periodic potential of the ions.

30. Weak correlations in real and momentum space Moving to many non-interacting electrons (add spin and 2 nd quantise): Band structure explains metals, semiconductors and insulators: Ground state simply fills up single-particle states:

31. Weak correlations The concept of a Fermi surface is crucial to non-interacting fermions: Landau’s Fermi-liquid theory shows that the Fermi surface survives non-zero interactions – System is weakly correlated and described by non-interacting quasi-particles, i.e. renormalised non-interacting electrons:

32. Strong correlations This picture fails for some materials like transition metal oxides, and the CuO 2 planes in high-T c superconductors. The reason is: ++ + “core-like” d or f valence orbitals small overlap = narrow bands confinement = large repulsion Gives the Hubbard model: Interactions significant – no simple quasi-particle picture. Half-filled strong-coupling limit is a Heisenberg anti-ferromagnet:

33. What do we want to compute? Given a Hamiltonian describing our strongly-correlated system we typically want to compute: These reduce to linear algebra problems we’ll review shortly … In both cases, we want to simulate local observables and long- ranged correlations and other properties. (2) Solve Dynamics of quenches, driving, … Find ground state and low-lying excitations (also thermal): (1) Solve gapped or gapless?

34. Analogue quantum simulation Such model Hamiltonian are now accurately realisable with cold atoms: Lasers BEC Optical Lattice Time of flight imaging: Superfluid : Mott-Insulator : quantum phase transition Bose Hubbard model