Simulating Physical Systems by Quantum Computers

1. Simulating Physical Systems by Quantum Computers J. E. Gubernatis Theoretical Division Los Alamos National Laboratory

2. Collaborators • Manny Knill (LANL/NIST-Boulder) • Raymond LaFlamme (LANL/Waterloo) • Camille Negrevergne (LANL/Bordeaux) • Gerardo Ortiz* (LANL) • Rolando Somma (LANL/Bariloche) *Special thanks for most of the drawings

3. Background • Feynman’s Puzzling Challenge “… the question is, If we wrote a Hamiltonian which involved these [Pauli] operators, locally coupled to corresponding operators on the other space-time points, could we imitate every quantum mechanical system which is discrete and has a finite number of degrees of freedom? I know, almost certainly, that we could do that for any quantum mechanical system which involvesBose particles. I’m not sure whether Fermiparticles could be described by such a system. So I leave that open …” (R. Feynman, 1982)

4. Background • The Puzzle: Feynman’s main thesis was quantum systems could not be efficiently imitated on classical systems. At the time of his statement • Bose systems were being simulated very well on classical computers using stochastic methods. • Fermi systems were/are having problems, the sign problem, but not for the sign problem mentioned by Feynman. • Negative probabilities (the sign problem) occur because of Fermi statistics and not because of Bell’s inequalities.

5. Background • In our first work [PRA 64, 22319 (2001)], we • Noted the existence of a general class of operator transformations that allow the mapping of any physical system to another. • If you can simulate Pauli (Bose) systems efficiently, you can simulate any other system efficiently provided you can implement the mapping efficiently. • Demonstrated that in many cases the dynamical sign problem, which plagues simulations on classical computers, will generally not occur on a quantum computer.

6. Background • In another work [PRA 65, 29902 (2002)], we addressed the question, Will a quantum computer simulate quantum systems more efficiently than a classical computer? • Do the algorithms scale with complexity polynomially? • What are the algorithms? • Can one efficiently simulate Fermi systems? • What are the quantum networks?

7. Outline • Universal Simulation • Models of computation  Algebra of operators • Example: spin-particle connection • Quantum Networks • One and two qubit operations • Quantum Simulation • Initialization • Time evolution • Measurement • Quantum Algorithm • Fermion simulation on a NMR quantum computer.

8. Universal Simulation of Physical Phenomena

9. Universal Simulation • Spin-Particle Connections

10. Spins ½ & 1D Spins N & n D Fermions Fermions Bosons Anyons Bosons Universal Simulation • Connections made explicit by the generalized Jordan-Wigner Transformation [Batista and Ortiz, PRL 86, 1082 (2001)]

11. Universal Simulation • Jordan-Wigner/Matsuda-Matsubara Transformations • Example: 1D Jordan-Wigner: Fermion  Spin-1/2

12. Universal Simulation • Two dimensional Extension

13. Universal Simulation • Anyon-Pauli Algebra Isomorphism

14. Universal Simulation • Anyon-Pauli Algebra Isomorphism

15. Quantum Computation • Quantum Control Model • The control Hamiltonian is implemented by a small number of quantum gates

16. Quantum Computation • Pauli spin representation • Universal gates

17. Quantum Computation • Fermion representation • Universal gates

18. Quantum Computation • Boson representation • Possibility of an infinite number of bosons occupying a state presents a problem • If Np is maximum number allowed for entire systems, then a solution is to restrict the boson operators for a given site to a finite basis of states

19. Quantum Computation • Boson Representation • The commutation relation • For a number of models the total number of Bosons is conserved. • Mapping is now between sets of states and is no longer between operator algebras. • Spin-1/2 gates

20. Quantum Computation • Boson representation • Example: Mapping chain of 5 sites and 7 bosons into a spin-1/2 state

21. Quantum Networks • Quantum Bit • Basis • Block sphere

22. Quantum Networks • Quantum Gates of the Block sphere

23. Quantum Networks • Hadamard gate

24. Quantum Networks • C-NOT gate

25. Quantum Networks

26. Quantum Networks • Controlled U

27. Quantum Networks • For any measurement • To an given initial state, add an ancilla qubit, • Express operators as sums of products of unitary operators, • Perform conditional evolutions by the unitary operators, • Measure state of ancilla qubit.

28. Quantum Networks • Advantages • Handles non-local observables, • “Non-demolition” measurement, • Knowledge of spectrum of operators or current state of system is not required.

29. Quantum Networks • 1 Qubit Measurement:

30. Quantum Networks • L Qubit Measurements:

31. Quantum Simulation • Three Stages • Preparation of initial state: |(0) • Propagation of initial state • Performance of measurements • Each stage requires controlling the elements of the quantum computer.

32. Quantum Simulation • Initial state preparation (fermions) • Encompass efficiently initial states of the form

33. Quantum Simulation • Initial state preparation • Preparation of |

34. Quantum Simulation • Initial state preparation • If gates and states are in different bases, exploits Thouless’s theorem (generalizes via the JW transformation)

35. Universal Simulation • Initial state preparation • Performing a sum of Slater determinants is involved. • Result is obtained probabilistically. • The basic steps are: • Add N extra ancilla

36. Universal Simulation • Initial state preparation • Generate • Apply the procedure to generate |

37. Universal Simulation • Initial state preparation • Generate • Probability of successful generation is • In general N attempts are necessary for success.

38. Quantum Simulation • Evolution of initial state

39. Quantum Simulation • Measurements of evolved state • Two classes were considered: • Correlation Function Measurements • Spectrum of a Hermitian operator 

40. Quantum Simulation • Correlation function:

41. Quantum Simulation • Details for

42. Quantum Simulation • Spectrum measurement of Hermitian operator :

43. Quantum Algorithm for a Quantum System • System to Simulate • Spinless fermion ring with an impurity site • Exactly solvable • Reducible to a three qubit problem: one ancilla and two “physical” qubits. • To measure:

44. Quantum Algorithm • Fourier transform modes • Spin-Fermion Mapping

45. Quantum Algorithm • Transformed H • Reduction to 2 Qubit Problem

46. Quantum Algorithm • Transform correlation function • Approximate unitary evolution • Generate initial state: “Fermi” sea

47. Quantum Algorithm

48. Quantum Simulation on a Quantum Computer • Implemented the algorithm on a classical computer • Reproduced the exact answer to controllable accuracy • Implemented the algorithm on a 7 qubit liquid state NMR quantum computer • Reproduced the exact result satisfactorily

49. Quantum Simulation • Experiment vs theory: spectrum of H: • One particle case

50. Quantum Simulation • Experiment vs Theory: