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 involves Bose particles. I’m not sure whether Fermi particles 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. Universal Simulation Connections made explicit by the generalized Jordan-Wigner Transformation [Batista and Ortiz, PRL 86, 1082 (2001)] Spins ½ & 1D FermionsBosons Anyons Spins N & n D Fermions

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 N p 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. 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 1. Preparation of initial state: |  (0)  2. Propagation of initial state 3. 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:

51. Concluding Summary We established connections between all languages of physical systems and the standard model of quantum computation. One in principle can simulate any physical system by any other physical system. We explored issues associated with efficient simulations of physical systems by a quantum network. Initialization, propagation and measurement steps were all proven to scale polynomially with complexity.

52. Concluding Summary We applied this technology to a dynamical model of lattice fermions. Problem scales exponentially on a classical computer. We successfully implemented this technology on a quantum computer. Considerable work on constructing efficient algorithms for measuring physical quantities remains undone. References: Phys. Rev. A 64, 22319 (2001). Phys. Rev. A 65, 29902 (2002). J. Quant. Information 1, 189 (2003).