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Simulating Physical Systems by Quantum Computers

This document explores the potential of quantum computers to simulate various physical systems, as inspired by Richard Feynman's challenges in quantum mechanics. It discusses efficient methods for simulating quantum systems, particularly Fermi and Bose particles, and the benefits of quantum computing in bypassing classical computational limitations. The work covers universal simulation techniques, including the generalized Jordan-Wigner transformation, and examines quantum networks and algorithms that enable the accurate representation and evolution of quantum states.

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Simulating Physical Systems by Quantum Computers

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  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:

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