Simulating Physical Systems by Quantum Computers

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Simulating Physical Systems by Quantum Computers

• J. E. Gubernatis
• Theoretical Division
• Los Alamos National Laboratory

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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
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Background

• Feynmans 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. Im not sure whether Fermi particles
could be described by such a system. So I leave
that open (R. Feynman, 1982)
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Background

• The Puzzle Feynmans 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
  Bells inequalities.

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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.

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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?

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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.

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Universal Simulation of Physical Phenomena
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Universal Simulation

• Spin-Particle Connections

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Universal Simulation

• Connections made explicit by the generalized
  Jordan-Wigner Transformation Batista and Ortiz,
  PRL 86, 1082 (2001)

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Universal Simulation

• Jordan-Wigner/Matsuda-Matsubara Transformations
• Example 1D Jordan-Wigner Fermion ? Spin-1/2

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Universal Simulation

• Two dimensional Extension

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Universal Simulation

• Anyon-Pauli Algebra Isomorphism

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Universal Simulation

• Anyon-Pauli Algebra Isomorphism

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Quantum Computation

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

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Quantum Computation

• Pauli spin representation
• Universal gates

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Quantum Computation

• Fermion representation
• Universal gates

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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

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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

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Quantum Computation

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

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Quantum Networks

• Quantum Bit
• Basis
• Block sphere

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Quantum Networks

• Quantum Gates of the Block sphere

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Quantum Networks

• Hadamard gate

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Quantum Networks

• C-NOT gate

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Quantum Networks
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Quantum Networks

• Controlled U

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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.

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Quantum Networks

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

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Quantum Networks

• 1 Qubit Measurement

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Quantum Networks

• L Qubit Measurements

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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.

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Quantum Simulation

• Initial state preparation (fermions)
• Encompass efficiently initial states of the form

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Quantum Simulation

• Initial state preparation
• Preparation of ??

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Quantum Simulation

• Initial state preparation
• If gates and states are in different bases,
  exploits Thoulesss theorem (generalizes via the
  JW transformation)

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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

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Universal Simulation

• Initial state preparation
• Generate
• Apply the procedure to generate ???

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Universal Simulation

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

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Quantum Simulation

• Evolution of initial state

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Quantum Simulation

• Measurements of evolved state
• Two classes were considered
• Correlation Function Measurements
• Spectrum of a Hermitian operator ?

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Quantum Simulation

• Correlation function

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Quantum Simulation

• Details for

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Quantum Simulation

• Spectrum measurement of Hermitian operator ?

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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

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Quantum Algorithm

• Fourier transform modes
• Spin-Fermion Mapping

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Quantum Algorithm

• Transformed H
• Reduction to 2 Qubit Problem

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Quantum Algorithm

• Transform correlation function
• Approximate unitary evolution
• Generate initial state Fermi sea

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Quantum Algorithm
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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

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Quantum Simulation

• Experiment vs theory spectrum of H
• One particle case

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Quantum Simulation

• Experiment vs Theory

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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.

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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).