Simulating Physical Systems by Quantum

Computers

J. E. Gubernatis

Theoretical Division

Los Alamos National Laboratory

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

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)

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.

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.

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?

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.

Universal Simulation of Physical Phenomena

Universal Simulation

Spin-Particle Connections

Universal Simulation

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

Spins½ & 1D

Fermions Bosons BosonsAnyons

SpinsN & n D

Fermions

Universal Simulation

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

1

1

1†

1

jl l

j zl

jl l

j zl

a

a

Universal Simulation

Two dimensional Extension

Universal Simulation

Anyon-Pauli Algebra Isomorphism

†

†

2 1

1

2

ii j

ij j

jj

jz j

i n

j

j j jx y

K a

a K

n

K e

i

Universal Simulation

Anyon-Pauli Algebra Isomorphism

†

† †

†

††

11

2

11

2

1

2

, , 0 , ,

, 1 1

,

ii z

j i ji j

ii z

j i ji j

zj j

ii j i j

ii j ij j

i j ij j

ea e S S

ea e S S

n S

a a a a A B A e B

a a e n

n a a

Quantum Computation

Quantum Control Model

The control Hamiltonian is implemented by a small number of quantum gates

,0 , 2 , ,0

, exp (P

U t U t t t U t t U t

U t t t i tH t

Quantum Computation

Pauli spin representation

Universal gates

,

1 0 0 1 0 1 0; ; ;

0 1 1 0 0 0 1

j y j

j j i jP x x y ij z z

j i j

x y z

j

jth factor

H t t t

iI

i

I I I

21 3, ,ii i j

zyx zii i

e e e

Quantum Computation

Fermion representation

Universal gates

†

† † † †

P j j j jj

ij i j j i ij i i j jij

H t a t a

t a a a a t a a a a

† † † †1 2 43, , ,i j j i i j j i i ji

i a a a a i a a a a i n ni ne e e e

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

1 2

† †

1, 1,

0

, , , with 1,2, ,

1

th

P

N i P

i

i factor

Nn i n i

n

n n n n N

b I I b I

n

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

†ij i j ij i j

ij

H b b n n

† †1, 0; , 1

!P P

N NPi j i j ij i i

P

Nb b b b b b

N

Quantum Computation

Boson representation Example: Mapping chain of 5 sites and 7 bosons

into a spin-1/2 state

Quantum Networks

Quantum Bit Basis

Block sphere

1 00 ; 1

0 1

cos 0 sin 12 2

ia e

Quantum Networks

Quantum Gates of the Block sphere

Quantum Networks

Hadamard gate

1 11

1 12

Quantum Networks

C-NOT gate

Quantum Networks

Quantum Networks

Controlled U

( )

0 0:

1 1

a az z

itQ

itQ

a aitQ

a a

U t e

U t e

CUe

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.

† , , unitaryi i i i ii

O aU V U V

Quantum Networks

Advantages Handles non-local observables, “Non-demolition” measurement, Knowledge of spectrum of operators or current

state of system is not required.

Quantum Networks

1 Qubit Measurement: †0 02 a U V

10 1

2

Quantum Networks

L Qubit Measurements: †0 0

1

12

2

Ma

i i ii

aU VN

Quantum Simulation

Three Stages1. Preparation of initial state: |(0)2. Propagation of initial state

3. Performance of measurements

Each stage requires controlling the elements of the quantum computer.

( ) 0

( ) exp

tiH t

t

t U t

U t iHt

Quantum Simulation

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

†

0

where is a Slater determinant

vacjj

c

b

Quantum Simulation

Initial state preparation Preparation of |

†

†

†2 2

2

†

0 0

Successive applications of

0 , up to a phase factor

m m

m m

i a a i

m

i a a

mm

e e a

e

a

Quantum Simulation

Initial state preparation If gates and states are in different bases, exploits

Thouless’s theorem (generalizes via the JW transformation)

†

†

† †

If vac and

M is a Hermitian matrix, then

, where

jj

ia M a

iM

a

e

b e a

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

0 0 0 vac 0 vaca

N

Universal Simulation

Initial state preparation Generate

Apply the procedure to generate |

1

vac

where has qbit being 1

N

a

1

N

a

Universal Simulation

Initial state preparation Generate

Probability of successful generation is

In general N attempts are necessary for success.

1

10 terms without 0

N

a aa

N

2

1

1 1| |

N

aN N

Quantum Simulation

Evolution of initial state

,

exp

where is in the form of the control Hamiltonian.

l

itH i tHl

l j

iH ti tHl

l l

l

H H U t e e

U t e i H t e

H

Quantum Simulation

Measurements of evolved state Two classes were considered:

Correlation Function Measurements

Spectrum of a Hermitian operator

( ) ( ) (0) iHt iHtABC t A t B e Ae B

2

2

( )

2 ( )

niti tn

n

n nn

U t e e

Quantum Simulation

Correlation function: †0 02 | |a T ATB

Quantum Simulation

Details for ABC t

Quantum Simulation

Spectrum measurement of Hermitian operator : 2 a itQe

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:

1† † †

1 10

1† †

0

( )

( )

N

i i i ii

N

i ii

H t c c c c b b

Vc b b c

N

†0 0 0G t b t b

Quantum Algorithm

Fourier transform modes Spin-Fermion Mapping

0 0

1 1

1 † 1

1 2 † 1 2

1 2 1 † 1 2 1( 1) ( 1)N N

k z k z

N N N N N Nk z z z k z z z

b b

c c

c c

Quantum Algorithm

Transformed H

Reduction to 2 Qubit Problem

11

0

12 1 2 1 2

0

ˆ2 1

( )

i

i

N

k zi

Ni

k z x x y yi

H

V

01 2 1 2 1 2( )2 2 2

kz z x x y y

VH

Quantum Algorithm

Transform correlation function

Approximate unitary evolution

Generate initial state: “Fermi” sea

1 1( ) iHt iHtG t e e

,xyzM

i tHiHt i t H i t Hz xye e e e H H H

Quantum Algorithm

1 1( ) iHt iHtG t e e

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

Quantum Simulation

Experiment vs theory: spectrum of H: One particle case

-14 -12 -10 -8 -6 -4 -2 0 2 4

0

50

100

150

200

FF

T

Frecuency

/itHe

Quantum Simulation

Experiment vs Theory: ABC t

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.

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).