View
220
Download
1
Embed Size (px)
Citation preview
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).