Quantum Informationand
the simulation of quantum systems
José Ignacio LatorreUniversitat de Barcelona
Buenos Aires, August 2007
I Simulation of quantum Mechanics
II Entanglement entropy
III Efficient representation of quantum systems
Physics
Theory 1 Theory 2
Exact solution
Approximated methods
Simulation
Classical Simulation
Quantum Simulation
Classical Theory
• Classical simulation• Quantum simulation
Quantum Mechanics
• Classical simulation• Quantum simulation
Classical simulation of Quantum Mechanics is related to our ability to supportlarge entanglement
Classical simulation may be enough to handle e.g. ground states
Quantum simulation needed for typical evolution of Quantum systems(linear entropy growth to maximum)
Classical computer
Quantum computer
?
IntroductionIntroduction
Is it possible to classically simulate faithfully a quantum system?
000 ψψ EH =
Heisenberg model0ψ
0)( ψtU
0210 ψψ OO∑ +⋅=i
ii SSH 1
represent
evolve
read
Introduction
Misconception: NO
• Exponential growth of Hilbert space
∑ ∑= =
〉=〉Ψd
i
d
inii
n
niic
1 11...
1
1...|...|
Classical representation requires dn complex coefficients
n
• A random state carries maximum entropy
ψψρ )( LnL Tr −=
( ) dLTrS LLL loglog)( ≈−≡ ρρρ
IntroductionIntroduction
Refutation
• Realistic quantum systems are not random
• symmetries (translational invariance, scale invariance)• local interactions• little entanglement
• We do not have to work on the computational basis
• use an entangled basis
IntroductionIntroduction
Plan
Measures of entanglement
Efficient description of slight entanglement
Entropy: physics vs. simulation
New ideas: MPS, PEPS, MERA
Measures of entanglement
One qubit
∑=
=+=1,0
1110
i
i icβαψ
Quantum superposition
Two qubits
∑=
=+++=1,0,
2121
2111100100ii
ii iicδγβαψ
Quantum superposition + several parties = entanglement
Measures of entanglement
Measures of entanglement
Bii
Aii
ii
ii iiciic 21,0,
11,0,
2121
21
21
21 ∑∑==
==ψ
• Separable states
BABii
Aii iic ζξψ == ∑
=2
1,0,1
21
21
( ) ( ) ( )BBAA
102110
2111100100
41 ++=+++=ψe.g.
• Entangled states
BABii
Aii iic ζξψ ≠= ∑
=2
1,0,1
21
21
( )100121 −=−ψe.g.
Measures of entanglement
Measures of entanglement
∑= BiAiip ζξρ
• Classically correlated states
• Entangled states
∑≠ BiAiip ζξρ
Separability problem: given ρ find whether it is entangled or not
Measures of entanglement
Pure states: Schmidt decomposition
BiAii
iAB p 〉〉=〉Ψ ∑=
ζξχ
|||1
BjA
B
ij
A
vuA iH
j
H
iAB 〉〉=〉Ψ ∑∑
==
|||dim
1
dim
1klkikij VUA
+= λ
A B
χ =min(dim HA, dim HB) is the Schmidt number
BA HHH ⊗=
Measures of entanglement
1>χ Entanglement
Diagonalize A
Measures of entanglement
BiAii
iAB p 〉〉=〉Ψ ∑=
ζξχ
|||1
Von Neumann entropy of the reduced density matrix
( ) Bi
iiAAA SppTrS =−=−= ∑=
χ
ρρ1
22 loglog
( ) ∑=
〉〈=Ψ〈〉Ψ=χ
ξξρ1
||||i
iiiABBA pTr
• χ=1 corresponds to a product state• Large χ implies large superpositions
• e-bit
ITrBA 21|| =Ψ〉〈Ψ== −−ρρ 1
21log
21
21log
21
22 =
+−== BA SS
Measures of entanglementMeasures of entanglement
Maximum Entropy for n-qubits
Strong subadditivity theorem
implies concavity on a chain of spins
nInn 221=ρ nS
n
innn ∑
=
=
−=
2
12 21log
21)(ρ
),(),()(),,( CBSBASBSCBAS +≤+
2MLML
LSSS −+ +≥
SL
SL-M
SL+M
Smax=n
Measures of entanglementMeasures of entanglement
Measures of entanglement
Other measures of entanglement:• concurrence• negativity• purity• ….
3-party entanglement
∑=
=1,0,,
321321
321
iii
iii iiicψ
5 invariants under local unitaries:
2Atrρ
2Btrρ
2Ctrρ ( )ABBAtr ρρρ ⊗ ( )ψdetH
n-party: 2n measures of entanglement
Measures of entanglement
3-party entanglement 5 invariants under local unitaries:
Measures of entanglement
Von Neumann entropy has an asymptotic meaning
A B
p
A and B share p entangled states
A and B perform LOCC to distill q singlets
( )pqS p ∞→= limψ
ψ
Measures of entanglement
Efficient description for slightly entangled states
BkAkk
kAB p 〉〉=〉Ψ ∑=
ζξχ
|||1
BA
H
i
H
iAB iic
B
ii
A
〉〉=〉Ψ ∑∑==
21
dim
1
dim
1
|||2
21
1
+=2121 kikkiii
VpUc
A BBA HHH ⊗=Schmidt decomposition
∑=
ΓΓ=χ
λ1
]2[]1[ 2121
k
ikk
ikiic
Efficient description
Retain eigenvalues and changes of basis
Efficient description
∑ ∑= =
〉=〉Ψd
i
d
inii
n
niic
1 11...
1
1...|...|
n
n
n
n
iniiiiic
][
...
]3[]2[]2[]1[]1[... 1
11
3
322
2
211
1
11....
−
−
ΓΓΓΓ= ∑ ααα
ααααααα λλ
Slight entanglement iff χ∼poly(n)
Matrix Product States
∑ ∑= =
〉=〉Ψd
i
d
inii
n
niic
1 11...
1
1...|...|
1
21
]1[ iA αα 232]2[ iA αα 343
]3[ iA αα 454]4[ iA αα 565
]5[ iA αα 676]6[ iA αα 787
]7[ iA αα
n
n
n
n
iniiiii AAAAc
][
...
]3[]2[]1[1... 1
12
3
43
2
32
1
21.... α
ααααααα∑
−
=
i
α
Approximate physical states with a finite χ MPS
IAA ii
i =+∑ ][][ ][][]1[][ iiii
i AA Λ=Λ −+∑canonical form PVWC06
λΓ=A
Efficient descriptionEfficient description
Graphic representation of a MPS χα ,,1=
di ,,1=jjj
ijA ][1+αα
ψ
Efficient computation of scalar products
operations2χd
3χnd
Efficient description
Local action on MPS
U
lkjiklijU δγδαδβγβαβ λλ ΓΓ=ΓΓ
~~~
Efficient description
n
n
n
n
iniiiii AAAAc
][
...
]3[]2[]1[1... 1
12
3
43
2
32
1
21.... α
ααααααα∑
−
=
Intelligent way to represent and manipulateentanglement
Classical analogy:I want to send 16,24,36,40,54,60,81,90,100,135,150,225,250,375,625Instruction: take all 4 products of 2,3,5 MPS= compression algorithm
n
n
n
n
iniiiiic
][
...
]3[]2[]2[]1[]1[... 1
11
3
322
2
211
1
11....
−
−
ΓΓΓΓ= ∑ ααα
ααααααα λλ
Efficient description
〉ΓΓ=
〉=〉
∑
∑
=
=
11,...,
)()1(
1
4
1...,...
...|....
...||
1
1
1
21
,1
1
ii
iic
nini
nii
iiimage
n
n
n
n
n
χ
αααααα
ψ
i1=1 i1=2
i1=3 i1=4
| i1 〉i2=1 i2=2
i2=3 i2=4
| i2 i1 〉105| 2,1 〉
Crazy ideas: Image compression
pixel addresslevel of grey
RG addressing
Efficient description
....
χ = 1PSNR=17
χ = 4PSNR=25
χ = 8PSNR=31
Max χ = 81
QPEG
• Read image by blocks• Fourier transform• RG address and fill• Set compression level: χ• Find optimal• gzip (lossless, entropic compression) •(define discretize Γ’s to improve gzip)• diagonal organize the frequencies and use 1d RG• work with diferences to a prefixed table
}{ )(aΓ
Efficient description
0),...,,(),...,,( 2121 =∂∂∂ nn xxxfO
)()...()()...(),...,,( 2121 2121
niiiiii
n xhxhxhAAAtrxxxf nn=
2}{min OfA
Constructed: adder, multiplier, multiplier mod(N)
Note: classical problems with a direct product structure!
Crazy ideas: Differential equations
Crazy ideas: Shor’s algorithm with MPS
Efficient descriptionEfficient description
Success of MPS will depend on how much entanglement is present in the physical state
Physics
exactS
Simulation
)(χS
If nSexact log>> MPS is in very bad shape
Back to the central idea: entanglement support
Physics vs. simulationPhysics vs. simulation
Exact entropy for a reduced block in spin chains
LcS LL 2log3 → ∞→ |1|log6 22/
λ−=∞→=cS NL
At Quantum Phase Transition Away from Quantum Phase Transition
Physics vs. simulationPhysics vs. simulation
Maximum entropy support for MPS
α
χ
αα λλ∑
=
−=1
logS
Maximum supported entanglement
χλα
1== ct
χlogmax, =≤ MPSSS
Physics vs. simulationPhysics vs. simulation
Faithfullness = Entanglement support
LcS LL 2log3 → ∞→
Spin chainsMPS
χχ
λα log1
max =→= S
Spin networks
LS LLxL → ∞→
Area law
Computations of entropies are no longer academic exercises but limits on simulations
PEPS
Physics vs. simulationPhysics vs. simulation
Physics
LcSL 2log3= VLRK02-03
LSL = For 3-SAT OL04
Simulation
0)(
S ~ .1 nNP-complete problems3-SAT Exact Cover
S ~ n log2 nFermionic systems?
S ~ r ~ nShor Factorization
S ~ nd-1/dSpin chains in d-dimensions
S ~ log2 nCritical spin chains
S ~ ctNon-critical spin chains
Local (12 levels), nearest neighbor H is QMA-complete!! AGK07
Physics vs. simulationPhysics vs. simulation
New ideas
MPS using Schmidt decompositions
Arbitrary manipulations of 1D systems
PEPS
2D, 3D systems
MERA
Scale invariant 1D, 2D, 3D systems
New ideas
New ideas
MPS for translational invariant spin chains (iTEBD)
0ψψε →− He
∑∑∑ +++ ⋅+⋅=⋅=iodd
iiieven
iii
ii SSSSSSH 111
commute commute
All even gates can be performe simultaneouslyAll odd gates can be performe simultaneouslyUse Trotter decomposition to combine them
ψ
New ideas
Bλ Aλ BλAΓ BΓBBjAAiBijβγβγαγααβ λλλ ΓΓ=Θ
ijklij
kl U αβαβ Θ=Θ~
la
Aa
ka
kl WV βααβ λ~~ =Θ
iB
Ai Vαβα
αβ λ1~ =Γ B
iBi Wβ
αβαβ λ1~ =Γ
Bλ Aλ~ BλAΓ~ BΓ~
New ideasNew ideas
Heisenberg model
060.443147182ln41
,0 −=−=exactE
S=.994-.44276223χ=8
S=1.26-.443094χ=16
S=.919-.44249501χ=6
S=.764-.44105813χ=4S=.486-.42790793χ=2
Trotter 2 order, δ=.001
New ideasNew ideas
New ideas
PEPS: Projected Entangled Pairs
iAγ
δβ
α
physical index
ancillae
Good: PEPS support an area law!!
Bad: Contraction of PEPS is #P
New results beat Monte Carlo simulations
New ideas
New ideas
MERA: Multiscale Entanglement Renormalization Ansatz
Intrinsic support for scale invariance!!
If MPS, PEPS, MERA are a good representation of QM
• Approach new problems
• PrecisionCan we do any better than DMRG?e.g.: Faithfull numbers for entropy? Exact solutions? Smaller errors?
• Can we simulate better than Monte Carlo?
• Are MPS, PEPS and MERA the best simulation solution?
Keep in mind:
• scaling of entropy: Area law
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