Entanglement entropy and the simulation of quantum systems Open discussion with pde2007
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Entanglement entropyand
the simulation of quantum systems
Open discussion with pde2007
José Ignacio LatorreUniversitat de Barcelona
Benasque, September 2007
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Physics
Theory 1 Theory 2
Exact solution
Approximated methods
Simulation
Classical Simulation
Quantum Simulation
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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: MPS, PEPS, MERA
Quantum simulation needed for time evolution of quantum systemsand for non-local Hamiltonians
Classical computer
Quantum computer
?
IntroductionIntroduction
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Is it possible to classically simulate faithfully a quantum system?
000 EH
Quantum Ising model
0
0)( tU
00 xj
xi
n
i
zi
n
i
xi
xiH
111
represent
evolve
read
IntroductionIntroduction
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Introduction
n
i
zi
n
i
xi
xiH
111
10
01
0
0
01
10 zyx
i
i
1
1
2
1
0
1
The lowest eigenvalue state carries a large superposition of product states
ccc Ex. n=3
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Naïve answer: NO
• Exponential growth of Hilbert space
d
i
d
inii
n
niic
1 11...
1
1...|...|
Classical representation requires dn complex coefficients
n
• A random state carries maximum entropy
)( LnL Tr
dLTrS LLL loglog)(
IntroductionIntroduction
computational basis
Is it possible to classically simulate faithfully a quantum system?
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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
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Plan
Measures of entanglement
Efficient description of slight entanglement
Entropy: physics vs. simulation
New ideas: MPS, PEPS, MERA
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Measures of entanglement
One qubit
1,0
1110
i
i ic
Quantum superposition
Two qubits
1,0,
211,0,
21
21
21
21
2111100100ii
BA
ii
ii
ii iiciic
Quantum superposition + several parties = entanglement
Measures of entanglement
2CH
22 CCH
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Measures of entanglement
Bii
A
ii
ii
ii iiciic 21,0,
11,0,
21
21
21
21
21
• Separable states
BABii
A
ii iic
21,0,
1
21
21
BBAA
102
110
2
111100100
2
1e.g.
• Entangled states
BABii
A
ii iic
21,0,
1
21
21
10012
1e.g.
Measures of entanglement
Local realism is droppedQuantum non-local correlations
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Measures of entanglement
Pure states: Schmidt decomposition = Singular Value Decomposition
BiAii
iAB p
|||1
BjA
B
ij
A
vuA i
H
j
H
iAB
|||dim
1
dim
1klkikij
VUA
A B
=min(dim HA, dim HB) is the Schmidt number
BA HHH
Measures of entanglement
1 Entangled state
Diagonalise A
Measures of entanglement
1 Separable state
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BiAii
iAB p
|||1
Von Neumann entropy of the reduced density matrix
Bi
iiAAA SppTrS
1
22 loglog
1
||||i
iiiABBA pTr
ITrBA 2
1|| 1
2
1log
2
1
2
1log
2
122
BA SS
Measures of entanglementMeasures of entanglement
1 Product state0S large S large Very entangled state
e-bit
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Maximum Entropy for n-qubits
Strong subadditivity theorem
implies entropy concavity on a chain of spins
nInn 22
1 nS
n
innn
2
12 2
1log
2
1)(
),(),()(),,( CBSBASBSCBAS
2MLML
L
SSS
SL
SL-M
SL+M
Smax=n
Measures of entanglementMeasures of entanglement
222 CCCH
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Efficient description for slightly entangled states
BkAkk
kAB p
|||1
BA
H
i
H
iAB iic
B
ii
A
21
dim
1
dim
1
|||2
21
1
2121 kikkiii VpUc
A BBA HHH Schmidt decomposition
1
]2[]1[ 21
21k
ikk
ikiic
Efficient description
Retain eigenvalues and changes of basis
Efficient description
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d
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....
Slight entanglement iff poly(n)<< dn
• Representation is efficient• Single qubit gates involve only local updating• Two-qubit gates reduces to local updating• Readout is efficient
Vidal 03: Iterate this process
ndndparameters 2#
efficient simulation
Efficient description
),(|
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Graphic representation of a MPS ,,1
di ,,1j
jj
ijA ][
1
Efficient computation of scalar products
operations2d
3nd
Efficient description
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Efficient computation of a local action
U
lklk
jiklij MU
~~~))((
Efficient description
3d
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Matrix Product States
d
i
d
inii
n
niic
1 11...
1
1...|...|
1
21
]1[ iA 2
32
]2[ iA 3
43
]3[ iA 4
54
]4[ iA 5
65
]5[ iA 6
76
]6[ iA 7
87
]7[ iA
n
n
n
n
iniiiii AAAAc ][
...
]3[]2[]1[1... 1
12
3
43
2
32
1
21....
i
α
Approximate physical states with a finite MPS
IAA i
i
i ][][ ][][]1[][ iii
i
i AA canonical form PVWC06
A
Efficient descriptionEfficient description
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n
n
n
n iniiiii AAAAc ][
...
]3[]2[]1[1 1
12
3
43
2
32
1
2
1 ....
Intelligent way to represent, manipulate, read-outentanglement
Classical simplified 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
1
1 ....
Efficient description
Adaptive representation for correlations among parties
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11,...,
)()1(
1
4
1...,...
...|....
...||
1
1
1
21
,1
1
ii
iic
nini
nii
iiimage
n
n
n
n
n
i1=1 i1=2
i1=3 i1=4
| i1 i2=1 i2=2
i2=3 i2=4
| i2 i1 105| 2,1
Spin-off: Image compression
pixel addresslevel of grey
RG addressing
Efficient description
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....
= 1PSNR=17
= 4PSNR=25
= 8PSNR=31
Max = 81
QPEG
• Read image by blocks• Fourier transform• RG address and fill• Set compression level: • 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
}{ )(a
Efficient description
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0),...,,(),...,,( 2121 nn xxxfO
)()...()()...(),...,,( 2121 21
21niii
iiin xhxhxhAAAtrxxxf
n
n2
}{min OfA
Note: classical problems with a direct product structure!
Spin-off: Differential equations
Efficient descriptionEfficient description
0),,(),,( 11 nn xxxx
D D DD
D
2222
2
2
Efficient description
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Matrix Product States for continuous variables
)(2
1 21
1
222
aa
n
aapm
dxdtH
)()()(.... 21][1
...
]2[]1[1 21
12
2
32
1
2 niiiinii xxxAAA
n
n
n
n
Harmonic chains
MPS handles entanglement Product basis
di ,,1
Truncate tr d tr
2,,1n
d
Efficient description
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][][ AHA
i
iiHH 1,Nearest neighbour interaction
][AH
][A
0][][
][][][
AA
AHA
A i
Minimize by sweeps
Choose Hermite polynomials for local basis )()exp()( 2 xhaxx ii
optimize over a
Efficient description
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Results for n=100 harmonic coupled oscillators(lattice regularization of a quantum field theory)
dtr=3 tr=3
dtr=4 tr=4
dtr=5 tr=5
dtr=6 tr=6
Newton-raphson on a
Efficient description
Errorin Energy
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Success of MPS will depend on how much entanglement is present in the physical state
Physics
exactS
Simulation
)(S
If nSexact log MPS is in very bad shape
Back to the central idea: entanglement support
Physics vs. simulationPhysics vs. simulation
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Exact entropy for a reduced block in spin chains
Lc
SLL 2log
3
|1|log6 22/ c
S NL
At Quantum Phase Transition Away from Quantum Phase Transition
Physics vs. simulationPhysics vs. simulation
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Maximum entropy support for MPS
2
1
2 log
S
Maximum supported entanglement
12 ct
logmax, MPSSS
Physics vs. simulationPhysics vs. simulation
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Faithfullness = Entanglement support
Lc
SLL 2log
3
Spin chainsMPS
log1
max S
Spin networks
LSLLxL
Area law
Computations of entropies are no longer academic exercises but limits on simulations
PEPS
Physics vs. simulationPhysics vs. simulation
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Exact Cover
A clause is accepted if 001 or 010 or 100
Exact Cover is NP-complete
0 1 1 0 0 1 1 0
For every clause, one out of eight options is rejected
instance
NP-complete
Entanglement for NP-complete problems
3-SAT is NP-completek-SAT is hard for k > 2.413-SAT with m clauses: easy-hard-easy around m=4.2n
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Adiabatic quantum evolution (Farhi,Goldstone,Gutmann)
H(s(t)) = (1-s(t)) H0 + s(t) Hp
Inicial hamiltonian Problem hamiltonian
s(0)=0 s(T)=1t
Adiabatic theorem:
if
E1
E0
E
t
gmin
NP-complete
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Adiabatic quantum evolution for exact cover
2)1( kjiC zzzH
|0> |0> |0>|0>|1> |1>|1> |1>
(|0>+|1>) (|0>+|1>) (|0>+|1>)….(|0>+|1>)
NP-complete
2
1 zi
iz
NP problem as a non-local two-body hamiltonian!
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n=100 right solution found with MPS among 1030 states
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Non-critical spin chains S ~ ct
Critical spin chains S ~ log2 n
Spin chains in d-dimensions S ~ nd-1/d
Fermionic systems? S ~ n log2 n
NP-complete problems3-SAT Exact Cover
S ~ .1 n
Shor Factorization S ~ r ~ n
Physics vs. simulationPhysics vs. simulation
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New ideas
MPS using Schmidt decompositions (iTEBD)
Arbitrary manipulations of 1D systems
PEPS
2D, 3D systems
MERA
Scale invariant 1D, 2D, 3D systems
New ideas
Recent progress on the simulation side
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2. Euclidean evolution
0 He
H
H
e
e
'
Non-unitary evolution entails loss of norm
oddeveni
ii HHhH 1,
evenoddevenoddeven HHHHH eeee 2/2/)(
oddeven HH , are sums of commuting pieces
Trotter expansion
MPS
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Ex: iTEBD (infinite time-evolving block decimation)
even
odd
A A AB B
B A A
A B
B
A B
Translational invariance is momentarily broken
MPS
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B A BA B BBjAAiBij
ijklij
kl U ~
la
Aa
ka
kl WV ~~
iB
Ai V
1~ B
iBi W
1~
B A~ BA~ B~
i)
ii)
iii)
iv)
MPS
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B A BA BL R
RLRL ,,
Schmidt decomposition produces orthonormal L,R states
MPS
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Moreover, sequential Schmidt decompositions produce isometries
B A BA BL
i
LL '
= AiAiB *2
are isometries
MPS
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),(),( H
Energy
Read out
Entropy for half chain
1
22 logS
MPS
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Heisenberg model
060.443147182ln4
1,0 exactE
=2 -.42790793 S=.486
=4 -.44105813 S=.764
=6 -.44249501 S=.919
=8 -.44276223 S=.994
=16 -.443094 S=1.26
Trotter 2 order, =.001
New ideasNew ideas
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entropy
energy
Convergence
MPS
Local observables are much easier to get than global entanglement properties
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S
M
Perfect alignment
1)( **
MPS
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New ideas
PEPS: Projected Entangled Pairs
iA
physical index
ancillae
Good: PEPS support an area law!!
Bad: Contraction of PEPS is #P
New results beat Monte Carlo simulations
New ideas
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A B
Entropy is proportional to the boundary
LSS BA
Contour A = L
“Area law”
Some violations of the area law have been identified
PEPS
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i
i
i AAE *
''
'' '
'
''
As the contraction proceeds, the number of open indices grows as the area law
PEPS
2D seemed out of reach to any efficient representation
Contraction of PEPS is #P
Building physical PEPS would solve NP-complete problems
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Yet, for translational invariant systems, it comes down to iTEBD !!
E E
Comparable to quantum Monte Carlo?
E
PEPSPEPS
E becomes a non-unitary gate
PEPS
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PEPS
r
rz
rx
rr
rx hH
][]'[
)',(
][
Results for 2D Quantum Ising model (JOVVC07)
)( hhAmz
)1(332.)1(06.33
)3(346.)1(10.32
5.41
h
h
h
MC 327.044.3 h
PEPS
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MERA
MERA: Multiscale Entanglement Renormalization Ansatz
Intrinsic support for scale invariance!!
),( U
MERA
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MERA All entanglemnent on one line
All entanglemnent distributed on scales
MERA
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Contraction = Identity
MERA
U
)(max UtrU
WV
WUU
Update
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If MPS, PEPS, MERA are a good representation of QM
• Approach hard problems
• PrecisionCan we simulate better than Monte Carlo?
• Are MPS, PEPS and MERA the best simulation solution?
• Spin-off?
Physics:
• Scaling of entropy: Area law << Volume law• Translational symmetry and locality reduce dramatically the amountof entanglement• Worst case (max entropy) remains at phase transition points
Physics vs. simulation
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Quantum Complexity Classes
QMA
11),(Pr/ xVLx yes
L is in QMA if there exists a fixed and a polynomial time verifier (V) such that
1),(Pr/ xVLx no
What is the QMA-complete problem?
Feynman idea (shaped by Kitaev)
..1 chttVH t
QMA
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QMA
QMA-complete problem
• Log-local hamiltonian• 5-body• 3-body• 2-body (non-local interactions) • 2-body (nearest neighbor 12 levels interaction)!
Given H on n-party decide ifbE
aE
0
0
)(
1
npolyba
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Open problems
• Separability problem (classification of completely positive maps)• Classification of entanglement (canonical form of arbitrary tensors)• Better descriptions of quantum many-body systems• Spin-off of MPS??• Rigorous results for PEPS, MERA• Need for theorems for gaps/correlation length/size of approximation• Exact diagonalisation of dilute quantum gases (BEC)• Classification of Quantum Computational Complexity classes• ….